Lp-theory of cylindrical boundary value problems : an operator-valued fourier multiplier and functional calculus approach

Lp-Theory of Cylindrical Boundary Value Problems Tobias Nau Lp-Theory of Cylindrical Boundary Value Problems An Oper...

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Lp-Theory of Cylindrical Boundary Value Problems

Tobias Nau

Lp-Theory of Cylindrical Boundary Value Problems An Operator-Valued Fourier Multiplier and Functional Calculus Approach

RESEARCH

Tobias Nau Konstanz, Germany

Dissertation der Universität Konstanz Tag der mündlichen Prüfung: 07.02.2012 1. Referent: Prof. Dr. Robert Denk 2. Referent: Prof. Dr. Jürgen Saal

ISBN 978-3-8348-2504-9 DOI 10.1007/978-3-8348-2505-6

ISBN 978-3-8348-2505-6 (eBook)

The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available in the Internet at http://dnb.d-nb.de.

Springer Spektrum © Vieweg+Teubner Verlag | Springer Fachmedien Wiesbaden 2012 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Cover design: KünkelLopka GmbH, Heidelberg Printed on acid-free paper Springer Spektrum is a brand of Springer DE. Springer DE is part of Springer Science+Business Media. www.springer-spektrum.de

To Rebecca and my parents Gerhard & Verena Nau

Acknowledgments This is to express my gratitude to several people who helped me during the time I worked on this thesis. First of all, I am very grateful to my supervisor Prof. Dr. Robert Denk for his support when challenging questions had to be faced. I have highly appreciated the pleasant working atmosphere over the last three years and the possibility to take part in teaching and research activities both at the University of Konstanz and at other universities. Moreover, I would like to thank him very much for the way he encouraged me as a young researcher. I express my gratitude to Prof. Dr. J¨ urgen Saal who always gave generously of his time when he was still part of the PDE research group in Konstanz. Furthermore, I am very grateful for his invitations to Darmstadt later on and for his great hospitality. Writing this thesis, I benefited a lot from what I have learned there. I am also very grateful to Prof. Dr. Wolfgang Arendt for helpful suggestions as well as for interesting discussions during my stays at the University of Ulm. Many thanks to the PDE research group in Konstanz. Special thanks to Mario Kaip, Johannes Schnur, and Tim Seger for their cooperativeness and the good times we had. Thanks to Springer Spektrum, in particular to Mrs. Ute Wrasmann and Mrs. Anita Wilke for their kind support. Also many thanks to Thomas Melzer and Siegfried Maier for their helpful comments on the manuscript. I thank my parents Gerhard and Verena Nau, who have continuously helped to make possible what I have enjoyed in my life so far. Special thanks also to my sister Dr. Stephanie Nau for good advice and assistance. Above all, I thank my wife Rebecca for enduring with a smile the times when I was preoccupied with mathematics and for all the support she has been giving me throughout the years. Tobias Nau

Zusammenfassung Die vorliegende Doktorarbeit behandelt Anfangs-Randwert-Probleme der Form ⎧ ⎪ in R+ × Ω, ⎨ ∂t u + A(x, D)u = f (ARP) B(x, D)u = 0 auf R+ × ∂Ω, ⎪ ⎩ in Ω u|t=0 = u0 in Lp -R¨ aumen, wobei Ω ein zylinderf¨ ormiges Gebiet ist. Das heißt Ω = U × V ist als das kartesische Produkt zweier Mengen gegeben. Dabei sind V und U Standardgebiete, was bedeutet, dass sich in diesen Gebieten Randwertprobleme mit bekannten Methoden der Lp -Theorie behandeln lassen (siehe z.B. [DHP03]). Ferner bezeichnet A(x, D) einen Differentialoperator in Ω, und B(x, D) repr¨ asentiert je nach Problemstellung verschiedene Operatoren auf dem Rand ∂Ω. Ziel der Arbeit ist es, f¨ ur den Operator AB im zugeordneten Cauchy-Problem u(t) ˙ + AB u(t) = f (t), t ∈ R+ , (CP) u(0) = u0 maximale (Lq -)Regularit¨ at in Lp -R¨ aumen (siehe Definition 5.1) nachzuweisen. Dazu wird ein Resultat aus [Wei01b] verwendet, das maximale Regularit¨ at unter passenden Voraussetzungen durch R-Sektorialit¨ at des entsprechenden Operators charakterisiert. Neben Aussagen zum Cauchy-Problem (CP) erm¨ oglicht die RSektorialit¨ at dank eines Resultats von Pr¨ uss zudem Aussagen zur maximalen Regularit¨ at des Operators AB im Rahmen von Volterra-Integralgleichungen (siehe [Pr¨ u93]). Damit r¨ uckt die Behandlung des zu (CP) geh¨ orenden Resolventenproblems in Form eines parameterabh¨ angigen Randwertproblems in den Vordergrund. Als zentrale Voraussetzung wird in dieser Arbeit die Anforderung gestellt, dass nicht nur Ω sondern das gesamte Anfangs-Randwert-Problem (ARP) zylinderf¨ ormig ist. Diese Anforderung beinhaltet zum einen, dass der Differentialoperator A in zwei Teile A = A1 + A2 zerf¨ allt, wobei A1 sowohl in den Koeffizienten als auch in den Differentialausdr¨ uangig ist. Andererseits soll cken nur von U , und A2 entspechend nur von V abh¨ sich auch der Randoperator B der Zylinderf¨ ormigkeit von Ω anpassen, indem er die Gestalt B1 , auf ∂U × V, B= B2 , auf U × ∂V

X

Zusammenfassung

aufweist. An dieser Stelle sei darauf hingewiesen, dass diese Rahmenbedingungen auf zahlreiche Problemstellungen in der Anwendung zutreffen. So hat zum Beispiel die W¨ armeleitungsgleichung mit Dirichlet- oder Neumann-Randbedingungen zylinderf¨ ormige Gestalt. Die geforderte Struktur erlaubt den Einsatz operatorwertiger Fouriermultiplikatoren und des operatorwertigen Dunford-Kalk¨ uls sektorieller Operatoren. Mithilfe dieser Methoden kann die R-Sektorialit¨ at oder ein ul von AB auf die entsprechende Eigenschaft des indubeschr¨ ankter H∞ -Kalk¨ uckgef¨ uhrt werden. Dieser Zugang er¨ offnet einen elezierten Operators A2B2 zur¨ ganten Weg maximale Regularit¨ at des Operators AB nachzuweisen. Operatorwertige Fouriermultiplikatoren werden sowohl im Kontext vektorwertiger Fouriertransformation als auch im Kontext vektorwertiger Fourierreihen eingesetzt. Dies erfolgt zun¨ achst auf abstrakter Ebene in den Kapiteln 6 und 7: Mit einem abgeschlossenen, linearen Operator A in einem Banachraum E werden Gleichungen der Form λu + P (D)u + Q(D)Au = f urfel in Lp (U, E) behandelt. Dabei stellt U entweder den Ganzraum Rn oder den W¨ (0, 2π)n dar, wobei im letzteren Fall zudem verallgemeinerte periodische Randbedingungen gestellt werden. Außerdem bezeichnen P (D) und Q(D) Differentialoperatoren, die von unterschiedlicher Ordnung sein k¨ onnen, und λ ist ein komplexer Parameter. Der Zusammenhang zu Resolventenproblemen zylinderf¨ ormiger Randwertprobleme wird nun hergestellt, indem der Operator A speziell als A2B2 im Banachraum ahlt wird. Unter geeigneten Voraussetzungen an die Koeffizienten ist Lp (V ) gew¨ anktem H∞ -Kalk¨ ul. Diese EigenA2B2 ein R-sektorieller Operator mit beschr¨ schaft l¨ asst sich u ¨ber die gewonnenen Resultate auf den gesamten Operator AB u ¨bertragen, sofern der Operator A1 (D) konstante Koeffizienten hat und eine passende Parameterelliptizit¨ at aufweist. Somit wird ein zylinderf¨ ormiges Modellanderter Gestalt eingeht. Obwohl sich die problem behandelt, in das A2B2 in unver¨ Ausgangslage dadurch deutlich von herk¨ ommlichen Modellproblemen im Ganzraum unterscheidet, k¨ onnen in ganz ¨ ahnlicher Weise nicht-konstante Koeffizienten von A1 durch Lokalisierung behandelt werden. Das Hauptresultat hierzu ist Theorem 8.10 in Kapitel 8. Ist Ω als das kartesische Produkt mehrerer Standardgebiete gegeben, so wird anstelle der operatorwertigen Fouriermultiplikatoren der operatorwertige DunfordKalk¨ ul und ein Resultat von Kalton und Weis eingesetzt (siehe [KW01]). Dieser Zugang wird in Kapitel 10 verfolgt. Theorem 10.5 zeigt, dass sich so unter passenden Vorausetzungen die gleichen Resultate erzielen lassen, wie sie in [DHP03] ur Randwertprobleme in Standardgebieten bewiesen werden. Insund [DDH+ 04] f¨ besondere lassen sich mit dieser Methode vektorwertige Randwertprobleme mit operatorwertigen Koeffizienten behandeln. Andererseits k¨ onnen auch neue Aussagen und vereinfachte Beweise zu klassischen Problemstellungen wie etwa der W¨ armeleitungsgleichung in zylinderf¨ ormigen

Zusammenfassung

XI

Gebieten gewonnen werden. Der verfolgte L¨ osungsansatz erlaubt es, gemischte Dirichlet-Neumann Randbedingungen auf unterschiedlichen Komponenten des Randes vorzugeben. Die entsprechenden Aussagen zum zugeh¨ origen Resolventenproblem des Laplace-Operators werden in Theorem 8.22, Theorem 10.9 und Theorem 10.13 bewiesen. Ein Anwendungsbereich dieser Ergebnisse ist die K¨ uhlung rechtwinkliger oder zylinderf¨ ormiger elektronischer Bauteile u ¨ ber K¨ uhlsysteme, die auf zwei gegenu ¨berliegenden Seiten oder auch nur auf einer Seite des Bauteils angebracht sind. Des Weiteren sind Anwendungen im Bereich der Zellbiologie gegeben. Hier werden Diffusionsprozesse in sogenannten Beobachtungsfenstern modelliert. Dabei handelt es sich um kleine, w¨ urfelf¨ ormige Bereiche innerhalb einer Zelle, wobei davon ausgegangen wird, dass sich die Strukur der Zelle u ¨ ber angrenzende Zellbereiche hinweg periodisch fortsetzt. Dem wird durch periodische Randbedingungen Rechnung getragen. Da im Rahmen dieser Arbeit auch Mischungen aus periodischen und Dirichlet-Neumann Randbedingungen behandelt werden, lassen sich dar¨ uber hinaus auch membran¨ ose Begrenzungen des Beobachtungsfensters modellieren. Als weitere Anwendung wird in Kapitel 9 das Resolventenproblem zur StokesGleichung in Schichten und in einer Klasse rechtwinkliger, zylinderf¨ ormiger Gebiete behandelt. Die Stokes-Gleichung geht durch Linearisierung aus der NavierStokes-Gleichung der Str¨ omungsdynamik hervor, weshalb zylinderf¨ ormige Gebiete von besonderem Interesse sind. Zun¨ achst werden periodische Randbedingungen behandelt. Insbesondere wird eine periodische Helmholtz-Zerlegung des zu¨ber Fouriermultiplikatoren hergeleitet. Durch den Einsatz geh¨ origen Lp -Raums u von Reflektionstechniken wird schließlich die Existenz der klassischen HelmholtzuckgeZerlegung des zugeh¨ origen Lp -Raums auf dieses periodische Analogon zur¨ f¨ uhrt. Die Hauptresultate dieses Kapitels sind in Theorem 9.15 und Theorem 9.17 zu finden.

Contents 1 Introduction and main results

1

2 Vector-valued Fourier transform and Fourier series

9

3 R-boundedness and operator-valued Fourier multiplier theorems 25 4 Classes of operators and Dunford functional calculus

41

5 Parabolic problems and maximal regularity

51

6 Fourier transform approach to operator-dependent problems 6.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 R-sectoriality and RH∞ -calculus . . . . . . . . . . . . . . . . . . .

55 55 61

7 Fourier series approach to operator-dependent problems 7.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 ν-periodic problems in cubical domains . . . . . . . . . . . . . . . . 7.3 R-sectoriality and RH∞ -calculus . . . . . . . . . . . . . . . . . . .

75 75 84 89

8 Application to cylindrical boundary value problems 99 8.1 Known results for bounded and exterior domains . . . . . . . . . . 99 8.2 Cylindrical boundary value problems . . . . . . . . . . . . . . . . . 103 8.3 A focus on the Laplacian . . . . . . . . . . . . . . . . . . . . . . . 126 9 Application to the Stokes equation 10 The 10.1 10.2 10.3

133

functional calculus approach 153 Operator-valued Dunford calculus . . . . . . . . . . . . . . . . . . . 153 Applications to cylindrical boundary value problems . . . . . . . . 154 A focus on the Laplacian . . . . . . . . . . . . . . . . . . . . . . . 164

A Notation and vector-valued function spaces

171

B Related topics in the literature

175

Bibliography

179

Index

187

List of Figures 3.1 3.2 3.3

Steps in the coarse decomposition of Z2 . . . . . . . . . . . . . . . . Steps in the fine decomposition of Z2 . . . . . . . . . . . . . . . . . The intermediate conditions. . . . . . . . . . . . . . . . . . . . . .

34 35 39

4.1 4.2

Relevant sets in the complex plane. . . . . . . . . . . . . . . . . . . Relation of angles in a Banach space of class HT . . . . . . . . . . .

41 47

6.1 6.2 6.3

Possible shift in case 0 ∈ ρ(A). . . . . . . . . . . . . . . . . . . . . Different paths of integration. . . . . . . . . . . . . . . . . . . . . . The path of integration in the extended Cauchy integral formula. .

64 69 70

7.1

The path of integration, discrete case. . . . . . . . . . . . . . . . .

95

8.1 8.2 8.3 8.4

The single steps during the reflection procedure. . . . . . Reflections depending on intersections with the boundary. Lipschitz domains with different ranges of p. . . . . . . . . Different observation windows in a biological cell. . . . . .

9.1

˜ and Ω. . . . . . . . . . . . . . . . . . . . . . . . . . 142 The domains Ω

. . . .

. . . .

. . . .

. . . .

. . . .

122 123 129 131

1 Introduction and main results The aim of this thesis is to develop a vector-valued Lp -approach to initial boundary value problems of the type (1.1)

∂t u + A(x, D)u B(x, D)u u|t=0

= = =

f 0 u0

in R+ × Ω, on R+ × ∂Ω, in Ω

on cylindrical domains Ω. Here cylindrical means that Ω is of the form Ω = U × V,

(1.2)

where V is a standard domain and U is given as a full space, a half space, a cube or a standard domain again. More than that, U itself can be given as the Cartesian product of finitely many domains of these types. For the sake of simplicity, throughout the introduction a standard domain is supposed to be a bounded, smooth domain. Furthermore, A(x, D) is a differential operator in Ω and B(x, D) is a multiple boundary operator, i.e. in general it represents several operators that act on the boundary ∂Ω of Ω. Our main interest is to prove maximal (Lq -)regularity on Lp -spaces (see Definition 5.1) of the operator AB which appears in the corresponding Cauchy problem (1.3)

u(t) ˙ + AB u(t) u(0)

= =

f (t), u0 .

t ∈ R+ ,

Unless specified otherwise, we consider 1 < p < ∞. To establish maximal regularity, we make use of the celebrated result of Weis in [Wei01b], roughly saying that AB enjoys the property of maximal regularity if and only if it is R-sectorial. Besides problem (1.3), integral equations of Volterra type involving AB are considered. Due to a result of Pr¨ uss (see Theorem 5.8) questions on maximal regularity are answered in terms of R-sectoriality also in this context. Let the underlying cylindrical domain Ω in (1.1) be replaced by a full space, a half space or standard domain V for a moment. With m ∈ N consider a differential operator aα (x)Dα (1.4) A(x, D) = |α|≤2m

of order 2m. Let B further represent m boundary operators bβ (x)Dβ , mj < 2m, j = 1, . . . , m. (1.5) Bj (x, D) = |β|≤mj

T. Nau, Lp-Theory of Cylindrical Boundary Value Problems, DOI 10.1007/978-3-8348-2505-6_1, © Vieweg+Teubner Verlag | Springer Fachmedien Wiesbaden 2012

2

1 Introduction and main results

By means of results in [DHP03] and [DDH+ 04], with a UMD space F , the corresponding Lp (V, F )-realization AVB with domain D(AVB ) := u ∈ W 2m,p (V, F ); B(·, D)u = 0 is known to enjoy the property of maximal regularity. In these works parameterellipticity and suitable assumptions on the coefficients are imposed to prove Rsectoriality and boundedness of the H∞ -calculus (see Propositions 8.3 and 8.4). The proofs rely on a localization of the boundary which makes the class of standard domains advisable. In this thesis we pursue a different strategy based on the restriction that not only Ω but the entire boundary value problem (1.1) is cylindrical. It implies that A resolves into two parts A = A1 + A2 such that A1 acts merely on U and A2 acts merely on V . Accordingly, B is assumed to resolve into two parts B=

B1 ,

on ∂U × V,

B2 ,

on U × ∂V

as well. Note that many standard systems such as the heat equation with Dirichlet or Neumann boundary conditions are of this form. We essentially take advantage of this cylindrical structure of the entire problem in order to employ operator-valued multiplier theory and an operator-valued Dunford functional calculus to address problem (1.1). By means of these methods R-sectoriality or boundedness of the H∞ -calculus of AB in Lp (Ω, F ) is reduced to the corresponding result on the induced operator A2B2 in Lp (V, F ). This approach reveals a short and elegant way to prove maximal regularity for boundary value problems of type (1.1) on cylindrical domains of the form (1.2). The operator-valued Fourier multiplier approach includes both multipliers in the context of vector-valued Fourier transform and Fourier series. In Chapters 6 and 7 it is introduced in the following rather abstract settings. Given a closed linear operator A in a Banach space E, we investigate equations of the form (1.6)

λu + P (D)u + Q(D)Au = f

in Lp (U, E), where U is given as the whole space Rn or the cube (0, 2π)n . Here P (D) and Q(D) define partial differential operators of possibly different orders and λ is a complex parameter. In the case U = Rn from (1.6) the operator-valued Fourier symbol of resolvent type (1.7)

m : Rn → L(E); m(ξ) := (λ + P (ξ) + Q(ξ)A)−1


3

occurs. Thus, we are faced with the question which conditions on m ensure that the associated Fourier operator F −1 mF defines the solution operator to (1.6) in Lp (Rn , E). In Chapter 6 we give an answer to this question by specifying more ˇ abstract conditions from [Wei01b]; see also [SW07] and [HHN02]. n In the case U = (0, 2π) equation (1.6) is supplemented with generalized periodic boundary conditions and Fourier transform is replaced by Fourier series. Consequently, m turns into its discretization M : Zn → L(E); M (k) := (λ + P (k) + Q(k)A)−1 and the E-valued distributions under consideration change from S (Rn , E) to the (Rn , E) of periodic distributions. In contrast to extensive theory on Espace Dper valued distributions in general and E-valued tempered distributions in particular, the contribution in literature to E-valued periodic distributions is rather sparse. We fill this gap in Chapter 2 providing useful results on this class of E-valued distributions. The first result is Theorem 2.7, a representation theorem for E-valued periodic distributions by means of multiple, E-valued distributional Fourier series. The latter enables us to prove a powerful uniqueness theorem on Fourier series in Corollary 2.9. Besides that, it serves as a starting point to give a comprehensive characterization of functions belonging to periodic Sobolev spaces in Theorem 2.14. The representation theorem is extended to partially periodic E-valued distributions in Theorem 2.19. Later on, the construction of a Helmholtz projection in Chapter 9 relies on this result. Thus, apart from their own interest, the results achieved in this chapter are indispensable for later applications. A first result on boundedness of the Fourier operator in Lp ((0, 2π)n , E) associated with M was given by Arendt and Bu in [AB02]; see also [BK04]. As pointed out by the authors their results can as well be deduced from a more involved multiˇ plier theorem in [SW07]. In so doing, we observe that the assumptions imposed in [AB02] and [BK04] can be relaxed. This result is made available in Theorem 3.24. The less restrictive assumptions of Theorem 3.24 are far from being artificial. In fact, this very improvement allows for a successful construction of the Helmholtz projection as indicated above. The conditions on m and M which all operator-valued multiplier theorems mentioned have in common involve R-boundedness of {m(ξ); ξ ∈ Rn \ {0}} and {M (k); k ∈ Zn }, respectively. In view of (1.7), R-sectoriality of A thus provides a starting point to approach equation (1.6). Already in this rather abstract setting the idea of transference becomes apparent. Under suitable assumptions on the Banach space E, let A define an appropriate realization of (1.6) in Lp (U, E). If A is R-sectorial in E, with the aid of the operator-valued multiplier theorems the same property is inferred for the operator A in Lp (U, E). Here the differential operators P and Q are assumed to be parameter-elliptic, where a Dore-Vennitype condition for the corresponding angles of parameter-ellipticity and the angle of R-sectoriality of A has to be satisfied.

4


For the application we have in mind, A is specified to be given as A2B2 in the Banach space E := Lp (V ). This leads to maximal regularity on Lp (Ω) for Cauchy problems associated with cylindrical boundary value problems that are partially endowed with generalized periodic boundary conditions. These types of boundary conditions appear, for instance, in the study of keratin network growth in biological cells (see [ABF+ 08]). Note that A2B2 is treated in full generality at once so that we are by no means restricted to constant coefficients in A2 . In order to deal with non-constant coefficients also in the first part A1 , a localization procedure is carried out. In contrast to existing literature, however, no whole space or half space but rather a cylindrical domain serves as a model problem here. The main result in this context is Theorem 8.10. One interesting outcome is that despite the splitting property of the differential operator A mixed derivatives of the solution with respect to all spacial directions belong to the underlying Lp -space. As mentioned

at the beginning, we can also achieve results on maximal regularity if Ω := N i=1 Vi is given as the Cartesian product of finitely many but more than just two or three domains. Then differential operators A = A1 + . . . + AN and multiple boundary operators of the form B = {Bi on V1 × . . . × Vi−1 × ∂Vi × Vi+1 × . . . × VN }i=1,...,N are considered (cf. Definition 10.3). Here each part Ai is given as a differential operator of order 2mi as introduced in (1.4) and each part Bi is a multiple boundary operator that represents mi boundary operators of type (1.5). Again each pair (Ai , Bi ) acts on Vi , respectively on its boundary, only. To investigate such type problems, we adopt the operator-valued Dunford functional calculus approach. It is based on the Kalton-Weis-Theorem (see [KW01]) which combines boundedness of the scalar H∞ -calculus with R-boundedness conditions on operator-valued holomorphic functions. Theorem 10.5 on problems of this type basically says that the operator-valued Dunford functional calculus approach yields exactly the same results as they are known for problems on standard domains, provided the induced boundary value problems on the different crosssections Vi match the requirements of [DHP03] and [DDH+ 04]. In particular, parameter-ellipticity is not imposed for the entire problem but rather for each induced problem on the single cross-sections Vi . The regularity of the solution on the other hand is not subordinate to the splitting property. For instance, if all orders of the differential operators coincide, i.e. m1 = . . . = mN =: m, maximal regularity results on the Lp (Ω, F )-realization AB can be derived where (1.8) D(AB ) := u ∈ W 2m,p (Ω, F ); B(·, D)u = 0 . Since non-smooth and non-convex domains Ω with non-compact boundary are covered, additional importance is attached to (1.8) and Theorem 10.5.


5

By virtue of this approach, for a class of Banach spaces F , we are thus able to consider F -valued solutions and to allow the coefficients of each part Ai and Bi to be L(F )-valued. Applications for equations with L(F )-valued coefficients are, for instance, given by coagulation-fragmentation systems (cf. [AW05b]) or spectral problems of parametrized differential operators in hydrodynamics (cf. [DMT02]). Besides boundary value problems involving L(F )-valued coefficients, we are able to establish new results and shorter proofs in the context of classical problems such as the scalar heat equation subject to Dirichlet, Neumann or mixed DirichletNeumann boundary conditions B = B(D). We focus on these problems which are covered choosing AB := −ΔB in Sections 8.3 and 10.3. Keep in mind that V being a standard domain by (1.8) always ensures that D(−ΔB ) = u ∈ W 2,p (Ω); B(D)u = 0 . Furthermore, we can handle the heat equation with mixed Dirichlet-Neumann conditions also in cylindrical domains built up by rough cross-sections Vi . To be more precise, the cross-sections Vi are allowed to be bounded Lipschitz domains. In that case, we apply results from [Woo07] to the operators acting on the single cross-sections to deduce R-sectoriality or even an R-bounded H∞ -calculus for the entire problem in Lp (Ω). As it is well-known, in the context of rough domains the values of p are severely limited. Moreover, knowledge on the regularity of solutions decreases in general. Consequently, a weak formulation of the negative Laplacian with D(−ΔB ) ⊂ u ∈ W 1,p (Ω); Δu ∈ Lp (Ω) is used. With regard on existing literature it is interesting to note that a large class of possibly unbounded Lipschitz domains Ω and simultaneously mixed DirichletNeumann type boundary conditions are covered. Moreover, new results on the range of p are established such that the Dirichlet Laplacian admits an R-bounded H∞ -calculus on Lp (Ω). In Chapter 9 we apply the Fourier multiplier techniques to the Stokes equation in Ω given as infinite layer or infinite rectangular cylinder. The Stokes equation arises by linearization from the Navier-Stokes equation which describes fluid dynamics. This makes these special types of cylindrical domains particularly rewarding. To solve the Stokes resolvent problem in Ω both Fourier transform and Fourier series are used. In order to define a Stokes operator in the space Lpσ (Ω) of solenoidal fields, we construct the Helmholtz projection P ∈ L(Lp (Ω)). To this end, we first establish a rather non-physical, periodic Helmholtz projection Pper as a Fourier multiplier operator in L(Lp (Ω2 )). Here the thickness of Ω2 compared to Ω is doubled. With suitable extension and restriction operators E and R, in Theorem 9.15 we prove the relation P = RPper E. Hence, besides existence of P a most elegant representation formula is obtained at the same time. For the special case of an infinite layer this gives an alternative

6


description of P to the ones in [Far03] and [Abe05a]/[Abe05b]. For infinite rectangular domains, as far as the author knows, no according result on the Helmholtz projection has been available up to now. This thesis is structured as follows. In Chapter 2 we recall Fourier transform and Fourier series in a vector-valued and distributional framework. Here periodic and partially periodic vector-valued distributions are treated extensively. Having Lp -applications in mind, we introduce ν-periodic Lp -Sobolev spaces. In Theorem 2.14 numerous characterizations for functions u of this space are deduced. Finally, we develop the concept of partial Fourier coefficients of a partially periodic distribution and prove a representation theorem in Theorem 2.19. In Chapter 3 we introduce continuous and discrete Fourier multipliers and establish regularity results on associated operators in Lemma 3.7 and Lemma 3.11. In order to present multiplier theorems, we first recall the notions of R-bounded operator families, Banach spaces of class HT , and property (α). With these concepts at hand, we state two operator-valued Michlin multiplier results from literature as ˇ Theorems 3.17 and 3.19. By means of a strong multiplier result due to Strkalj and Weis we are able to weaken the rather strict R-boundedness condition of Theorem 3.19 which is done in Theorem 3.24. Moreover, we discuss an intermediate condition that combines flexibility and strength in a way most suitable for later purposes. In Chapter 4 classes of operators in Banach spaces are recalled. In particular, classes of pseudo-sectorial and sectorial operators are introduced. The latter allow for a Dunford calculus which is discussed rather extensively. We define boundedness of the H∞ -calculus and briefly comment on the class BIP of operators that admit bounded imaginary powers. By means of R-boundedness introduced in Chapter 3 the classes of (pseudo-)R-sectorial operators and of operators that admit an R-bounded H∞ -calculus are defined. In the sequel well-known relations of these classes are recalled. Once again the notions from Banach space geometry introduced in Chapter 3 are needed. At the end of Chapter 4 the Dore-VenniTheorem and a more recent theorem due to Kalton and Weis on the sum of two closed operators are presented. Chapter 5 collects definitions and results on parabolic problems. We briefly comment on maximal regularity in the context of Cauchy problems and state the mentioned result of Weis on an equivalent description of maximal regularity by means of R-sectoriality. Sufficient conditions for maximal regularity of a generator of an analytic semigroup are recalled for later use. The notion of maximal regularity is further defined for Volterra integral equations and sufficient conditions for this property in terms of R-boundedness due to Pr¨ uss are presented. Given a closed operator A in a Banach space X, Chapters 6 and 7 carry out the Fourier multiplier approach to A-dependent partial differential equations in


7

the whole space Rn and in the cube (0, 2π)n . In the latter case, ν-periodic boundary conditions are imposed. In both cases, properties like R-sectoriality and a bounded H∞ -calculus are transferred from A to the corresponding realizations of these problems in the respective Lp -spaces. This is done in Section 6.2 for the case of the whole space, and in Section 7.3 for the case of the cube, respectively. In Section 7.2, moreover, a characterization of unique solvability of A-dependent problems in (0, 2π)n is derived (see Theorem 7.15). This result implies sufficient conditions for unique solvability of the corresponding DirichletNeumann problem, provided all differential operators under consideration admit an appropriate structure. Employing reflection arguments, this is proved in Proposition 7.16. All of these results rely on verified multiplier conditions, that is, on verified R-boundedness conditions. This is carried out in Sections 6.1 and 7.1, respectively. The crucial tool to estimate R-bounds is known as the contraction principle of Kahane (see Lemma 3.2). It is combined with parameter-ellipticity assumptions on the differential operators and properties of A. Therefore, estimates for parameter-elliptic polynomials inferred from homogeneity arguments are summarized in Lemma 6.5. In addition, various representation formulas for continuous and discrete derivatives of operator-valued functions are needed. The corresponding results are Lemma 6.1 and Lemma 7.1. A proof of the latter includes tedious technical calculations, for one has to keep track of numerous shifts in Zn at the same time. Thereby, a wide class of resolvent-type Fourier multipliers is deduced in Propositions 6.8 and 7.8. In Chapter 8 these results are applied to parameter-elliptic, cylindrical boundary value problems in general and the Laplacian on Lipschitz cylinders in particular. In Section 8.1 we first collect known results from literature on parameter-elliptic boundary value problems in standard domains V . The Laplacian is also considered on bounded domains V of Lipschitz type. In the sequel Lp -realizations of these problems replace the abstract operator A considered in Chapter 6 and Chapter 7. This allows to treat problems with constant coefficients in unbounded cylindrical domains Ω := Rn1 × (0, 2π)n2 × V . Non-constant coefficients are treated later on with the aid of a localization procedure. Recall that in contrast to existing literature no full space or half space but a cylindrical domain serves as a model problem. In Section 8.3 we focus on the Laplacian in cylindrical domains augmented with mixed ν-periodic and Dirichlet-Neumann boundary conditions. Furthermore, the modeling of keratin network growth in biological cells by the Laplacian with pure periodic as well as mixed periodic and Dirichlet-Neumann boundary conditions is discussed. The main results of this chapter are Theorem 8.10 and Theorem 8.22. Chapter 9 considers an application to the Stokes problem in infinite layers and infinite rectangular domains Ω. First we investigate the Stokes resolvent problem subject to ν-periodic boundary conditions. Here partial Fourier series with respect to bounded coordinate directions and Fourier transform with respect to unbounded coordinate directions of the domain are employed. We recourse to the results from the previous chapters to prove unique solvability in Lp (Ω) for

8


1 < p < ∞. The precise spaces for velocity field and pressure in the ν-periodic setting are formulated in Theorem 9.5. A deeper investigation of the Stokes resolvent problem yields a ν-periodic analogue of the well-known Helmholtz projection in Lp (Ω) in terms of a Fourier multiplier operator. This rather non-physical projection turns out to be closely related to the standard Helmholtz projection in that space. A representation formula which reveals the connection of both projections is established in Theorem 9.15. Again appropriate reflection techniques come into play. The Helmholtz projection allows for a definition of the Stokes operator in the space of solenoidal fields Lpσ (Ω). Supplemented with pure-slip boundary conditions, R-boundedness of the H∞ -calculus for the Stokes operator is proved in Theorem 9.17. In Chapter 10 we employ the operator-valued Dunford calculus to investigate cylindrical boundary value problems in cylindrical domains built up by finitely many standard domains. Here operator-valued Fourier multipliers are replaced by the Kalton-Weis-Theorem. An essential assumption of this theorem is that extensions of operators, first defined on the single cross-sections, are resolvent commuting. This to prove is the main task in the application to cylindrical boundary value problems in Section 10.2. The main result is Theorem 10.5. The Laplacian in Lipschitz cylinders is brought into focus once more in Section 10.3. Using weaker commutator conditions the approach outlined above is also applied to a class of operators whose resolvents do not commute. This can be used to investigate heat conduction in a Lipschitz cylinder with either in longitudinal directions or in cross-sections non constant heat conductivity coefficient. Basic notation used in this thesis as well as definitions and facts on vector-valued function spaces and vector-valued distributions are collected in Appendix A. In Appendix B we finally comment on topics in the literature which are related to parts of this thesis.

2 Vector-valued Fourier transform and Fourier series We start this chapter by stating the main definitions and results on the vectorvalued Fourier transform as presented e.g. in [Ama95] and [ABHN01]. For a comprehensive introduction to the topic we refer to the mentioned monographs and the references therein. Later on, we establish the according results for vector-valued Fourier series. Let E denote an arbitrary Banach space and let f ∈ L1 (Rn , E). Then the E-valued Fourier transform of f is the function F f ∈ L∞ (Rn , E) defined by 1 (2.1) Ff (ξ) := e−ixξ f (x)dx. n (2π) 2 Rn

To be more precise, formula (2.1) induces a mapping in L L1 (Rn , E), C∞ (Rn , E) where the space C∞ (Rn , E) consists of the continuous functions vanishing at infinity. In (2.1) we have used the abbreviation xξ := x, ξ :=

n

xj ξj

j=1

for the standard scalar product in Rn . Restricted to S(Rn , E), the Fourier transform defines an isomorphism in this space whose inverse is given by 1 eixξ f (x)dx. (2.2) F −1 f (ξ) := n (2π) 2 Rn This property extends to the space S (Rn , E) of E-valued tempered distributions, where the extension of the Fourier transform is defined via duality, i.e. (2.3)

F u(f ) := u(F f )

(u ∈ S (Rn , E), f ∈ S(Rn )).

For u ∈ L1 (Rn , E) understood as a tempered distribution it is easily seen that this definition coincides with the one given in (2.1). The following result is of technical interest in view of coefficients in differential expressions which involve closed operators. Lemma 2.1. Let A be a closed operator in E. For u ∈ Lp (Rn , D(A)) it holds that Fu ∈ S (Rn , D(A)) and F Au = AFu. Proof. Closedness of A yields u(x)F f (x)dx = A Rn

Rn

Au(x)Ff (x)dx

(f ∈ S(Rn ))

(see e.g. [ABHN01, Proposition 1.1.7]). Since Au ∈ Lp (Rn , E) defines a tempered distribution in S (Rn , E), the claim follows from the definition in (2.3). T. Nau, Lp-Theory of Cylindrical Boundary Value Problems, DOI 10.1007/978-3-8348-2505-6_2, © Vieweg+Teubner Verlag | Springer Fachmedien Wiesbaden 2012

10


Remark 2.2. In Lemma 2.1 we can obviously replace F by F −1 . When dealing with differential operators and their resolvents, it is important to be aware of a relation between the Fourier transform of a Sobolev space function and the Fourier transforms of its partial derivatives. Even for tempered distributions u ∈ S (Rn , E) this relation is given by the fundamental property (2.4)

F (Dα u)(ξ) = ξ α F u(ξ)

(u ∈ S (Rn , E), α ∈ Nn 0 ),

where Dα := D1α1 . . . Dnαn and Dj := −i∂j . This is obvious for u ∈ S(Rn , E) due to the integration by parts formula and extends to u ∈ S (Rn , E) via (2.3). We turn our attention to E-valued Fourier series, starting with the well-known definition of E-valued Fourier coefficients as presented in [ABHN01, Section 4.2.G] for continuous E-valued functions. This definition copies verbatim to the context of functions in Lp (Qn , E), where Qn := (0, 2π)n (cf. [AB02]). More precisely, for f ∈ Lp (Qn , E) and k ∈ Zn 1 e−ikx f (x)dx fˆ(k) := (2π)n Qn is a well-defined element of E which is called the k-th Fourier coefficient of f . Without further calculations, the respective analogue of Lemma 2.1 follows. Lemma 2.3. Let A be a closed operator in E. For f ∈ Lp (Qn , D(A)) it holds that fˆ(k) ∈ D(A) and Afˆ(k) = (Af )ˆ(k) for all k ∈ Zn . Proof. As above, closedness of A yields A e−ikx f (x)dx = e−ikx Af (x)dx Qn

(k ∈ Zn ).

Qn

For k ∈ Zn and η ∈ E set ek (x) := eikx and (ηeik· )(x) := (η ⊗ ek )(x) := ηek (x) for x ∈ Rn . Given arbitrary multi-indices α, β ∈ Zn , we define α ≤ β by αj ≤ βj for all j = 1, . . . , n. With [α, β] := {k ∈ Zn ; α ≤ k ≤ β} and ηk ∈ E, f= ηk eik· = ηk ⊗ e k k∈[α,β]

k∈[α,β]

defines the trigonometric polynomial given by f (x) = ηk eikx (x ∈ Rn ). k∈[α,β]

It satisfies fˆ(k) = 0 if k ∈ / [α, β] and fˆ(k) = ηk else. We denote the class of Evalued trigonometric polynomials on Qn by T(Qn , E). It is well-known that the multiple series of Fourier coefficients (fˆ(k))k∈Zn of a function f ∈ Lp (Qn , E) does


11

not converge to f in Lp (Qn , E) in general. However, two strategies seem to be natural in order to overcome this problem. The first one is to change the series under consideration. This is done by summing up the Cesaro means of the Fourier coefficients instead of the Fourier coefficients themselves. For m ∈ N0 and N ∈ N recall the one dimensional Dirichlet and Fejer kernel Dm (t) :=

m

eikt ,

FN (t) :=

k=−m

1 Dm (t) N +1

(t ∈ R).

m≤N

Furthermore, recall the multiple Fejer kernel Fn,N (x) :=

n

FN (xj )

(x ∈ Rn ).

j=1

The proof of the following result is well-known for scalar-valued functions and copies verbatim to E-valued functions. Proposition 2.4. Let f ∈ Lp (Qn , E). Then Fn,N ∗ f ∈ T(Qn , E) for each N ∈ N and Fn,N ∗ f → f in Lp (Qn , E) for N → ∞. In particular, the space of trigonometric polynomials T(Qn , E) is dense in Lp (Qn , E). The second strategy is closely related to the extension of the Fourier transform from S(Rn , E) to S (Rn , E) as the idea is once more to extend the definition of Fourier coefficients to a certain subspace of D (Rn , E). In the present case, this subspace is given by the periodic distributions in D (Rn , E) which includes Lp (Qn , E) in some appropriate sense. Replacing Lp -convergence by convergence in D (Rn , E), we will see that each periodic distribution has a representation as the limit of its Fourier series. Since contributions in the literature to this topic in an E-valued setting seem to be rather sparse, we give a more detailed presentation of this theory. The results are well-known in scalar valued context (see e.g. [Wal94]) and thanks to the extensive E-valued distribution theory in [Ama03] the proofs can be copied along the lines. A distribution T ∈ D (Rn , E) is called 2π-periodic, or merely periodic, if T (ϕ) = T (τ2πk ϕ)

(ϕ ∈ C0∞ (Rn ), k ∈ Zn ),

where τ denotes the translation operator from Appendix A. We denote the sub space of all periodic distributions by Dper (Rn , E). Given g ∈ Lp (Qn , E), the 2πperiodic extension gper to the whole space Rn is locally integrable. For the sake of simplicity we write g = gper whenever we do not want to point out periodicity (Rn , E). In explicitly. Thus, g defines a regular, periodic distribution Tg ∈ Dper that sense, the class ∞ Cper (Qn , E) := {u|Qn ; u ∈ C ∞ (Rn , E) : u(x) = u(x + 2πk) (k ∈ Zn )}

12


(Rn , E). Basically, this is due to the well-known fact that there is dense in Dper exist (ϕε )ε ⊂ C0∞ (Rn ) such that T ∗ ϕε → T ∗ δ = T in D (Rn , E) for any distri (Rn , E) and ϕ ∈ C0∞ (Rn ), T ∗ ϕ ∈ C ∞ (Rn , E) bution T . Indeed, given T ∈ Dper is periodic again. This is clear since

ˇ = T (τx+2πk ϕ) ˇ = (T ∗ ϕ)(x + 2πk) (T ∗ ϕ)(x) = T (τx ϕ)

(k ∈ Zn ).

The most important observation towards the definition of Fourier coefficients for periodic distributions is the existence of a function α ∈ C0∞ (Rn ) such that ∗ α := k∈Zn τ2πk α = 1, i.e. (2.5) α∗ (x) := α(x + 2πk) = 1 (x ∈ Rn ). k∈Zn

Note that the sum is finite for each x ∈ Rn . In case n = 1 consider ϕ ∈ C0∞ (R) non-negative, such that ϕ(x) ≥ c > 0 for |x| ≤ π. Then ϕ∗ (x) > 0 for all x ∈ R and β := ϕ/ϕ∗ enjoys property (2.5), i.e. β ∗ = 1. For arbitrary n ∈ N we simply set α(x) := β(x1 ) · . . . · β(xn ). If we additionally let supp ϕ ⊂ (−2π, 2π), we can achieve β(0) = α(0) = 1. In virtue of (2.5), the values of Tg for g ∈ Lp (Qn , E) are explicitly given by (g, ϕ) = g(x)ϕ(x)dx = g(x)ϕ∗ (x)dx. Rn

Note that

k∈Zn

Qn

g(x)ϕ(x)dx = 2πk+Qn

g(x)ϕ(x + 2πk)dx

Qn k∈Zn

as all sums are finite. In particular, for fixed k ∈ Zn and f ∈ Lp (Qn , E) we have 1 1 fˆ(k) = e−ikx f (x)α(x)dx = (f, αe−ik· ). (2π)n Rn (2π)n (Rn , E) by means This gives rise to the definition of Fourier coefficients of T ∈ Dper of

(2.6)

Tˆ(k) :=

1 T (αe−ik· ). (2π)n

(Rn , E) defined by For instance, consider the periodic Dirac distribution δ2π ∈ Dper nˆ −ik· δ2π (ϕ) = q∈Zn ϕ(2πq). Then (2π) δ2π (k) = δ2π (αe ) = q∈Zn α(2πq) = 1, that is, 1 (k ∈ Zn ). δˆ2π (k) = (2π)n ∞ (Qn , E) independence of Tˆ(k) of the particular choice of α can By density of Cper easily be deduced from the fact that fˆ(k) does not depend on α at all.


13

Remark 2.5. Since independence can as well be shown in a direct and, in view of the definition in (2.5), a very representative manner, we carry it out briefly. Let α, β ∈ C0∞ (Rn ) both fulfill (2.5). Then β = α∗ β = q∈Zn α(· + 2πq)β ∈ C0∞ (Rn ) and the sum is de facto finite. This allows for the calculation α(· + 2πq)βe−ik· ) = T (α(· + 2πq)βe−ik· ). T (βe−ik· ) = T ( q∈Zn

q∈Zn

Due to periodicity of T we have T (α(· + 2πq)βe−ik· ) = T (β(· − 2πq)αe−ik· ) and therefore T (β(· + 2πq)αe−ik· ) = T (β ∗ αe−ik· ) = T (αe−ik· ). T (βe−ik· ) = q∈Zn

The fact that the Fourier coefficients of f ∈ Lp (Qn , E) are uniformly bounded for k ∈ Zn persists in the weaker form of polynomial boundedness for general (Rn , E). This is an easy observation in view of (A.1) since for arbitrary T ∈ Dper n K ⊂ R with supp α ⊂ K there exists m ∈ N0 such that (2.7)

(2π)n Tˆ(k) E = T (αe−ik· ) E ≤ Cp(m,K) (αe−ik· ) ≤ C|k|m

for k = 0. Lemma 2.6. Let (ηk )k∈Zn ⊂ E. Let C > 0 and m ∈ N0 such that ηk E ≤ C|k|m

(k ∈ Zn \ {0}).

ηk eik· converges in D (Rn , E), i.e. T ∈ Dper (Rn , E). Proof. With |k| = |k|2 = k12 + . . . + kn2 and N ∈ N we consider ηk eikx . f (x) = |k|2N m n

Then T :=

k∈Zn

k∈Z \{0}

Then |k|2N m = (k12 + . . . + kn2 )N m and ηk E 1 ≤ C 2N . |k|2N m |k| Thus, the sum on the right-hand side is uniformly convergent, provided N is chosen to be large enough. Hence, f is continuous and convergence is also given in D (Rn , E). Moreover, there exists a differential operator L = L(∂) such that ηk LTf (ϕ) = |k|2N m (eik· , ϕ) = ηk (eik· , ϕ). 2N m |k| n n k∈Z \{0}

k∈Z \{0}

This shows existence of η0 ∈ E such that T = η0 + LTf ∈ D (Rn , E). Since periodicity is obvious, the proof is finished.

14


With the help of Lemma 2.6 it is no longer difficult to prove the mentioned representation theorem. ik· ˆ (Rn , E) we have T = in Theorem 2.7. For every T ∈ Dper k∈Zn T (k)e n D (R , E). Proof. Due to the estimate given in (2.7) and Lemma 2.6 the series on the righthand side exists in D (Rn , E) and it remains to show equality. To this end, we find ˇ ikx e−ik· ) = eikx (2π)n Tˆ(k) T ∗ αeik· (x) = T (τx (αeik· )ˇ) = T ((τx α)e ˇ for fixed x ∈ Rn fulfills (2.5) again. This gives since τx α 1 1 T ∗ αeik· = T ∗ αeik· = T ∗ αδ2π . Tˆ(k)eik· = n (2π) (2π)n n n n k∈Z

k∈Z

k∈Z

Note that the convolutions are well-defined because of supp αeik· ⊂ supp α for all k ∈ Zn . Since α ∈ C0∞ (Rn ) can be chosen such that supp α ⊂ (−2π, 2π)n and α(0) = 1, it follows that T ∗ αδ2π = T ∗ δ = T and the proof is complete. Remark 2.8. In the last proof we took the liberty to choose α in a suitable manner. (Rn , E). To In fact, any α subject to (2.5) fulfills T ∗ αδ2π = T for every T ∈ Dper n see this, let ϕ ∈ D(R ) be arbitrary and set (αδ2π ∗ ϕ)(x) = δ2π (α(·)ϕ(x − ·)) = α(2πk)ϕ(x − 2πk) =: ψ(x). k∈Zn

Then (T ∗ ψ)(y) = T (

α(2πk)ϕ(y + 2πk − ·)) =

k∈Zn

α(2πk)T (ϕ(y + 2πk − ·))

k∈Zn

which, by periodicity of T , is equal to α(2πk)T (ϕ(y − ·)) = T (ϕ(y − ·)) α(2πk) = T (ϕ(y − ·))α∗ (0) = T ∗ ϕ. k∈Zn

k∈Zn

Hence, (T ∗ αδ2π ) ∗ ϕ = T ∗ (αδ2π ∗ ϕ) = T ∗ ϕ for arbitrary ϕ ∈ D(Rn ) which shows the claim. An immediate consequence is an important uniqueness result for Fourier series. ηq eiq· with (ηq )q∈Zn ⊂ E, then Namely, if T = q∈Zn

T (αe−ik· ) =

q∈Zn

ηq (eiq· , αe−ik· ) =

q∈Zn

eiqx e−ikx dx = (2π)n ηk .

ηq Qn

Hence, Tˆ(k) = ηk for all k ∈ Zn . On the other hand, from equality of Fourier coefficients of periodic distributions T and F on certain subsets of Zn important relations between T and F themselves can be deduced.


15

(Rn , E). Corollary 2.9. Let T, F ∈ Dper (Rn , E). (i) If Tˆ(k) = Fˆ (k) for all k ∈ Zn , then T = F in Dper

(ii) If Tˆ(k) = Fˆ (k) for all k ∈ Zn \ {0}, then T = F + η with η := Tˆ(0) − Fˆ (0). (iii) If Tˆ(k) = Fˆ (k) for all k ∈ Zn such that kj = 0 for some j ∈ {1, . . . , n}, then (Rn , E) independent of xj such that T = F + G. there exists G ∈ Dper Proof. Due to the representation theorem we have

Tˆ(k) − Fˆ (k) eik· T −F = k∈Zn

which implies (i) and (ii) immediately. If Tˆ(k) = Fˆ (k) for all k ∈ Zn such that kj = 0 as demanded in (iii), we get

Tˆ(k) − Fˆ (k) eik· =

Tˆ((k , 0)) − Fˆ ((k , 0)) ei(k ,0)· .

k ∈Zn−1

k∈Zn

Hence, T − F is independent of xj for ei(k

,0)·

enjoys this property.

As a further consequence the converse assertion of Lemma 2.3 follows. Corollary 2.10. Let A be a closed operator in E. Let (ηk )k∈Zn ⊂ D(A), C > 0, and m ∈ N0 such that ηk (D(A)) ≤ C|k|m

Then T :=

ηk eik· and F :=

k∈Zn

k∈Zn

(k ∈ Zn \ {0}).

Aηk eik· fulfill AT = F in Dper (Rn , E).

Proof. Let ϕ ∈ D(Rn ) be arbitrary. Then there exist ηT , ηF ∈ E such that

Rn

ηk eikx ϕ(x)dx → ηT

k∈[−N,N ]n

and, by closedness of A, ikx A ηk e ϕ(x)dx = Rn

k∈[−N,N ]n

Rn

Aηk eikx ϕ(x)dx → ηF

k∈[−N,N ]n

in E for N → ∞ due to convergence in D (Rn , E). Using the closedness of A another time yields ηT ∈ D(A) and AηT = ηF which proves the claim.

16


In case of Lp (Qn , E)-functions this assertion can also be proved by means of Fejer’s theorem (cf. [AB02, Lemma 3.1]). Indeed, let f, g ∈ Lp (Qn , E) such that fˆ(k) ∈ D(A) and Afˆ(k) = gˆ(k) for all k ∈ Zn . Then f ∈ Lp (Qn , D(A)) and Af = g in Lp (Qn , E). Finally, the representation theorem implies the crucial relation most similar to (Rn , E) (2.4) between Fourier coefficients of distributional derivatives of T ∈ Dper and Fourier coefficients of T itself. Switching again to Dα := (−i)|α| ∂ α , it reads as (2.8)

(Dα T )ˆ(k) = k α Tˆ(k)

(Rn , E), k ∈ Zn , α ∈ Nn (T ∈ Dper 0 ).

Besides uniqueness of Fourier coefficients we have used the fact that derivation acts as a continuous operator in D (Rn , E), i.e. Dα Tˆ(k)eik· = Tˆ(k)Dα eik· . k∈Zn

k∈Zn

Equation (2.8) will help to characterize periodic Sobolev spaces later on. m,p (Qn , E) Definition 2.11. Let m ∈ N0 . The E-valued periodic Sobolev space Wper of order m consists of all u ∈ W m,p (Qn , E) such that

(j = 1, . . . , n; 0 ≤ < m).

∂j u|xj =0 = ∂j u|xj =2π

0,p (Qn , E) = Lp (Qn , E) and for m ∈ N Note that Wper

W m,p (Qn , E) → Lp (Qn−1 , W m,p ((0, 2π), E)) → Lp (Qn−1 , C m−1 ([0, 2π], E)). m,p Hence, all traces in the definition of Wper (Qn , E) are well-defined by continuity.

Remark 2.12. Obviously, for all m ∈ N0 the space of trigonometric polynomials m,p (Qn , E). is a subset of Wper x Lemma 2.13. Let g ∈ Lp (Rn , E), j ∈ {1, . . . , n}, and f (x) := 0 j g(x , s)ds. Then for r > 0 we have f |Rn−1 ×(−r,r) ∈ Lp (Rn−1 , C((−r, r), E)) and ∂j Tf = Tg in D (Rn , E). Proof. Let r > 0, and ω := f |Rn−1 ×(−r,r) , i.e. ωx (t) := ω(x , t) :=

t 0

g(x , s)ds

(t ∈ (−r, r)).

Then H¨ older’s inequality gives p

p

ωx (t) E ≤ g(x , ·) 1,(−r,r),E ≤ |t| p−1 g(x , ·) p,(−r,r),E ≤ |t| p−1 g(x , ·) p,R,E and therefore ω ∈ Lp (Rn−1 , C((−r, r), E)). An easy application of Fubini’s theorem proves ∂j Tf = Tg in D (Rn , E).


17

Theorem 2.14. Let m ∈ N, u ∈ W m,p (Qn , E), and let uper denote its periodic extension to L1loc (Rn , E). Then the following assertions are equivalent: m,p (Qn , E). (i) u ∈ Wper m,p (ii) uper ∈ Wloc (Rn , E).

(iii) (Dα uper )ˆ(k) = kα u ˆper (k) for |α| ≤ m and k ∈ Zn . (iv) For each 0 < |α| ≤ m and each j = 1, . . . , n such that αj = 0 there exists ω ∈ Lp (Qn , E) which is independent of xj such that Dα−ej u = ω + V , where x V ∈ Lp (Qn , E) defined by V (x , xj ) := 0 j Dα u(x , s)ds fulfills V (·, 2π) ≡ 0 p in L (Qn−1 , E). (v) Dα u|xj =0 = Dα u|xj =2π for j = 1, . . . , n and |α| < m. Proof. (i) ⇒ (ii): Let j ∈ {1, . . . , n}, x = (x , xj ), and u ∈ W m,p (Qn , E). Then H(· + 2πk) uper (x , ·) = gx + ax k∈Z

x -almost everywhere, where H denotes the Heaviside function and gx ∈ C(R, E) is absolutely continuous on (2πk, 2π(k + 1)) for each k ∈ Z. Furthermore, ax ∈ E is precisely defined by ax := limt→0 u(x , t) − limt→2π u(x , t). Evidently, the condition u(·, 0) = u(·, 2π) yields ax = 0, hence ∂j uper (x , ·) = gx . The mentioned properties of gx allow for the integration by parts formula to find gx (t)ϕ (t)dt = − gx (t)ϕ(t)dt (ϕ ∈ C0∞ (R)). R

R

This shows gx ∈ L1loc (R, E). Since j was arbitrary, uper ϕ ∈ W 1,p (Rn , E) for m,p (Rn , E) follows. These arguments applied arbitrary ϕ ∈ C0∞ (Rn ), i.e. uper ∈ Wloc to derivatives up to order m prove (ii). m,p (Rn , E) implies Dα Tuper = TDα uper for all |α| ≤ m. Now (ii) ⇒ (iii): uper ∈ Wloc (iii) follows from (2.8). (iii) ⇒ (iv): Let 0 < |α| ≤ m be arbitrary and j ∈ {1, . . . , n} such that αj = 0. First note that V ∈ Lp (Qn , E) by Lemma 2.13. Calculating Fourier coefficients of V yields 1 (Dα u)ˆ(k) (k ∈ Zn , kj = 0), Vˆ (k) = kj which gives Vˆ (k) = (Dα−ej u)ˆ(k) for all k ∈ Zn such that kj = 0 by (iii). Hence, (Rn , E) independent of xj such that TDα−ej u = G + TV there exists G ∈ Dper by means of Corollary 2.9. In particular, since Dα−ej u, V ∈ Lp (Qn , E), there exists ω ∈ Lp (Qn , E) independent of xj such that G = Tω . Finally, V (·, 2π) is well-defined since V ∈ Lp (Qn−1 , C([0, 2π], E)) again by Lemma 2.13. We calculate (V (·, 2π))ˆ(k ) = Vˆ (k , 0) = 0 for all k ∈ Zn−1 , that is, V (·, 2π) ≡ 0 in

18


Lp (Qn−1 , E) and (iv) is proven. (iv) ⇒ (v): V (·, 0) = V (·, 2π) ≡ 0 by (iv) implies Dα u|xj =0 = ωα,j = Dα u|xj =2π , where ωα,j ∈ Lp (Qn , E) for j = 1, . . . , n and |α| < m. Finally, (v) ⇒ (i) is obviously true and the proof is complete. m,p As seen in the previous proof, u ∈ Wper (Qn , E) implies Dα Tuper = TDα uper for |α| ≤ m. Thus, each derivative up to order m is given as a regular distribution. The following proposition proves the converse assertion to be valid. In fact, even more is true since each periodic distribution T such that all derivatives Dα T of highest order |α| = m define regular distributions is regular itself. Moreover, it is m,p (Qn , E). In that sense, periodic Sobolev spaces induced by a function u ∈ Wper and homogeneous periodic Sobolev spaces coincide. Proposition 2.15. Let m ∈ N and T ∈ Dper (Rn , E). For each α ∈ Nn subject to p |α| = m let vα ∈ L (Qn , E) be given such that vˆα,per (k) = k α Tˆ(k) for all k ∈ Zn . m,p (Qn , E) such that Dα u = vα for |α| = m and T = Tu . Then there exists u ∈ Wper

Proof. First note that vˆα,per (k) = kα Tˆ(k) = (Dα T )ˆ(k) for all k ∈ Zn . This gives Dα T = Tvα,per for all |α| = m due to Corollary 2.9. Let β be given such that |β| = m − 1. Set Tj := Tvj := vj := vβ+ej ,per and define S := D β T . Then D j S = Tj

(j = 1, . . . , n).

Recall that any S ∈ D (R , E) subject to these equations is uniquely determined except for a constant η ∈ E. We show that S is a regular distribution Sf , where f |Qn ∈ Lp (Qn , E) fulfills (iv) of Theorem 2.14. Obviously S1 (ϕ) := −T1 (ψϕ ) as defined in (A.2) fulfills the equation for j = 1. The definition of ψϕ shows x1

n

−T1 (ψϕ ) = TV (ϕ) + Tω (ϕ),

where V (x1 , x ) := 0 v1 (s, x )ds. Here ω is independent of x1 and both V and ω restricted to Qn define functions in Lp (Qn , E). In particular S1 is regular. Let 1 < k < n and assume existence of a regular distribution Sk ∈ D (Rn , E) such that Dj Sk = Tj holds true for all j = 1, . . . , k. Following [Jan71, Satz 23.7], a regular distribution S ∗ ∈ D (Rn , E) can be found such that Sk+1 := Sk + S ∗ solves all equations with j ≤ k +1. Consequently, Sk+1 = Sk+1,f is regular as well. Moreover, it is independent of x1 , . . . , xk and f |Qn ∈ Lp (Qn , E). By induction we find that Dβ T = S := Sn,f is a regular distribution for which f |Qn ∈ Lp (Qn , E) and (iv) of Theorem 2.14 is fulfilled at least for j = 1. Since we could have started with any coordinate j ∈ {1, . . . , n}, by uniqueness of S except for a constant, the claim follows for |α| = 1 and by iteration for |α| = m. More generally, with ν ∈ Cn we will consider ν-periodic Sobolev spaces denoted m,p (Qn , E). They represent the spaces of all u ∈ W m,p (Qn , E) such that by Wν,per (Dα u)|xj =2π = e2πνj (Dα u)|xj =0

(j = 1, . . . , n, |α| < m).


19

m,p m,p (Qn , E) = W0,per (Qn , E). The following Lemma provides useIn particular, Wper m,p (Qn , E). ful characterizations of ν-periodic Sobolev spaces Wν,per

Lemma 2.16. The following assertions are equivalent: m,p (Qn , E). (i) u ∈ Wν,per m,p (ii) There exists v ∈ Wper (Qn , E) such that u = eν· v.

(iii) u ∈ W m,p (Qn , E) and for all |α| ≤ m it holds that (e−ν· Dα u)ˆ(k) = (k − iν)α (e−ν· u)ˆ(k)

(k ∈ Zn ).

Proof. (i) ⇔ (ii): This follows from the fact that the operator of multiplication with e±ν· is a bijection in W m,p (Qn , E). m,p (Qn , E) (ii) ⇔ (iii): Let ν ∈ Cn and m ∈ N be arbitrary. Then v := e−ν· u ∈ Wper and Theorem 2.14 yields (Dα v)ˆ(k) = kα vˆ(k)

(|α| ≤ m, k ∈ Zn ).

By means of the Leibniz rule α α α ν· β ν· α−β D (e v) = v= D e D (−i)β ν β eν· Dα−β v β β β≤α β≤α α = eν· (−iν)β Dα−β v. β β≤α Thus, α (−iν)β (Dα−β v)ˆ(k) β β≤α α = (−iν)β kα−β vˆ(k) = (k − iν)α vˆ(k). β β≤α

e−ν· Dα (eν· v) ˆ(k) =

On the other hand, let u ∈ W m,p (Qn , E) be given such that (e−ν· Dα u)ˆ(k) = (k − iν)α (e−ν· u)ˆ(k)

(|α| ≤ m, k ∈ Zn )

holds true. Then v := e−ν· u fulfills α α α α −ν· β −ν· α−β u= D e D (iν)β e−ν· Dα−β u D v = D (e u) = β β β≤α β≤α

20


which yields

α (D v)ˆ(k) = (iν)β e−ν· Dα−β u ˆ(k) β β≤α α = (iν)β (k − iν)α−β (e−ν· u)ˆ(k) = k α vˆ(k). β β≤α α

m,p (Qn , E) by Theorem 2.14 and the proof is complete. Hence, v ∈ Wper Accordingly, we define ν-periodic distributions Dν,per (Rn , E) by means of the condition

(2.9)

T (eν· ϕ) = T (eν· τ2πk ϕ)

(ϕ ∈ C0∞ (Rn ), k ∈ Zn ).

Then, as indicated in the previous lemma, by definition of eν· T we find (Rn , E) = eν· T ; T ∈ Dper (Rn , E) Dν,per (Rn , E) if and only if and T ∈ Dν,per

(e−ν· Dα T )ˆ(k) = (k − iν)α (e−ν· T )ˆ(k)

(k ∈ Zn ).

For applications later on, in what follows we define partial Fourier series for a distribution T ∈ D (Rn × Rm , E) which is periodic with respect to Rn only. We (Rn+m , E) for this class of distributions to avoid any mix-up with write Dper,n (Rn+m , E) = Dper,n+m (Rn+m , E). Accordingly, we write Dν,per,n (Rn+m , E) Dper for the class of partially ν-periodic distributions. The definition 1 T (αe−ik· ) (k ∈ Zn ) Tˆ(k) := (2π)n (Rn , E) as given in (2.6) is extended to T ∈ Dper,n (Rn+m , E) by for T ∈ Dper setting

1 T(x,y) (α(x)e−ikx ϕ(y)) (ϕ ∈ C0∞ (Rm ), k ∈ Zn ). Tˆ(k,y) (ϕ) := (2π)n

We write T = T(x,y) as well as T (ϕ) = T (ϕ(x, y)) = T(x,y) (ϕ(x, y)) in the sequel to indicate the dependency on particular variables. Along the lines of Remark 2.5 we see that the definition of Tˆ(k,y) is independent of the particular choice of α subject to equation (2.5). In order to work with partial Fourier series efficiently, we have to establish a representation theorem extending Theorem 2.7. Throughout the remaining part of the chapter, let α satisfy equation (2.5). Given w ∈ L1loc (Rn , E) we frequently make use of the notation [w] = [w]x for the regular distribution Tw ∈ D (Rn , E). Finally, we set ∞ (Rn+m , E) := u ∈ C ∞ (Rn+m , E); u(x, y) = u(x + 2πk, y) (k ∈ Zn ) . Cper,n


21

(Rn+m , E), f ∈ D(Rn ) and g ∈ D(Rm ). Then for Lemma 2.17. Let T ∈ Dper,n all k ∈ Zn it holds that

(2.10) T(u,v) [αeik· ]x f (x + u) · g(v) = (2π)n Tˆ(k,y) g(y) · [eik· ]x f (x) . ∞ (Rn+m , E). Proof. By density it suffices to consider T = TΨ , where Ψ ∈ Cper,n This allows for the calculation

ik· TΨ(u,v) [αe ]x f (x + u) · g(v) = Ψ(u, v) α(x)eikx f (x + u)dx g(v)d(u, v) Rn+m Rn = Ψ(u, v) α(x − u)eik(x−u) f (x)dx g(v)d(u, v) Rn+m Rn eikx Ψ(u, v)α(x − u)e−iku g(v)d(u, v) f (x)dx. = Rn

Rn+m

Since α(u) ˜ = τx α(u) for fixed x ∈ Rn satisfies (2.5) again, we conclude

ik· −iku T(u,v) [αe ]x f (x + u) · g(v) = eikx TΨ(u,v) α(u)e g(v) f (x)dx ˜ n R

ikx nˆ e (2π) TΨ(k,v) g(v) f (x)dx = (2π)n TˆΨ(k,v) g(v) · [eik· ]x f (x) = Rn

and the proof is complete. Lemma 2.18. For T ∈ Dper,n (Rn+m , E) and all k ∈ Zn it holds that

(2.11)

T ∗ [αeik· ]x ⊗ δy = (2π)n Tˆ(k,y) ⊗ [eik· ]x .

Proof. It suffices to consider ϕ ∈ D(Qn × Rm ) such that ϕ(x, y) = f (x)g(y) with f ∈ D(Rn ) and g ∈ D(Rm ) due to density of D(Rn ) × D(Rm ) in D(Rn+m ) (see e.g. [Ama03, Theorem 1.8.1]). In that case

T ∗ [αeik· ]x ⊗ δy f (x)g(y) = T [αeik· ]x ⊗ δy ∗ f (x)g(y) ˇ ˇ

= T [αeik· ]x ⊗ δy ∗ f (−x)g(−y) ˇ

= T(u,v) [αeik· ]x ⊗ δy f (x − u)g(y − v) ˇ

= T(u,v) [αeik· ]x f (x − u) · g(−v) ˇ = T(u,v) [αeik· ]x f (x + u) · g(v)

22


and Lemma 2.17 yields

f (x)g(y) = (2π)n TˆΨ(k,v) g(v) · [eik· ]x f (x) T ∗ [αeik· ]x ⊗ δy

= (2π)n Tˆ(k,y) ⊗ [eik· ]x f (x)g(y) .

Now we are in the position to extend Theorem 2.7 to partially periodic distri (Rn+m , E). butions T ∈ Dper,n (Rn+m , E) it holds that Theorem 2.19. For T ∈ Dper,n (2.12) T = Tˆ(k,y) ⊗ [eik· ]x . k∈Zn

Proof. First note that T = T ∗ δ(x,y) = T ∗ (δx ⊗ δy ) = T ∗ (αδx,2π ⊗ δy ) if α = α(x) is chosen in such a way that supp α ⊂ (−2π, 2π)n and α(0) = 1. By means of Lemma 2.18

(2π)−n T ∗ [αeik· ]x ⊗ δy . Tˆ(k,y) ⊗ [eik· ]x = k∈Zn

k∈Zn

−n ik· [e ]x . Because of separate continuity of both the Recall δx,2π = k∈Zn (2π) convolution and the tensor product, the sum on the right-hand side converges in D (Rn+m , E) and ik·

−n (2π) T ∗ [αe ]x ⊗ δy = T ∗ (αδx,2π ⊗ δy ) = T. k∈Zn

Lemma 2.20. For T ∈ Dper,n (Rn+m , E) and all k ∈ Zn it holds that

1 2 1 2 Dxα Dyα T ˆ(k,y) = kα Dα Tˆ(k,y) .

Proof. This follows from the fact that α1 α2

1 2 Dx Dy Tˆ(k,y) ⊗ [eik· ]x Dxα Dyα T = k∈Zn

=

α1 α2

2 1 Dyα Tˆ(k,y) ⊗ Dxα [eik· ]x = k Dy Tˆ(k,y) ⊗ [eik· ]x .

k∈Zn

k∈Zn


23

In particular tangential derivation and calculation of partial Fourier coefficients commute. Finally, we extend the previous lemma to the context of ν-periodicity. (Rn+m , E) and all k ∈ Zn it holds Lemma 2.21. Let ν ∈ Cn . For T ∈ Dν,per,n that

−ν· α1 α2 1 2 e Dx Dy T ˆ(k,y) = (k − iν)α Dyα (e−ν· T(k,y) )ˆ .

Proof. Let ϕ ∈ D(Rn+m ). Then

1

1 2 1 2 1 2 e−ν· Dxα Dyα T (ϕ) = Dxα Dyα T (e−ν· ϕ) = (−1)|α | Dyα T Dxα (e−ν· ϕ) 1 α |α1 | α2 β 1 −ν· α1 −β 1 = (−1) Dy T e D ϕ (iν) x β1 1 1 β ≤α

by the Leibniz rule. From T (e−ν· Dxα

1

−β 1

ϕ) = e−ν· T (Dxα

1

−β 1

ϕ) = (−1)|α

we further deduce

1 2 1 2 e−ν· Dxα Dyα T (ϕ) = (−1)|β | Dyα

=

2 Dyα

β 1 ≤α1

β 1 ≤α1

α1 β1

α1 β1

1

−β 1 |

Dxα

1

−β 1 −ν·

e

1

(iν)β Dxα

1

(−iν)β Dxα

1

1

−β 1 −ν·

−β 1 −ν·

e

T (ϕ)

e

T (ϕ)

T (ϕ).

By means of Dxα

1

−β 1 −ν·

e

1 1 T = Dxα −β e−ν· T ˆ(k,y) ⊗ [eik· ]x =

k∈Zn

k

α1 −β 1 −ν·

e

T ˆ(k,y) ⊗ [eik· ]x

k∈Zn

we find

1 2 1 2 (k − iν)α Dyα e−ν· T ˆ(k,y) ⊗ [eik· ]x (ϕ) e−ν· Dxα Dyα T (ϕ) =

k∈Zn

which proves the assertion.

3 R-boundedness and operator-valued Fourier multiplier theorems In this chapter we present results on operator-valued Fourier multipliers both in the context of Fourier transform and Fourier series. They employ the concept of Rboundedness which we introduce next. With R-boundedness at hand, conditions can be deduced which make sure that a function defines a Fourier multiplier. Besides that, R-boundedness is as well involved in necessary conditions for Fourier multipliers. We refer to [DHP03] and [KW04] for a comprehensive introduction to the notion of R-bounded operator families and restrict ourselves in this thesis to the definition and some basic properties only. Definition 3.1. A family T ⊂ L(X, Y ) is called R-bounded if there exist a C > 0 and a p ∈ [1, ∞) such that for all N ∈ N, Tj ∈ T , xj ∈ X and all independent symmetric {−1, 1}-valued random variables εj on a probability space (G, M, P ) for j = 1, . . . , N , we have that (3.1)

N

εj Tj xj Lp (G,Y ) ≤ C

j=1

N

εj xj Lp (G,X) .

j=1

The smallest C > 0 such that (3.1) is satisfied is called R-bound of T and is denoted by Rp (T ). For our purposes there is no need to distinguish the p-dependent R-bounds. Hence, we omit the index p and merely write R(T ). Observe that R-boundedness implies uniform norm boundedness. In Hilbert spaces both concepts are equivalent (see e.g. [KW04]). The following two results on R-boundedness will be used frequently in subsequent proofs. The first one shows that R-bounds behave like uniform bounds concerning sums and products. This follows as a direct consequence of the definition of R-boundedness. The second one is known as the contraction principle of Kahane. A proof can be found in [KW04] or [DHP03]. The contraction principle of Kahane and R-boundedness of the sum of R-bounded families yield R-boundedness also for the union of R-bounded families. Lemma 3.2. a) Let X, Y, and Z be Banach spaces and let T , S ⊂ L(X, Y ) as well as U ⊂ L(Y, Z) be R-bounded. Then T ∪ S ⊂ L(X, Y ), T + S ⊂ L(X, Y ) and U T ⊂ L(X, Z) are R-bounded as well. More precisely, we have R(T ∪ S), R(T + S) ≤ R(S) + R(T )

and

R(U T ) ≤ R(U)R(T ).


3 R-boundedness and operator-valued Fourier multiplier theorems

26

Furthermore, if T denotes the closure of T with respect to the strong operator topology, we have R(T ) = R(T ). b) [Contraction principle of Kahane] Let 1 ≤ p < ∞. Then for all N ∈ N, xj ∈ X, and εj as above, and for all aj , bj ∈ C with |aj | ≤ |bj | for j = 1, . . . , N we have

(3.2)

N

aj εj xj Lp (G,X) ≤ 2

j=1

N

bj εj xj Lp (G,X) .

j=1

We present two further results on R-boundedness we will need in the sequel. The first one (see e.g. [KW04, Corollary 2.14]) is of importance for applications which involve R-bounds of integral operator families. In particular, special families including Dunford integral representations are captured. Lemma 3.3. Let T ⊂ L(X) be R-bounded, let G ⊂ Rn be a domain and C > 0. Given N ∈ L∞ (G, L(X)) such that N (G) ⊂ T and g ∈ L1 (G) we set TN,g f := g(x)N (x)f dx (f ∈ X). G

Then {TN,g ; N, g as above, g 1 ≤ C} ⊂ L(X) is R-bounded. For the moment we denote by mϕ the operator of multiplication with a function ϕ in Lp -spaces. The upcoming result is of particular interest in view of localization procedures we will have to carry out while treating non-constant coefficients in boundary value problems. A proof can be found e.g. in [DHP03, Corollary 3.7]. Lemma 3.4. Let 1 ≤ p < ∞, let E and F be Banach spaces and let G ⊂ Rn be a domain. Set X := Lp (G, E) and Y := Lp (G, F ) and let T ⊂ L(X, Y ) be R-bounded. Then for each C > 0 the set {mϕ T mψ ; T ∈ T , ϕ, ψ ∈ L∞ (G), ϕ ∞ , ψ ∞ ≤ C} is R-bounded. With R-boundedness in mind, we first turn to continuous Fourier multipliers, whereas discrete Fourier multipliers are considered later on. Let E and F denote arbitrary Banach spaces and let m ∈ L∞ (Rn , L(E, F )). Then the operator Tm f := F −1 mF f

(f ∈ S(Rn , E))

is a well-defined mapping from S(Rn , E) to S (Rn , F ) since mFf ∈ L∞ (Rn , F ) defines a regular distribution in S (Rn , F ). Definition 3.5. Let 1 ≤ p < ∞. A function m ∈ L∞ (Rn , L(E, F )) is called a continuous, operator-valued, (Lp -)Fourier multiplier, if Tm f ∈ Lp (Rn , F ) for all f ∈ S(Rn , E) and if C > 0 exists such that Tm f p,F ≤ C f p,E

(f ∈ S(Rn , E)).

In that case Tm ∈ L(Lp (Rn , E), Lp (Rn , F )) by density of S(Rn , E) ⊂ Lp (Rn , E). The operator Tm is called the Fourier multiplier operator associated with m.


27

Starting with f ∈ F −1 (D(Rn , E)), the assumption m ∈ L∞ (Rn , L(E, F )) can be replaced by the weaker condition m ∈ L1loc (Rn , L(E, F )) (cf. [DHP03]). However, we stick to L∞ -functions right from the definition since it is well-known that Fourier multipliers are necessarily essentially bounded. The corresponding proof carried out for n = 1 as well as comments on the extension to arbitrary n ∈ N ˇ can be found in [DHP03], [SW07], or [KW04]. Proposition 3.6. Let E, F be Banach spaces, m ∈ L1loc (Rn , L(E, F )) and let Lm ⊂ Rn denote the set of Lebesgue points of m. If m defines a continuous Fourier multiplier, then {m(ξ); ξ ∈ Lm } is R-bounded. In applications often questions on regularity of Tm f arise. The following lemma provides a condition on m which is equivalent for W ,p -regularity of Tm f . Lemma 3.7. Let 1 ≤ p < ∞, ∈ N0 and let m ∈ L∞ (Rn \ {0}, L(E, F )). Then the following assertions are equivalent: (i) Tm ∈ L(Lp (Rn , E), W ,p (Rn , F )). (ii) For each |α| ≤ the function mα : ξ → ξ α m(ξ) defines a continuous Fourier multiplier. Proof. (i) ⇒ (ii): Set κα (ξ) := ξ α . For arbitrary f ∈ S(Rn , E) we have F Dα Tm f = κα F F −1 mF f = mα Ff in S (Rn , F ). Hence, F −1 mα F f = Dα Tm f and there exists C > 0 such that F −1 mα F f p,F = Dα Tm f p,F ≤ C f p,E for all |α| ≤ and all f ∈ S(Rn , E). (ii) ⇒ (i): For arbitrary f ∈ S(Rn , E) first note that Tm f ∈ Lp (Rn , F ) and that FDα Tm f = κα F Tm f = mα Ff holds true in S (Rn , F ). Applying F −1 yields Dα Tm f = Tmα f and Dα Tm f p,F = Tmα f p,F ≤ C f p,E for all |α| ≤ and all f ∈ S(Rn , E). Hence, Dα Tm extends to a bounded operator in L(Lp (Rn , E), Lp (Rn , F )). In close analogy, the notion of a Fourier multiplier is defined in the context of Fourier series. We refer e.g. to [CW77] for useful results on how continuous ˇ and discrete multipliers are linked to each other. See also [SW07, Lemma 3.3] ∞ n and the discussion on Theorem 3.23 below. Let M ∈ (Z , L(E, F )), that is M : Zn → L(E, F ) uniformly bounded. Then the operator TM defined by (TM f )ˆ(k) = M (k)fˆ(k)

(k ∈ Zn )

28


(Rn , F ). This mapping represents a well-defined mapping from Lp (Qn , E) to Dper has to be understood in the sense that there exists C > 0 such that

M (k)fˆ(k) F ≤ C and therefore TM f :=

(k ∈ Zn )

M (k)fˆ(k)eik· ∈ Dper (Rn , F )

k∈Zn

by Lemma 2.6. Definition 3.8. Let 1 ≤ p < ∞. A function M ∈ ∞ (Zn , L(E, F )) is called a discrete, operator-valued, (Lp -)Fourier multiplier, if C > 0 exists such that TM f p,F ≤ C f p,E

(f ∈ T(Qn , E)).

In that case TM ∈ L(Lp (Qn , E), Lp (Qn , F )) by density of T(Qn , E) ⊂ Lp (Qn , E). The operator TM is called the Fourier multiplier operator associated with M . In the definition of TM , uniform boundedness of M can be relaxed to the assumption of M being polynomially bounded for Lemma 2.6 can be applied if M (k)fˆ(k) F ≤ C|k|m

(k ∈ Zn ).

When restricted to the space of trigonometric polynomials T(Qn , E), the operator TM maps to T(Qn , F ) of course without any assumptions on M : Zn → L(E, F ). However, the following result shows the assumption M ∈ ∞ (Zn , L(E, F )) to be a natural choice. A proof carried out for n = 1 can be found in [AB02]. Comments ˇ on the extension to arbitrary n ∈ N are given in [BK04] and [SW07]. Proposition 3.9. Let E, F be Banach spaces and M : Zn → L(E, F ). If M defines a discrete Fourier multiplier, then {M (k); k ∈ Zn } is R-bounded. It is worth noting that the property of M being a

discrete Fourier multiplier is characterized by the set inclusion TM Lp (Qn , E) ⊂ Lp (Qn , F ) without any continuity condition. We make this result more precise in the following lemma. Lemma 3.10. For every function M : Zn → L(E, F ) the following two statements are equivalent: (i) M is a discrete Lp -multiplier. (ii) For each f ∈ Lp (Qn , E) there exists g ∈ Lp (Qn , F ) such that gˆ(k) = M (k)fˆ(k)

(k ∈ Zn ).

Proof. (i) ⇒ (ii): As already mentioned in the definition, TM first defined for trigonometric polynomials only extends uniquely to TM : Lp (Qn , E) → Lp (Qn , F ) by continuity. (ii) ⇒ (i): We define TM f = g, where g fulfills gˆ(k) = M (k)fˆ(k) for k ∈ Zn .


29

As D(TM ) = Lp (Qn , E), the closed graph theorem yields continuity of TM if we can show that TM is closed. To this end, let fn → f in Lp (Qn , E) as well as gn := TM fn → g in Lp (Qn , F ). Since Lp (Qn , E) ⊂ L1 (Qn , E), Lebesgue’s theorem of dominated convergence shows fˆn (k) → fˆ(k) and gˆn (k) → gˆ(k) for all k ∈ Zn . Since further M (k) ∈ L(E, F ), we additionally have gˆn (k) → M (k)fˆ(k). Hence, gˆ(k) = M (k)fˆ(k) for all k ∈ Zn and Corollary 2.9 yields g = TM f . With Lemma 3.10 at hand, questions on regularity of TM f can be answered more easily than those on regularity of Tm f (cf. Lemma 3.7). This aspect is made available in the following lemma. Lemma 3.11. Let 1 ≤ p < ∞, ∈ N0 and let M ∈ ∞ (Zn , L(E, F )). Then the following assertions are equivalent: ,p (Qn , F )). (i) TM ∈ L(Lp (Qn , E), Wper ,p (ii) TM ∈ L(Lp (Qn , E), Lp (Qn , F )) maps Lp (Qn , E) into Wper (Qn , F ).

(iii) Mα : k → kα M (k) defines a discrete Fourier multiplier for each |α| ≤ . (iv) Mα : k → kα M (k) defines a discrete Fourier multiplier for each |α| = . Proof. (i) ⇒ (ii): Obvious. ,p (ii) ⇒ (iii): For arbitrary f ∈ Lp (Qn , E) we have TM f ∈ Wper (Qn , F ) and (Dα TM f )ˆ(k) = kα M (k)fˆ(k) = Mα (k)fˆ(k) by (2.8). Since Dα TM f ∈ Lp (Qn , F ) it follows that Mα defines a Fourier multiplier for all |α| ≤ by Lemma 3.10. (iii) ⇒ (iv): Obvious. (iv) ⇒ (i): Let f ∈ Lp (Qn , E) and α subject to |α| = be arbitrary. By assumption we have vα := TMα f ∈ Lp (Qn , F ). Thus there exists C > 0 such that Mα (k)fˆ(k) F ≤ C. Set ˜ (α) (k) := M

0,

kα = 0,

M (k),

kα = 0.

Since |kα | ≥ 1 we deduce ˜ (α) (k)fˆ(k) F ≤ kα M (k)fˆ(k) F ≤ C M

(k ∈ Zn ).

In particular, there exists C > 0 such that this estimate is valid for all α = ej , where j = 1, . . . , n. This shows M (k)fˆ(k) F ≤ C for all k ∈ Zn \ {0} and further M (k)fˆ(k) F ≤ C

(k ∈ Zn ).

30


Lemma 2.6 yields T :=

k∈Zn

(Rn , F ) and by (2.8) we get M (k)fˆ(k)eik· ∈ Dper

Dα T =

Mα (k)fˆ(k)eik· = vα .

k∈Zn ,p (Qn , F ) such that T = Tg Proposition 2.15 therefore implies existence of g ∈ Wper α ˆ and (D g)ˆ(k) = M (k)f (k) for all |α| ≤ . Lemma 3.10 now finishes the proof.

We head for multiplier results and return to continuous Fourier multipliers first. The question under what circumstances a function defines an operator-valued Fourier multiplier had been unanswered for a long time. A satisfying answer eventually was given by Weis in [Wei01b], who has been able to prove an operatorvalued version of the Michlin multiplier theorem. Many other contributions, both ˇ in the context of Fourier transform and Fourier series, such as [AB02], [SW07], [HHN02], and [BK04] followed. In order to present these Fourier multiplier results, a few preparations are in order. The idea Weis pursued to prove the operatorvalued version of the Michlin multiplier theorem in case n = 1 was to build up a class of Fourier multipliers out of one basic multiplier function, namely m : R \ {0} → L(E); m(ξ) := ξ|ξ|−1 idE . However, there exist Banach spaces E such that this function fails to define a Fourier multiplier. Thus, the condition that it fulfills enters as a restriction on E in the multiplier theorem of Weis. A commonly used description of this property is given in terms of the Hilbert transform H : S(R, E) → S (R, E) defined by Hf := F −1 mi F f , where mi (ξ) := iξ|ξ|−1 idE . Definition 3.12. A Banach space E is called Banach space of class HT if there

exists a q ∈ (1, ∞) such that H extends to a bounded operator in L Lq (R, E) . Remark 3.13. a) The property of a Banach space E to be of class HT is equivalent to E being a UMD space. This in turn is equivalent to the condition that E is ζ-convex. For more information on these equivalences and on ζ-convexity in particular we refer to [Dor93, Section 3] and the survey article [RdF86] referred to therein. b) As a consequence of ζ-convexity, each Banach space of class HT is superreflexive and therefore reflexive ([RdF86, Proposition 2]). We collect some more properties of the class HT from [Ama95, Theorem 4.5.2] which we will use frequently without further comments. Lemma 3.14. a) Hilbert spaces are spaces of class HT . b) Every Banach space isomorphic to a space of class HT is of class HT . c) Finite products of spaces of class HT are of class HT . d) The spaces Lp (G, F ) are of class HT for 1 < p < ∞ if F is of class HT . e) Closed linear subspaces of spaces of class HT are of class HT .


31

The whole process of building larger classes of multiplier functions out of the basic multiplier function is carried out in detail in [KW04]. With a second Banach space F , functions of type m : R \ {0} → L(E, F ); m(ξ) := χ(a,b] (ξ) ⊗ B are established as Fourier multipliers, where χ(a,b] denotes the characteristic function of the interval (a, b] and B ∈ L(E, F ) is a fixed bounded linear operator. This allows for L(E, F )-valued functions to be considered. It finally ends with the operator-valued Michlin multiplier result due to Weis which relies on Rboundedness. To present the Fourier multiplier results in their full strength, in addition to class HT , a further notion from Banach space geometry called property (α) is needed. One situation where it comes into play is when a whole family of multiplier functions is considered and if the family of associated operators is required to be R-bounded again. Definition 3.15. A Banach space X is said to have property (α) if there exists a C > 0 such that for all n ∈ N, αij ∈ C with |αij | ≤ 1, all xij ∈ X, and all independent symmetric {−1, 1}-valued random variables ε1i on a probability space (G1 , M1 , P1 ) and ε2j on a probability space (G2 , M2 , P2 ) for i, j = 1, . . . , N , we have

G1

G2

N

ε1i (u)ε2j (v)αij xij X dudv ≤ C

G1

i,j=1

G2

N

ε1i (u)ε2j (v)xij X dudv.

i,j=1

The following facts on property (α) can be found in [KW04]. Lemma 3.16. For n ∈ N, Cn enjoys property (α) and, if E enjoys property (α), for 1 ≤ p < ∞ the spaces Lp (G, E) enjoy property (α) as well. Now we are in the position to state the Michlin multiplier theorem in operatorvalued context due to Weis which gives sufficient conditions on m to define a ˇ Fourier multiplier. See [Wei01b], [HHN02], and [SW07]. Recall that 0 ≤ γ ≤ 1 is defined by means of the different components, i.e. 0≤γ≤1

:⇐⇒

0 ≤ γi ≤ 1

(i = 1, . . . , n).

Theorem 3.17. a) Let 1 < p < ∞, let E and F be Banach spaces of class HT and let T ⊂ L(E, F ) be R-bounded. If m ∈ C n (Rn \ {0}, L(E, F )) satisfies |γ| γ |ξ| D m(ξ); ξ ∈ Rn \ {0}, 0 ≤ γ ≤ 1 ⊂ T , (3.3) then m defines a Fourier multiplier. b) If E and F additionally enjoy property (α), then γ γ (3.4) ξ D m(ξ); ξ ∈ Rn \ {0}, 0 ≤ γ ≤ 1 ⊂ T

32


is sufficient. In this case, the set

{Tm ; m satisfies condition (3.4)} ⊂ L Lp (Rn , E), Lp (Rn , F ) is R-bounded again. The additional result on R-boundedness of operators associated with a family of multipliers in part b) is due to Girardi and Weis [GW03]. For further information on operator-valued Fourier multipliers we refer to [KW04] and [DHP03]. Remark 3.18. In case n = 1 the weights |ξ||γ| in a) and |ξ γ | in b) are equal. For n > 1 the weight |ξ||γ| is larger in the sense that for | · |i ∈ {| · |1 , | · |2 } we have

|γ| |γ| max |ξj | ≥ |ξj | = |ξ γ |. |ξ|i ≥ |ξ||γ| ∞ ≥ j, γj =0

j, γj =0

The upcoming result is the analogue of Theorem 3.17 in the context of discrete Fourier multipliers. It was first proved by Arendt and Bu in [AB02] for n = 1. For arbitrary n it is due Bu and Kim ([BK04]). In the article [Bu06], Bu gives a shorter proof for arbitrary n by means of induction based on the result for n = 1. Instead of derivatives as above, in the context of Fourier series discrete derivatives are involved which we introduce next. Let E and F be Banach spaces and let G ⊂ Zn . For a function M : Zn → L(E, F ) the restriction of M to G is defined by M (k), k ∈ G, MG (k) := 0, k∈ / G.

n In particular MZn = M . Let α, β ∈ Z ∪ {−∞, ∞} with α ≤ β, i.e. αj ≤ βj for j = 1, . . . , n and let ej denote the j-th unit vector in Zn . The difference operators Δej are defined as M[α,β] (k) − M[α,β] (k − ej ), kj = αj , ej Δ M[α,β] (k) := 0, kj = αj , where k ∈ [α, β]. For arbitrary γ ∈ {0, 1}n we set (3.5)

Δ0 M[α,β] = 0,

Δγ M[α,β] := Δγ1 e1 . . . Δγn en M[α,β] .

We will sometimes refer to Δγ as the discrete derivative of order γ. Theorem 3.19. a) Let 1 < p < ∞, let E and F be Banach spaces of class HT and let T ⊂ L(E, F ) be R-bounded. If M : Zn → L(E, F ) satisfies |γ| γ |k| Δ M (k); k ∈ Zn , 0 ≤ γ ≤ 1 ⊂ T , (3.6) then M defines a Fourier multiplier. b) If E and F additionally enjoy property (α), then γ γ k Δ M (k); k ∈ Zn , 0 ≤ γ ≤ 1 ⊂ T (3.7)


33


{TM ; M satisfies condition (3.7)} ⊂ L Lp (Qn , E), Lp (Qn , F )

is R-bounded again. ˜ defined Remark 3.20. In [BK04] Theorem 3.19 is stated with discrete derivatives Δ γ γ ˜ in such a way that Δ M (k + γ) = Δ M (k). However, as for fixed γ ∈ {0, 1}n there exist c, C > 0 such that c|k − γ| ≤ |k| ≤ C|k − γ| for k ∈ Zn \ {0, 1}n , the contraction principle of Kahane shows our formulation to be equivalent to the one given in [BK04] (cf. condition (3.12) below). As mentioned by Arendt and Bu, their result can as well be deduced from a ˇ ˇ discrete multiplier theorem due to Strkalj and Weis in [SW07], see Theorem 3.23 below. The latter also serves to prove the Michlin multiplier theorem in the multidimensional case. Thus, it can be used as baseline to prove both Theorem 3.17 ˇ and Theorem 3.19. We briefly sketch the expressions and ideas used in [SW07] in order to present this result. Based on this, we can quickly derive a version of the multiplier result of Bu and Kim with a slight but useful weakening of the assumptions on M . Given a bounded subset T ⊂ L(E, F ), the Minkowski functional of the absolute convex hull N N λj Tj ; N ∈ N, Tj ∈ T , λj ∈ C with |λj | = 1 aco(T ) := j=1

j=1

is defined as · T : L(E, F ) → [0, ∞]; T → inf{t > 0; T ∈ t aco(T )}. Let M : Zn → L(E, F ) and T ⊂ L(E, F ) be bounded. Then the T -variation of M in the interval [α, β] is defined as Δγk M[α,β] (k) T , var M[α,β] := [α,β],T

k∈[α,β]

where γk = (γk1 , . . . , γkn ) with

γkj :=

1,

kj = αj ,

0,

kj = αj .

The following decompositions of Zn are crucial. Definition 3.21. a) D0 := {0} ⊂ Zn and for d = nr + j, r ∈ N0 , j ∈ {1, . . . , n} Dd := {k ∈ Zn ; |k1 |, . . . , |kj−1 | < 2r+1 , 2r ≤ |kj | < 2r+1 , |kj+1 |, . . . , |kn | < 2r } is called the coarse decomposition of Zn . r−1 ≤ || < 2r } for r ∈ N, and b) For ν = (ν1 , . . . , νn ) ∈ Nn 0 , Ir = { ∈ Z; 2 I0 := {0} Dν := Iν1 × . . . × Iνn is called the fine decomposition of Zn .


34

(a)

(b)

(c)

(d)

(e)

Figure 3.1: Steps in the coarse decomposition of Z2 : (a): D0 to (e):

4 d=0

Dd

Note that Dd is the union of sd ≤ 2 intervals [αd,i , βd,i ]. Respectively, Dν is the union of sν ≤ 2n intervals [αν,i , βν,i ]. One thus defines the T -variation of M : Zn → L(E, F ) with respect to the coarse decomposition by var M :=

Dd ,T

sd i=1

var

[αd,i ,βd,i ],T

M[αd,i ,βd,i ]

and with respect to the fine decomposition by var M :=

Dν ,T

sν i=1

var

[αν,i ,βν,i ],T

M[αν,i ,βν,i ] .

In case n = 1 we have Dd = Dν for d = ν ∈ N0 . In case n > 1 for each ν ∈ Nn 0 there exists d = dν ∈ N0 such that Dν ⊂ Dd . See Figure 3.1 for an illustration of the coarse decomposition and Figure 3.2 for an illustration of the fine decomposition of Z2 . Definition 3.22. M is said to be of bounded T -variation with respect to the coarse decomposition Dd , respectively the fine decomposition Dν , if there exists an Rbounded subset T ⊂ L(E, F ) such that the condition (3.8)

sup var M < ∞,

d∈N0 Dd ,T


35

respectively sup var M < ∞,

(3.9)

Dν ,T ν∈Nn 0

is fulfilled.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

Figure 3.2: Steps in the fine decomposition of Z2 : (a): D(0,0) to (i):

(i)

ν≤(2,2)

Dν

With this notation at hand, we can now state the discrete multiplier theorem ˇ ˇ due to Strkalj and Weis from [SW07]. Theorem 3.23. a) Let 1 < p < ∞ and let E and F be Banach spaces of class HT . Let M : Zn → L(E, F ) be of bounded T -variation with respect to the coarse decomposition. Then M defines a discrete Lp -multiplier.

36


b) Let E and F additionally enjoy property (α) and let M : Zn → L(E, F ) be of bounded T -variation with respect to the fine decomposition. Then M defines a discrete Lp -multiplier. The authors of [BK04] also consider the coarse and fine decomposition of Zn . Therefore, it is not astonishing that their result can be deduced from Theorem 3.23. Doing that, it is obvious that the assumptions (3.6) and (3.7) on M can be relaxed. While the readability of the weaker condition decreases, the restrictions for M anyhow similarly do so. Due to the definition of the coarse and fine decomposition most of the benefit is gained close to zero. It will turn out that this relaxation helps to avoid undesirable shifts of operators in applications (see e.g. Lemma 9.4 and Proposition 7.25). For this reason we present the following version of Theorem 3.19. Theorem 3.24. a) Let 1 < p < ∞, let E and F be be Banach spaces of class HT and let T ⊂ L(E, F ) be R-bounded. If M : Zn → L(E, F ) satisfies (3.10)

{|k||γk | Δγk MDd (k); d ∈ N0 , k ∈ Dd } ⊂ T ,

then M defines an Lp -multiplier. b) If E and F additionally enjoy property (α), then (3.11)

{kγk Δγk MDν (k); ν ∈ Nn 0 , k ∈ Dν } ⊂ T


{TM ; M satisfies condition (3.11)} ⊂ L Lp (Qn , E), Lp (Qn , F ) is R-bounded again. In particular, (3.11) is fulfilled if (3.12) {M (k); k ∈ Zn } ∪ {kγ Δγ M (k); k ∈ Zn \ {−1, 0, 1}n , 0 = γ ≤ 1 ⊂ T . Proof. a) By Theorem 3.23 it suffices to show that M is of bounded T -variation with respect to the coarse decomposition. Let d ∈ N0 with d = rn + j be arbitrary and Dd = [αd,− , βd,− ] ∪ [αd,+ , βd,+ ], where kj ≤ 0 for k ∈ [αd,− , βd,− ] and kj ≥ 0 for k ∈ [αd,+ , βd,+ ]. Then Δγk MDd (k) T ≤

1 |k||γk |

follows by assumption (3.10). The definition of [αd,± , βd,± ] and γk shows sup var ≤ 2 sup

d∈N0 Dd ,T

d∈N0

k∈[αd,− ,βd,− ]

1 . |k||γk |


37

We rearrange the sum and use the fact that | · | ≥ | · |∞ for | · | ∈ {| · |1 , | · |2 } to get k∈[αd,− ,βd,− ]

1 1 1 = ≤ . |γ | |γ| |γ| |k| k |k| γ≤1 k;γk =γ γ≤1 k;γk =γ |k|∞

Clearly, for 0 ≤ ≤ n we have {γ ∈ Nn 0 ; 0 ≤ γ ≤ 1, |γ| = } =

n .

As the length of any edge of [αd,− , βd,− ] is smaller than 2r+2 and |k|∞ ≥ 2r for d ≥ 1 we have

1

γ≤1 k;γk =γ

|γ|

|k|∞

l n n 2r+2 n n l ≤ ≤ 4 ≤ 23n . 2r =0 =0

Since this estimate is independent of d, the proof of part a) is complete. b) Again by Theorem 3.23 it suffices to show that M is of bounded T -variation with respect to the fine decomposition. Let ν = (ν1 , . . . , νn ) ∈ Nn 0 be arbitrary and sν [αν,i , βν,i ]. D ν = Iν 1 × . . . × I ν n = i=1

By assumption (3.11) Δγk MDν (k) T ≤

1 |kγk |

holds true. Let i− ∈ {1, . . . , sν } such that αν,i− ≤ 0. By definition of γk and Dν we then have 1 1 ≤ , γ |k k | |kγk | k∈[αν,i ,βν,i ]

k∈[αν,i− ,βν,i− ]

hence,

sup var = sup ν∈Nn 0

Dν ,T

ν∈Nn 0

≤ sup

sν

(Δγk M[αν,i ,βν,i ] )(k) T

i=1 k∈[αν,i ,βν,i ] sν

ν∈Nn 0 i=1 k∈[αν,i ,βν,i ]

1 ≤ 2n sup |kγk | ν∈Nn 0


1 . |kγk |


38

Again we rearrange the sum to get

1 = γ k |k |

|γ|≤1 k;γk =γ


=1+

n

j=1 k;γk =ej

... +

n

1 |kγk |

n 1 ej + |k | j,i=1

j=1 k;γk =1−ej

1 + ... |kej +ei |

k;γk =ej +ei

1 1 + . |k1−ej | k;γ =1 |k1 | k

For arbitrary ∈ {1, . . . , n} we find k;γk =ej1 +...+ej

1 = |kej1 +...+ej | k;γ

k =ej1 +...+ej

=

ν −1 2 j1 −2

k ;γ =ej2 +...+ej i=1

As

ν −1 2 j −2

r=0

1 ν −1 2 j +r

ν −1 2 ji −2

r=0

1

1 1 · ... · |kj1 | |kj |

2νj1 −1 + r

r=0

k

=

1 1 · ... · |kj2 | |kj |

1 2νji −1 + r

.

≤ 1 for all ν ∈ Nn 0 and all j = 1, . . . , n, this gives n n = 22n sup var ≤ 2 Dν ,T j ν∈Nn 0 j=0 n

independent of ν ∈ Nn 0. The result on R-boundedness of operators associated with a family of multipliers is now proved as in [GW03, Theorem 3.2]. Remark 3.25. As long as we investigate finitely many functions with regard to being discrete Fourier multipliers, it suffices to check conditions (3.10) or (3.11) for large k ∈ Zn , i.e. for k ∈ Zn \ G with a finite set G ⊂ Zn . This is due to the fact that a family of finitely many bounded linear operators as well as the union of finitely many R-bounded families is R-bounded. By our choice of the intermediate condition (3.12) it is now apparent what ’benefit close to zero’ exactly means. Within the cube {−1, 0, 1}n only the values of M itself enter into the R-boundedness condition, whereas no values of the discrete derivatives Δγ M have to be considered. In particular, since γ ≤ 1, the value M (0) does not occur in any expression resulting from shifts of M as a consequence of discrete derivation of order γ. For an infinite family {Mλ ; λ ∈ Λ} of functions


(a) The coarse decomposition.

39

(b) The fine decomposition.

Figure 3.3: The intermediate conditions. The balls in the center indicate that here the intervals [αd,i , βd,i ] and [αν,i , βν,i ] do only contain one single point k ∈ Z2 . Hence, kj = αj for j = 1, 2 and γk = 0. This implies that no discrete derivatives of order γ = 0 enter in the calculation of varDd ,T M and varDν ,T M .

under consideration recall that the family {Mλ (k); k ∈ {−1, 0, 1}n , λ ∈ Λ} of operators cannot be neglected (see Proposition 3.9). Similarly, an intermediate condition can be defined for part a) of the multiplier theorem. However, we cannot exclude the whole cube {−1, 0, 1}n in case γ = 0. Still, the value M (0) does not occur in any expression resulting from shifts of M as a consequence of discrete derivation of order γ (see Figure 3.3).

4 Classes of operators and Dunford functional calculus This chapter provides important classes of linear operators in Banach spaces. Each class plays a crucial role in the investigation of parabolic problems. For later application we will distinguish between injective and non-injective operators. It starts with the classes of pseudo-sectorial and sectorial operators which allow for a Dunford functional calculus. With R-boundedness from Chapter 3 at hand, the class of operators admitting an R-bounded H∞ -calculus is discussed. This class is of particular interest in view of our purposes later on. We refer to [DHP03], [KW04], and [Haa06] for a deeper investigation of the different classes. Definition 4.1. A closed, densely defined linear operator A in a Banach space X is called pseudo-sectorial if (−∞, 0) ⊂ ρ(A) and if there exists C > 0 such that (4.1)

t(t + A)−1 L(X) ≤ C

(t > 0).

In this case, it is well-known that there exists a φ ∈ (0, π) such that the uniform estimate extends to all λ ∈ Σπ−φ := {z ∈ C\{0}; | arg(z)| < π − φ}, see e.g. [DHP03]. The number φA := inf φ; ρ(−A) ⊃ Σπ−φ ,

sup λ∈Σπ−φ

λ(λ + A)−1 L(X) < ∞

is called spectral angle of A. The class of pseudo-sectorial operators is denoted by ΨS(X). A pseudo-sectorial operator A is called sectorial if additionally R(A) is dense in X and N (A) = {0}. The class of sectorial operators is denoted by S(X).

(a) Spectra of A and −A.

(b) The sector Σπ−φ .

Figure 4.1: Relevant sets in the complex plane.


42

4 Classes of operators and Dunford functional calculus

For A ∈ ΨS(X) observe that σ(A) ⊂ ΣφA . It is well-known that −A generates a bounded analytic C0 -semigroup on X in case that φA < π2 . The definition of sectoriality as presented above follows [DHP03]. It collects all additional properties of an operator A subject to condition (4.1) which allow for an intensive functional calculus. Pseudo-sectoriality, however, is an advisable weakening that is motivated, for instance, by the Neumann Laplacian on bounded domains which obviously fails to be injective. Besides pseudo-sectoriality and sectoriality, a third notion called quasi-sectoriality is used in literature. It defines operators which are sectorial after a suitable shift (cf. [Haa06]). Hence, pseudosectorial operators are a special class of quasi-sectorial operators. For the sake of convenience we distinguish between pseudo-sectorial and sectorial operators only. Note that injectivity of A could have been canceled in the definition of sectoriality for it is implied by the assumption R(A) = X. Moreover, if X is known to be reflexive, most additional properties besides (4.1) come for free. The following proposition comments on these relations briefly. We refer to [Haa06] and [DHP03] for proofs and comprehensive introductions to (pseudo-)sectorial operators. Proposition 4.2. Let A define a closed operator subject to condition (4.1). a) It holds that N (A) ∩ R(A) = {0}, hence, density of R(A) implies N (A) = {0}. b) If X is reflexive, we have D(A) = X and X = N (A) ⊕ R(A). Consequently, A ∈ ΨS(X). Moreover, density of R(A) and N (A) = {0} are equivalent. Hence, each of the latter conditions implies A ∈ S(X). Next we introduce a holomorphic functional calculus for sectorial operators. This will lead to the notion of an H∞ -calculus and a first important subclass of S(X). For a comprehensive introduction to this concept we refer to [CDMY96], [KW01], [DHP03], [KW04], [DV05], and [Haa06]. For σ ∈ (0, π] we denote by H∞ (Σσ ) the commutative algebra of bounded, holomorphic functions on Σσ , that is, H∞ (Σσ ) := {f : Σσ → C; f is holomorphic, |f |σ∞ < ∞} , where |f |σ∞ := sup{|f (z)|; z ∈ Σσ }. Using ρ(z) :=

z (1+z)2

we define the subalgebra

H0∞ (Σσ ) := {f ∈ H∞ (Σσ ); there exist C, ε > 0 such that |f (z)| ≤ C|ρ(z)|ε for all z ∈ Σσ }. Let A be a pseudo-sectorial operator in X. Pick σ ∈ (φA , π] and ψ ∈ (φA , σ). The path Γ := (∞, 0]eiψ ∪ [0, ∞)e−iψ oriented counterclockwise, i.e. the positive real axis R+ lies to the left, stays with the only possible exception at zero in the resolvent set of A. Hence, by Cauchy’s


43

integral formula and the pseudo-sectoriality of A, for every f ∈ H0∞ (Σσ ) the Bochner integral 1 f (μ)(μ − A)−1 dμ f (A) := 2πi Γ represents a well-defined element in L(X). Moreover, the above formula defines an algebra homomorphism (4.2)

ΦA : H0∞ (Σσ ) → L(X);

f → f (A)

known as the Dunford calculus. Example 4.3. It holds that ρ(A) :=

1 2πi

Γ

ρ(μ)(μ − A)−1 dμ = A(1 + A)−2 .

If A is sectorial, one can extend the Dunford calculus to arbitrary f ∈ H∞ (Σσ ) if one accepts that f (A) is possibly unbounded. To see this, we make use of the fact that ρf ∈ H0∞ (Σσ ) for f ∈ H∞ (Σσ ). Furthermore, since A ∈ S(X) is injective, the same is true for ρ(A)−1 . This is sufficient to define f (A) := ρ(A)−1 (ρf )(A),

D(f (A)) := {x ∈ X; (ρf )(A)x ∈ D(A) ∩ R(A)},

which gives rise to a closed, densely defined operator in X. By Cauchy’s theorem this definition is consistent with the former one if f ∈ H0∞ (Σσ ). Moreover, if A is sectorial, functions f of polynomial growth at 0 and at infinity can be handled. In that case the above definition has to be modified. More precisely, there exists k ∈ N such that f (A) := ρ(A)−k (ρk f )(A),

D(f (A)) := {x ∈ X; (ρk f )(A)x ∈ D(Ak ) ∩ R(Ak )},

gives rise to a closed, densely defined operator in X (see [DHP03, Theorem 2.1] or [Haa06, Proposition 2.3.13]). For μ ∈ C the complex powers fμ (z) := z μ are holomorphic in the sliced complex plane Σπ and admit polynomial behavior at 0 and at infinity. Thus, Aμ is well-defined as a closed, densely defined operator in X. In particular, fractional powers Aα , 0 < α < 1 are captured. It is worthwhile to mention that pseudo-sectoriality only is not sufficient to apply these kinds of extension procedures for the injectivity of A cannot be dropped. Indeed, if A is not injective and if g ∈ H0∞ (Σσ ) exists such that gf ∈ H0∞ (Σσ ) and g(A)−1 is injective, then f has to show a certain limit behavior at 0. In fact, λ ∈ C and s > 0 have to exist, such that f (z) − λ tends to zero by the rate of |z|s as z → 0 (see [Haa06, Lemma 2.3.8]). This, however, still allows for the consideration of the subalgebra E(Σσ ) := H0∞ (Σσ ) ⊕ (1 + z)−1 ⊕ 1 := {g(z) + c(1 + z)−1 + d;

g ∈ H0∞ (Σσ ), c, d ∈ C}

44


which weakens the condition of decay at infinity. For f ∈ E(Σσ ) one sets f (A) := g(A) + c(1 + A)−1 + d whenever f (z) = g(z) + c(1 + z)−1 + d with g ∈ H0∞ (Σσ ) and c, d ∈ C. Given a pseudo-sectorial operator A, the Dunford calculus extends to an algebra homomorphism ΦA : E(Σσ ) → L(X); f → f (A) (see [Haa06, Theorem 2.3.3]). For many other special classes of functions f a definition of f (A) is meaningful in case of a pseudo-sectorial operator A; for instance, there exists g ∈ E(Σσ ) such that the definition f (A) := g(A)−1 (gf )(A) extends the Dunford calculus to holomorphic f of polynomial growth at infinity which fulfill the necessary limit behavior at 0 ([Haa06, Proposition 2.3.11]). Definition 4.4. Let A ∈ ΨS(X) and let F ∈ {H0∞ , E, H∞ }. We say that the Fcalculus of A is bounded if there exists σ > φA such that f (A) ∈ L(X) for all f ∈ F (Σσ ) and ΦA : F(Σσ ) → L(X); f → f (A) is bounded with respect to the topologies on H∞ (Σσ ) and L(X), that is, if there exists C > 0 such that (4.3)

f (A) L(X) ≤ Cσ |f |σ∞

(f ∈ F (Σσ )).

The bound for ΦA in general depends on σ. The infimum over all σ > φA such that this bound remains finite is called F-angle of A and is denoted by φF A. We denote the class of operators A ∈ ΨS(X) which admit a bounded H0∞ -calculus on X by ΨH∞ (X) and the class of operators A ∈ S(X) which admit a bounded F H∞ -calculus on X by H∞ (X). In both cases, we write as well φ∞ A instead of φA . Remark 4.5. Instead of (4.3) we can equivalently require (4.4)

f (A) L(X) ≤ Cσ

(f ∈ F (Σσ ), |f |σ∞ ≤ 1).

Moreover, the E-calculus of an operator A ∈ ΨS(X) is bounded if and only if the H0∞ -calculus is bounded. To see this, consider for a moment all f ∈ E(Σσ ) such that |f |σ∞ ≤ 1, say f (z) = gf (z) + cf (1 + z)−1 + df with gf ∈ H0∞ (Σσ ) and cf , df ∈ C. Then c = cf , d = df and g = gf are bounded independently of f . On the one hand, since g(z) → 0 and c(1 + z)−1 + d → c + d for z → 0 it follows that |c + d| ≤ 1. On the other hand, since g(z) + c(1 + z)−1 → 0 for |z| → ∞ it follows that |d| ≤ 1, hence |c| ≤ 2. Altogether |g|σ∞ ≤ 4 follows and this estimate does not depend on f . The following result known as the convergence lemma ([CDMY96, Lemma 2.1], see also [Haa06, Proposition 5.1.4]) is the crucial tool to get a grip on f (A) in case f∈ / E(Σσ ).


45

Lemma 4.6. Let A ∈ ΨS(X), σ ∈ (φA , π) and let (fn )n∈N ⊂ H∞ (Σσ ) such that fn (A) is well-defined for n ∈ N. Suppose that supn∈N fn ∞ < ∞, let the limit f (z) := limn→∞ fn (z) exists pointwise on Σσ , and assume that f (A) is welldefined. Then fn (A)x → f (A)x for all x ∈ D(A) ∩ R(A). If additionally A ∈ S(X), fn (A) ∈ L(X), and supn∈N fn (A) L(X) < ∞, then f (A) ∈ L(X) and fn (A) → f (A) strongly. In case A ∈ S(X) ∩ ΨH∞ (X), thanks to the convergence lemma and the following result (see e.g. [CDMY96, Corollary 2.2]), ΦA extends boundedly from H0∞ (Σσ ) to H∞ (Σσ ). Lemma 4.7. Let f ∈ H∞ (Σσ ) and let ρn ∈ H0∞ (Σσ ) be defined by ρn (z) :=

1 1 n2 z − z − = . 1 + z/n 1 + zn (1 + nz)(n + z)

Then fn := ρn f and f fulfill the assumptions of the convergence lemma. Corollary 4.8. For A ∈ S(X) we additionally have A ∈ ΨH∞ (X) if and only if A ∈ H∞ (X). An important subclass of H∞ (X) is given by the class of sectorial operators which admit bounded imaginary powers. Definition 4.9. Let A ∈ S(X). Then A is said to admit bounded imaginary powers if Ais ∈ L(X) for all s ∈ R and if there is a constant C > 0 such that Ais L(X) ≤ C

(|s| ≤ 1).

The class of such operators is denoted by BIP(X). Moreover, the quantity θA := lim|s|→∞

1 log Ais L(X) |s|

is called power angle of A ∈ BIP(X). Equivalently, A ∈ BIP(X) if there exists M ≥ 1 and θ such that (4.5)

Ais L(X) ≤ M eθ|s|

(s ∈ R).

Moreover, θA = inf{θ; (4.5) is valid} (see [DHP03], [KW04], or [Haa06]). The non-trivial implication follows from the group property of imaginary powers Ais and the convergence lemma. Let s ∈ R and σ ∈ (0, π). For the imaginary power function f : Σσ → C; f (z) := z is we then have |f (z)| = |z is | = |eis ln z | = e|s| arg(z) ≤ eσ|s|

(z ∈ Σσ ).

Thus |f |σ∞ ≤ eσ|s| which particularly shows θA ≤ φ∞ A . Altogether we have H∞ (X) ⊂ BIP(X) ⊂ S(X)

46

and


φA ≤ θA ≤ φ∞ A.

One remarkable property of operators which belong to the class BIP(X) is the following result on fractional powers (see e.g. [Tri78]). It describes the spaces

Xα := XAα := D(Aα ), · α , x α := x X + Aα x X (0 < α < 1) by means of complex interpolation. Here the embeddings D(A) ⊂ Xα ⊂ X

(0 < α < 1)

are valid. Proposition 4.10. Suppose A ∈ BIP(X). Then for 0 < α < 1 the space Xα is isomorphic to [X, D(A)]α . So far uniform norm boundedness of different families of operators has been considered. With the stronger concept of R-boundedness at hand (see Chapter 3), the above definition of sectoriality can be adjusted to R-sectoriality. Definition 4.11. A pseudo-sectorial operator A ∈ ΨS(X) is called pseudo-Rsectorial if there exist an angle φ ∈ (0, π) and a constant Cφ > 0 such that (4.6)

R({λ(λ + A)−1 ; λ ∈ Σπ−φ }) ≤ Cφ .

The class of pseudo-R-sectorial operators is denoted by ΨRS(X) and we call φRS A given as the infimum over all angles φ such that (4.6) holds the R-angle of A. If in addition A ∈ S(X), then A is called R-sectorial and we write A ∈ RS(X). As R-boundedness is stronger than the uniform boundedness with respect to operator norm in general, R-sectoriality always implies the sectoriality of an operator A and we have φA ≤ φRS A . Accordingly, we can strengthen the condition of a bounded H∞ -calculus by means of R-boundedness. Definition 4.12. Let A ∈ ΨS(X) and let F ∈ {H0∞ , E, H∞ }. We say that the F-calculus of A is R-bounded if there exists σ > φA such that f (A) ∈ L(X) for all f ∈ F(Σσ ) and if there exist a σ > φA and a constant Cσ > 0 such that (4.7)

R({f (A); f ∈ F (Σσ ), |f |σ∞ ≤ 1}) ≤ Cσ .

The infimum over all σ > φA such that (4.7) holds true with some Cσ > 0 is called the RF -angle of A and is denoted by φRF A . We denote the class of operators A ∈ ΨS(X) which admit an R-bounded H0∞ calculus on X by ΨRH∞ (X) and the class of operators A ∈ S(X) which admit an R-bounded H∞ -calculus on X by RH∞ (X). In both cases, we write as well instead of φRF φR∞ A A .


47

Remark 4.13. Observe that the E-calculus of an operator A ∈ ΨS(X) is Rbounded if and only if A ∈ ΨRH∞ (X). In particular, A ∈ ΨRH∞ (X) implies A ∈ ΨRS(X) (cf. Remark 4.5). In order to compare all classes of operators introduced above, the notions of class HT and property (α) are employed. For instance, if X is of class HT , then it is shown in [CP01] that bounded imaginary powers imply R-sectoriality. Proposition 4.14. Let X be a Banach space of class HT and let A ∈ BIP(X) with power angle θA . Then A ∈ RS(X) and φR A ≤ θA . To sum it up, in a Banach space X of class HT the inclusions RH∞ (X) ⊂ H∞ (X) ⊂ BIP(X) ⊂ RS(X) ⊂ S(X) are valid and the corresponding angles fulfill ∞ R∞ φ A ≤ φR A ≤ θA ≤ φA ≤ φA .

Figure 4.2: Relation of angles in a Banach space of class HT .

In a Banach space enjoying property (α) more can be derived. Indeed, boundedness and R-boundedness of the H0∞ -calculus coincide. Equality of boundedness and R-boundedness of the H∞ -calculus can be found in [KW01, Theorem 5.3]. In fact, all steps of the proof carry over to the case of an H0∞ -calculus. Proposition 4.15. Let X have property (α). Then A ∈ ΨH∞ (X) if and only if A ∈ ΨRH∞ (X). Accordingly, A ∈ H∞ (X) if and only if A ∈ RH∞ (X). In these R∞ cases φ∞ A = φA . Furthermore, if A is known to be R-sectorial and to admit a bounded H∞ calculus, equality of according angles follows ([KW01, Proposition 5.1]). In fact, the proof applies to the according assertion for pseudo-sectorial operators. Here no restriction on the underlying Banach space is needed. Proposition 4.16. Let X be a Banach space. If A ∈ H∞ (X) ∩ RS(X), respecR tively A ∈ ΨH∞ (X) ∩ ΨRS(X), then φ∞ A = φA .

48


For a long time it remained an open question under what circumstances in a Banach space X the sum of two closed operators A and B is closed again. A first suitable answer was given by the celebrated result of Dore and Venni in [DV87], roughly saying that the assertion is valid if both operators have bounded imaginary powers and if their resolvents commute, i.e. if

λ ∈ ρ(A), μ ∈ ρ(B) . (λ − A)−1 (μ − B)−1 = (μ − B)−1 (λ − A)−1 In fact, even the property of having bounded imaginary powers is inherited by A + B. In their paper, Dore and Venni imposed 0 ∈ ρ(B) ∩ ρ(A). This restriction could be removed in [PS90, Theorem 4]. Theorem 4.17. Let X be a Banach space of class HT . Let A, B ∈ BIP(X) be resolvent commuting and assume θA + θB < π. Then A + B with domain D(A + B) := D(A) ∩ D(B) is closed and sectorial. Moreover, there exists C > 0 such that

x ∈ D(A + B) Ax X + Bx X ≤ C (A + B)x X is satisfied and 0 ∈ ρ(A) ∪ ρ(B) implies 0 ∈ ρ(A + B). Let θA = θB . Then A + B ∈ BIP(X) and θA+B ≤ max{θA , θB }. Another answer to the question under what conditions A+B is closed was given by Kalton and Weis in [KW01]. They were able to establish closedness of A + B if one of the operators is R-sectorial and if the other one admits a bounded H∞ calculus ([KW01, Theorem 6.3]). Here no condition on the underlying Banach space X is needed. Besides closedness, R-sectoriality is established for A + B provided the underlying Banach space enjoys property (α). Theorem 4.18. Let X be a Banach space and let A ∈ H∞ (X) and B ∈ RS(X) R be two resolvent commuting operators such that φ∞ A + φB < π. Then A + B with domain D(A + B) := D(A) ∩ D(B) is closed and sectorial. Moreover, there exists C > 0 such that

x ∈ D(A + B) Ax X + Bx X ≤ C (A + B)x X is satisfied and 0 ∈ ρ(A) ∪ ρ(B) implies 0 ∈ ρ(A + B). ∞ R If X enjoys property (α), then A + B ∈ RS(X) and φR A+B ≤ max φA , φB . This result is particularly important for the situation of Cauchy problems, i.e. B = ∂t , since then the assumption for A of having bounded imaginary powers is reduced to the weaker property of R-sectoriality. In many other situations such as in the investigation of Volterra integral equations the latter can be employed successfully. In view of our applications, we want sums A + B or as well products AB not only to be R-sectorial again, but even to admit an H∞ -calculus. This leads to the following result which is obtained in the same way as Theorem 4.18 (see [NS11a]).


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Proposition 4.19. a) Let X be a Banach space of class HT with property (α). ∞ Let A, B ∈ H∞ (X) be two resolvent commuting operators such that φ∞ A + φB < π. R,∞ ∞ ∞ ∞ Then A + B ∈ RH (X) and φA+B ≤ max {φA , φB }. ∞ ∞ b) If additionally 0 ∈ ρ(A), then AB ∈ RH∞ (X) and φR,∞ AB ≤ φA + φB . Remark 4.20. By iteration it readily follows that the assertions remain true for finite sums (respectively finite products) as long as in each step the condition for the H∞ -angles and commutativity of the resolvents is satisfied. Accordingly, Proposition 4.18 applies in iteration to B ∈ RS(X) and A1 , . . . , AN ∈ H∞ (X). In application, we also employ a corresponding result to Proposition 4.19a) for the non-commuting case from [PS07]. Indeed, the same assertion holds if the following so-called Labbas-Terreni condition is satisfied. ⎧ Let 0 ∈ ρ(A) and let there exist constants c > 0, 0 ≤ α < β < 1, ⎪ ⎪ ⎪ ⎪ ⎨ ψA > φA , ψB > φB , ψA + ψB < π, (4.8) ⎪ such that for all λ ∈ Σπ−ψA , μ ∈ Σπ−ψB it holds that ⎪ ⎪ ⎪ ⎩ A(λ + A)−1 [A−1 , (μ + B)−1 ] ≤ c/(1 + |λ|)1−α |μ|1+β . Here [S, T ] = ST − T S. The result given in [PS07] then reads as follows. Proposition 4.21. Let X be a Banach space of class HT with property (α) and ∞ let A, B ∈ H∞ (X) . Suppose that (4.8) holds for some angles ψA > φ∞ A , ψB > φB with ψA + ψB < π. Then there exists δ ≥ 0 such that A + B + δ is invertible and such that A + B + δ ∈ RH∞ (X) and φ∞ A+B+δ ≤ max{ψA , ψB }. In case that the resolvents commute or if c in (4.8) is small enough, we can take δ = 0. Remark 4.22. Again iteration is possible, provided the angle and commutator conditions are satisfied in every step. Remark 4.23. Both Proposition 4.19 and Proposition 4.21 exist in slightly different versions if X is an arbitrary Banach space, (cf. [KW01] and [PS07]). In our applications, however, the assumptions that the underlying Banach space is of class HT and that it enjoys property (α) are always satisfied.

5 Parabolic problems and maximal regularity This chapter recalls the notion of maximal regularity both in the context of Cauchy problems (see e.g. [Dor93] and [KW04]) and Volterra integral equations (see e.g. [Pr¨ u93]). In both cases, sufficient conditions for maximal regularity in terms of properties of linear operators from the previous chapter are derived. First we turn our attention to Cauchy problems u˙ + Au

=

f

in (0, T ),

u(0)

=

0,

where T ∈ (0, ∞]. Here A : X ⊃ D(A) → X is supposed to be a closed and densely defined operator. Recall that in case A ∈ ΨS(X), φA < π2 , the operator −A generates a bounded analytic C0 -semigroup on X. For a suitable treatment of related nonlinear problems, however, the generation of an analytic semigroup might not be enough. Then the stronger property of maximal regularity is required which is defined as follows. Definition 5.1. Let 1 ≤ q ≤ ∞, let X be a Banach space, and let A be closed and densely defined. Then A is said to have maximal Lq -regularity on X if for each f ∈ Lq (R+ , X) there is a unique solution u : R+ → D(A) of the Cauchy problem u˙ + Au u(0)

(5.1)

= =

f in R+ , 0,

satisfying the estimate u ˙ Lq (R+ ,X) + Au Lq (R+ ,X) ≤ C f Lq (R+ ,X) with a C > 0 independent of f ∈ Lq (R+ , X). Due to a result of Sobolevskii ([Sob64], see also [Dor93, Theorem 4.2]), the class of operators having Lq -maximal regularity does not depend on q. Indeed, if A as Lq -maximal regularity for one q ∈ (1, ∞), then A as Lq -maximal regularity for all q ∈ (1, ∞). We therefore drop the indication and agree to speak of maximal regularity of A, only. Furthermore, if the operator A has the property of maximal regularity, then −A is known to be the generator of an analytic semigroup (see e.g. [Dor93, Theorem 2.1] and the references given there). Remark 5.2. Note that maximal regularity does not imply u ∈ W 1,q (R+ , X) for the solution of the Cauchy problem. In fact, in case A has maximal regularity, the solution u of (5.1) fulfills u ∈ W 1,q (R+ , X) and u W 1,q (R+ ,X) + Au Lq (R+ ,X) ≤ C f Lq (R+ ,X) T. Nau, Lp-Theory of Cylindrical Boundary Value Problems, DOI 10.1007/978-3-8348-2505-6_5, © Vieweg+Teubner Verlag | Springer Fachmedien Wiesbaden 2012

52


if and only if the condition 0 ∈ ρ(A) is added (see e.g. [KW04]). Since there is no common differentiation in terminology, we agree to speak of A having strong maximal regularity if A has maximal regularity and if 0 ∈ ρ(A). Moreover, if A has the property of (strong) maximal regularity, A has also the property of (strong) maximal regularity if R+ in (5.1) is replaced by any finite interval (0, T ), T < ∞ (see e.g. [KW04]). Note that maximal regularity and strong maximal regularity coincide if R+ is replaced by any finite interval (0, T ), T < ∞. In applications, sometimes maximal regularity is established for A + δ, δ > 0 only but not for A itself. Restricted to finite intervals (0, T ), T < ∞ this still yields maximal regularity of A substituting f by e−δ· f and u by eδ· u ∈ W 1,p ((0, T ), X). The following result due to Weis ([Wei01b, Theorem 4.2]) characterizes the property of maximal regularity on Banach spaces of class HT by means of pseudoR-sectoriality of the operator under consideration. Theorem 5.3. Let X be a Banach space of class HT and let A generate a bounded analytic semigroup on X. Then A has the property of maximal regularity if and π only if A ∈ ΨRS(X) and φR A < 2. As a consequence of Theorem 5.3 we can use pseudo-R-sectoriality to reformulate the result given in [Wei01a, Corollary 4d)]. Corollary 5.4. Let 1 < p < ∞ and let G ⊂ Rn be an arbitrary domain. If −A is the generator of a positive analytic semigroup of contractions on Lp (G), then π A ∈ ΨRS(Lp (G)) and φR A < 2. By definition A ∈ RS(Lp (G)) follows, provided A is injective. In order to deduce a bounded H∞ -calculus for A, the condition of injectivity has to be added necessarily. In fact, it is the only additional condition which has to be imposed. This result relies on Corollary 5.4, Proposition 4.16 and a result of Duong ([Duo90]) and Hieber and Pr¨ uss ([HP98]) based on the transference principle. In the absence of injectivity the proof in [Duo90] still applies. In that case, of course only an H0∞ -calculus for A can be deduced. Corollary 5.5. Let 1 < p < ∞ and let G ⊂ Rn be an arbitrary domain. If −A is the generator of a positive analytic semigroup of contractions on Lp (G), π then A ∈ ΨH∞ (Lp (G)) and φ∞ A < 2 . If in addition A is injective, it holds that π < . A ∈ H∞ (Lp (G)) and φ∞ A 2 As a second application of R-sectoriality in the context of parabolic equations, we consider the abstract Volterra equation t b(t − s)Au(s)ds = f (t) (t ∈ J) (5.2) u(t) + 0

in a Banach space X. Here A : X ⊃ D(A) → X is again a closed and densely defined operator, J = [0, T ] with T ∈ (0, ∞), and b ∈ L1loc (R+ ) is a scalar-valued kernel.


53

This equation is often referred to as the strong Volterra equation. The mild formulation of equation (5.2) reads as t b(t − s)u(s)ds = f (t) (t ∈ J). (5.3) u(t) + A 0

In order to discuss maximal Lq -regularity for abstract Volterra equations, we introduce the spaces Hqα (R, X) by means of Fourier transform. For α ∈ R+ we define (see [Tri78, Definition 4.2.1]) α

Hqα (R, X) := {f ∈ S (R, X); ∃fα ∈ Lq (R, X) : F fα (ξ) = (1 + |ξ|2 ) 2 Ff (ξ)} and f α,q := f α,q,R := fα q . For J = [0, T ] with T ∈ (0, ∞) we set Hqα (J, X) := {f |J ; f ∈ Hqα (R, X)} and f α,q := f α,q,J :=

inf

g : g|J =f, g∈Hqα (R,X)

gα q .

Definition 5.6. Let 1 < q < ∞, J = [0, T ] with T ∈ (0, ∞), and b ∈ L1loc (R+ ). We say that equations (5.2) and (5.3) have maximal Lq -regularity if there exists α ≥ 0 such that (i) for each f = b∗g where g ∈ Lq (J, X) and b as above there is a unique solution u ∈ Hqα (J, X) ∩ Lq (J, D(A)) of (5.2), and there is a constant C(T ) > 0 such that u α,q + Au q ≤ C(T ) g q , (ii) for each f ∈ Lq (J, D(A)) there is a unique solution u ∈ Lq (J, D(A)) of (5.2) with u − f ∈ Hqα (J, X), and there is a constant C(T ) > 0 such that

u q + u − f α,q + Au q ≤ C(T ) f q + Af q , (iii) for each f ∈ Lq (J, X) there is a unique solution u ∈ Lq (J, X) of (5.3) with b ∗ u ∈ Hqα (J, X) ∩ Lq (J, D(A)), and there is a constant C(T ) > 0 such that u q + b ∗ u α,q + Ab ∗ u q ≤ C(T ) f q . To present a result on maximal Lq -regularity of equations (5.2) and (5.3) based on R-sectoriality of A, some definitions of useful properties of b are in order. First recall that a function b ∈ L1loc (R+ ) is of subexponential growth if for each ε > 0 ∞ e−εt |b(t)|dt < ∞. 0

If b is of subexponential growth, the Laplace transform Lb of b is defined to be ∞ e−zt b(t)dt (Re z > 0). Lb(z) := 0

54


Definition 5.7. Let b ∈ L1loc (R+ ) be of subexponential growth. (i) Let k ∈ N. Then b is called k-regular if there is a constant c > 0 such that |z n Lb(n) (z)| ≤ c|Lb(z)|

(Re z > 0, 0 ≤ n ≤ k).

(ii) Let Lb(z) = 0 for all Re z > 0. Then b is called sectorial with angle φb if

φb := sup{| arg Lb(z) |; Re z > 0} < π. If b ∈ L1loc (R+ ) is of subexponential growth and 1-regular, for each ξ ∈ R \ {0} the limit Lb(λ) Lb(iξ) := lim Reλ>0, λ→iξ

exists ([Pr¨ u93, Lemma 8.1]). Now we are in the position to state the result on maximal Lq -regularity that we already mentioned. Theorem 5.8. Let X be a Banach space of class HT . Let b ∈ L1loc (R+ ) be of subexponential growth, 1-regular, sectorial, and let lims→∞ |Lb(s)|sα < ∞ for some α ≥ 0. Let A ∈ RS(X) such that the parabolicity condition φb + φR A < π is fulfilled. In addition, let either 0 ∈ ρ(A) or b ∈ L1 (R+ ). Then for each 1 < q < ∞ equations (5.2) and (5.3) have maximal Lq -regularity. This result due to Pr¨ uss is basically taken from [Pr¨ u93, Theorem 8.7] where it is proved for A ∈ BIP(X) and φR A replaced by θA . This is due to the fact that the proof there is based on Theorem 4.17 by Pr¨ uss and Sohr. Hence, it is proved in [Pr¨ u93, Theorem 8.6] that the multiplier operator B associated with m(ξ) :=

1 Lb(iξ)

admits bounded imaginary powers with θB = φb . Since furthermore B ∈ H∞ (X) and φ∞ B = φb , the result in the version as stated can now be deduced from the more recent Theorem 4.18. In the subsequent chapters we establish pseudo-R-sectoriality for numerous operators under consideration. If possible, we always emphasize that the R-angle is less than π2 . However, we do not formulate the implications on the parabolic problems presented above each time.

6 Fourier transform approach to operator-dependent problems In this chapter we employ the continuous Fourier multiplier result Theorem 3.17 to investigate partial differential equations in the whole space Rn . These equations are allowed to depend on a closed linear operator in a suitable way. Apart from its own interest, this chapter provides the first part of an abstract background for the treatment of cylindrical boundary value problems later on.

6.1 Preliminaries The multiplier conditions of Theorem 3.17 rely on derivatives of operator-valued functions. Hence, we start this section with appropriate representation formulas. Given α ∈ Nn 0 \ {0}, let r ωj = α (6.1) Zα := W = (ω 1 , . . . , ω r ); 1 ≤ r ≤ |α|, 0 < ω j ≤ α, j=1

denote the set of all additive decompositions of α into r = rW multi-indices. For the sake of consistence we set Z0 := {∅} and r∅ := 0. The following lemma collects well-known representation formulas for derivatives of operator-valued smooth functions, where m, r ∈ N and X, Y, Z as well as Xj for j = 0, . . . , r denote Banach spaces. Lemma 6.1. a) [Leibniz rule] Let T ∈ C m (Rn , L(X, Y )), S ∈ C m (Rn , L(Y, Z)). Then ST ∈ C m (Rn , L(X, Z)) and for α ∈ Nn 0 such that |α| ≤ m it holds that α (Dα−β S)(Dβ T ). Dα (ST ) = β β≤α

b) Let

Sj ∈ C 1 (Rn , L(Xj−1 , Xj )) for j = 1, . . . , r. Then rj=1 Sj ∈ C 1 (Rn , L(X0 , Xr )) and D ei

r j=1

r r e

l−1

Sj = Sj (D i Sl ) Sj . l=1

j=1

j=l+1

c) Let S ∈ C (R , L(X, Y )) such that S (x) := (S(x))−1 exists for all x ∈ Rn . Then S −1 ∈ C m (Rn , L(Y, X)) and for α ∈ Nn 0 such that |α| ≤ m it holds that rW j (−1)rW S −1 Dα (S −1 ) = (Dω S)S −1 . m

−1

n

W∈Zα

j=1


56

6 Fourier transform approach to operator-dependent problems

Proof. The formulas follow by induction from the basic formulas for derivatives of vector-valued functions as presented e.g. in [Ama03, Section 2.4]. In the sequel let E denote a Banach space and A : E ⊃ D(A) → E a closed operator. We turn our attention to multipliers of resolvent type. Definition 6.2. Let P and Q define C-valued polynomials. We call a continuous Fourier multiplier a continuous multiplier of resolvent type if it is built by sums and compositions of operator-valued functions ξ → P (ξ) + Q(ξ)A : D(A) → E and their pointwise inverses

−1 : E → D(A). ξ → P (ξ) + Q(ξ)A

In the situation described above D := D(A), · A is a Banach space, where · A denotes the graph norm of A. If we set AD : D → E; x → Ax, then AD ∈ L(D, E). Right after the following lemma we will no longer distinguish between A and AD Lemma 6.3. Let P and Q define C-valued polynomials. Set S : Rn → L(D, E); ξ → P (ξ) + Q(ξ)AD . Then S ∈ C ∞ (Rn , L(D, E)) and Dα S(ξ) = Dα P (ξ) + Dα Q(ξ)AD

(α ∈ Nn 0 ).

To establish a suitable class of resolvent type multipliers, we will consider elliptic α of an polynomials P and Q. Recall the principal part P # (ξ) := |α|=m aα ξ α arbitrary polynomial P (ξ) := |α|≤m aα ξ with aα ∈ C, |α| ≤ m as well as the degree of P given by deg P := m. Definition 6.4. Let P : Rn → C; ξ → P (ξ) define a polynomial and let P # denote its principal part. a) P is called elliptic if P # (ξ) = 0 for ξ ∈ Rn \ {0}. b) Let φ ∈ (0, π). Then P is called parameter-elliptic in Σπ−φ if λ + P # (ξ) = 0 for (λ, ξ) ∈ Σπ−φ × Rn \ {(0, 0)}. In this case, ϕP := inf{φ ∈ (0, π); P is parameter-elliptic in Σπ−φ } is called the angle of parameter-ellipticity of P .

6.1 Preliminaries

57

Lemma 6.5. a) P is parameter-elliptic in Σπ−φ if and only if for all polynomials N with deg N ≤ deg P there exist C > 0 and a bounded subset G ⊂ Rn such that the estimate |ξ|m |N (ξ)| ≤ C|λ + P (ξ)| holds for all λ ∈ Σπ−φ , all ξ ∈ Rn \ G, and all 0 ≤ m ≤ deg P − deg N . b) P is elliptic if and only if the assertion in a) is valid for λ = 0. c) If P is parameter-elliptic in Σπ−φ , for each polynomial N with deg N ≤ deg P and each ε > 0 there exists C > 0 such that the estimate |ξ|m |N (ξ)| ≤ C|λ+P # (ξ)| holds for all λ ∈ Σπ−φ , all ξ ∈ Rn \ Bε (0), and all 0 ≤ m ≤ deg P − deg N . d) If P is elliptic, the assertion in c) is valid for λ = 0. Proof. a) First assume P to be parameter-elliptic in Σπ−φ . By the triangle inequality we can assume that N is given as a monomial. The function

|ξ|m N (ξ) κ : Σπ−φ × Rn \ (0, 0) → C; (λ, ξ) → λ + P # (ξ) is continuous and quasi-(deg P, 1)-homogeneous of degree m + deg N − deg P . If m + deg N equals deg P , it is therefore quasi-(deg P, 1)-homogeneous of degree zero, i.e. κ(sρ λ, sξ) = κ(λ, ξ) (s > 0), where ρ := deg P . Hence, it is bounded. To see this, we set K := {(λ, ξ) ∈ Σπ−φ × Rn ; |λ| + |ξ|ρ = 1}. By the parameter-ellipticity condition we obtain λ + P # (ξ) = 0

((λ, ξ) ∈ K).

Consequently, κ is a continuous function on the compact set K and we obtain |κ(λ, ξ)| ≤ M

((λ, ξ) ∈ K).

By the quasi-homogeneity of κ this implies |κ(sρ λ, sξ)| ≤ M

((λ, ξ) ∈ K, s > 0).

Due to |sρ λ| + |sξ|ρ = sρ (|λ| + |ξ|ρ )

((λ, ξ) ∈ K, s > 0)

ρ −1/ρ

and the particular choice s = (|λ| + |ξ| ) (sρ λ, sξ) ∈ K

we thus deduce

((λ, ξ) ∈ Σπ−φ × Rn ).

Therefore |κ(λ, ξ)| = |κ(sρ λ, sξ)| ≤ M

((λ, ξ) ∈ Σπ−φ × Rn ).

58


If m + deg N < deg P , for each ε > 0 boundedness remains true for κ restricted to Σπ−φ × (Rn \ Bε (0)). Moreover, the upper bound tends to zero as ε is getting large. In particular, for L := P − P # and ε large enough

|L(ξ)| ≤ 12 |λ + P # (ξ)| (λ, ξ) ∈ Σπ−φ × (Rn \ Bε (0)) follows. Therefore

|λ + P (ξ)| = |λ + P # (ξ) + L(ξ)| ≥ |λ + P # (ξ)| − |L(ξ)| ≥ 12 |λ + P # (ξ)|

and for each fixed ε > 0 there exists C > 0 such that |ξ|m |N (ξ)| ≤ C|λ + P # (ξ)| ≤ 2C|λ + P (ξ)| holds true for all (λ, ξ) ∈ Σπ−φ × (Rn \ Bε (0)). # Conversely, let this estimate hold

true for some ε > 0. Assume λ0 + P (ξ0 ) = 0 n for a fixed (λ0 , ξ0 ) ∈ Σπ−φ × R \ (0, 0). Then with ρ := deg P tρ (λ0 + P # (ξ0 )) = tρ λ0 + P # (tξ0 ) = 0 follows. The particular choice m = ρ and N ≡ 1 now implies tρ |ξ0 |m = |tξ0 |m |N (tξ0 )| ≤ C|tρ λ0 + P (tξ0 )| = |L(tξ0 )| which gives a contradiction for large t > 0. Assertion c) has just been proved in part one of the proof of a). With dependence of κ on λ being neglected, assertions b) and d) follow along the lines. Remark 6.6. For |α| ≤ deg P the polynomial Dα P defines a polynomial of degree not greater than deg P − |α|. Hence, the assertions of Lemma 6.5 particularly apply to m := |α| and N := D α P . Remark 6.7. Recall that deg P has to be even in case P is elliptic and n > 1. In the following proposition we investigate multipliers of resolvent type involving elliptic polynomials. Note that we do not require deg P = deg Q when saying ’P and Q elliptic’ and that existence of (λ + μA)−1 for λ, μ ∈ C is meant to imply both (λ + μA)−1 (E) = D(A) and (λ + μA)−1 ∈ L(E). Proposition 6.8. Let A be a closed operator in a Banach space E of class HT

−1 is and let P, Q : Rn → C denote elliptic polynomials such that P (ξ) + Q(ξ)A well-defined for all ξ ∈ Rn \ {0}. Let N define an arbitrary polynomial subject to deg N ≤ deg P and assume that the families

−1

−1 P (ξ) P (ξ) + Q(ξ)A ; ξ ∈ Rn \ {0} , P (ξ) + Q(ξ)A ; ξ ∈ Rn \ {0} , and

−1 A P (ξ) + Q(ξ)A ; ξ ∈ Rn \ {0}

are R-bounded. Then for 1 < p < ∞

−1 m : Rn \ {0} → L(E); ξ → N (ξ) P (ξ) + Q(ξ)A

defines an Lp -multiplier.

6.1 Preliminaries

59

Proof. Since the subset of invertible operators in L(E) is known to be open,

−1 existence of P (ξ) + Q(ξ)A for all ξ ∈ Rn \ {0} implies smoothness of m. Lemma 6.1 and Lemma 6.3 therefore yield γ

−1 |γ| γ |ξ| D m(ξ) = (−1)rW |ξ||γ−β| (Dγ−β N )(ξ) P (ξ) + Q(ξ)A β W∈Z β≤γ β

·

rW

−1 . |ξ||ωj | Dωj P (ξ) + Dωj Q(ξ)A P (ξ) + Q(ξ)A

j=1

Recall deg Dγ−β N ≤ deg N − |γ − β| from Remark 6.6. Hence, ellipticity of P and Lemma 6.5 imply |ξ||γ−β| |Dγ−β N (ξ)| ≤ C|P (ξ)| for ξ ∈ Rn \ GN with a bounded set GN ⊂ Rn . By Kahane’s contraction principle we obtain the R-boundedness of

−1 |ξ||γ−β| Dγ−β N (ξ) P (ξ) + Q(ξ)A ; ξ ∈ Rn \ G N . Along the same lines R-boundedness of

−1 |ξ||ωj | Dωj P (ξ) P (ξ) + Q(ξ)A ; ξ ∈ Rn \ G P follows. Since

−1

−1 = idE −P (ξ) P (ξ) + Q(ξ)A , Q(ξ)A P (ξ) + Q(ξ)A R-boundedness of

−1 ; ξ ∈ Rn \ G Q |ξ||ωj | Dωj Q(ξ)A P (ξ) + Q(ξ)A finally follows from the ellipticity of Q. Setting G := GN ∪ GP ∪ GQ , it is left to prove R-boundedness of the above families for ξ ∈ G \ {0} for then the assertion follows from Lemma 3.2 and Theorem 3.17. To this end, we make use of the additional R-boundedness conditions for the families

−1 ; ξ ∈ Rn \ {0} P (ξ) + Q(ξ)A and

−1 A P (ξ) + Q(ξ)A ; ξ ∈ Rn \ {0} .

By means of the contraction principle of Kahane, for a fixed M > 0, they show the families

−1 ; ξ ∈ Rn \ {0}, |δ| ≤ M (P (ξ) + δ) P (ξ) + Q(ξ)A and

−1 (Q(ξ) + δ)A P (ξ) + Q(ξ)A ; ξ ∈ Rn \ {0}, |δ| ≤ M

60


to be R-bounded as well. Now let |δ| be large enough, such that P (ξ) + δ = 0 and Q(ξ) + δ = 0 for all ξ ∈ G. Then there exists C > 0 such that |ξ||γ−β| |Dγ−β N (ξ)| ≤ C, |P (ξ) + δ| and

|ξ||ωj | |Dωj P (ξ)| ≤ C, |P (ξ) + δ|

|ξ||ωj | |Dωj Q(ξ)| ≤C |Q(ξ) + δ|

for all ξ ∈ G due to continuity. Applying the contraction principle of Kahane, the proof is complete. Corollary 6.9. Let A be a closed operator in a Banach space E of class HT , let −1 P, Q : Rn → C denote homogeneous elliptic polynomials, and let P (ξ)+Q(ξ)A n be well-defined for all ξ ∈ R \{0}. Let N define a homogeneous polynomial subject to deg N = deg P and assume that the family

−1 ; ξ ∈ Rn \ {0} P (ξ) P (ξ) + Q(ξ)A is R-bounded. Then for 1 < p < ∞

−1 m : Rn \ {0} → L(E); ξ → N (ξ) P (ξ) + Q(ξ)A defines an Lp -multiplier. Proof. Because of P = P # , N = N # subject to deg N = deg P , and Q = Q# we can set G = GN = GP = GQ = {0} (cf. Lemma 6.5). Therefore we can neglect R-boundedness of

−1 ; ξ ∈ Rn \ {0} P (ξ) + Q(ξ)A and

−1 A P (ξ) + Q(ξ)A ; ξ ∈ Rn \ {0} .

Example 6.10. Let A ∈ ΨRS(E), i.e. (0, ∞) ⊂ ρ(−A) and

R {t(t + A)−1 ; t ∈ (0, ∞)} < ∞. Let α ∈ Nn 0 such that |α| = 2. Then, due to Corollary 6.9 m : Rn \ {0} → L(E); ξ → ξ α (|ξ|2 + A)−1 defines a Fourier multiplier. In view of Lemma 3.7, the previous example indicates that parameter-ellipticity of P and pseudo-R-sectoriality of A are likely to define an appropriate setting to apply the theory of resolvent type multipliers successfully.

6.2 R-sectoriality and RH∞ -calculus

61

6.2 R-sectoriality and RH∞ -calculus Let E be a Banach space of class HT and let A be a closed operator in E. In what follows we consider the parameter-dependent and A-dependent problem given by (6.2)

λu + A(D)u = f

Here A(D) := P (D) + Q(D)A :=

in Rn .

pα D α +

qα D α A

|α|≤m2

|α|≤m1

with m1 , m2 ∈ N0 and constant coefficients pα , qα ∈ C. The following results are restricted to homogeneous polynomials. To some extent, they can be improved in order to cover non-homogeneous polynomials as well. However, as indicated by Proposition 6.8, the assumptions needed are less convenient to present. Moreover, in view of the applications we have in mind we can relinquish full generality. Lemma 6.11. Let ϕP , ϕQ , φ, ϑ ∈ [0, π) be given. If λ + P (ξ) ∈ Σπ−φ Q(ξ)

(λ ∈ Σπ−ϑ , ξ ∈ Rn \ {0})

holds for all homogeneous polynomials P and Q which are parameter-elliptic with angles of parameter-ellipticity ϕP and ϕQ , then ϕP + ϕQ + φ < π. In that case, ϑ ∈ (max{ϕP , ϕQ + φ}, π − min{ϕP , ϕQ + φ}) implies λ + P (ξ) ∈ Σπ−φ Q(ξ)

(λ ∈ Σπ−ϑ , ξ ∈ Rn \ {0}).

Proof. Let ϕ > ϕP . By definition, parameter-ellipticity of P yields P (ξ)+λ = 0 for (λ, ξ) ∈ Σπ−ϕ × Rn \ {(0, 0)}, that is, P (ξ) = λ for (λ, ξ) ∈ C \ Σϕ × Rn \ {(0, 0)} respectively P (ξ) ∈ Σϕ . Since the range of P is closed, this ensures P (ξ) ∈ ΣϕP . Accordingly we deduce Q(ξ) ∈ ΣϕQ and moreover Q(ξ) 1 ∈ ΣϕQ = Q(ξ) |Q(ξ)|2

(ξ ∈ Rn \ {0}).

Recall that e.g. ξ → ρe±iθ |ξ|2 with ρ > 0 and θ ∈ [0, π) define parameter-elliptic polynomials with angles of parameter-ellipticity equal to θ. For fixed ϑ ∈ [0, π) the assertion λ + P (ξ) ∈ Σπ−φ (λ ∈ Σπ−ϑ , ξ ∈ Rn \ {0}) Q(ξ) for all parameter-elliptic polynomials P , Q with angles of parameter-ellipticity ϕP and ϕQ is therefore equivalent to arg(λ + e±iϕP ) + ϕQ < π − φ

(λ ∈ Σπ−ϑ ).

62


This implies π − ϑ + ϕP + ϕQ ≤ π − φ with π − ϑ > 0. Hence, ϕP + ϕQ + φ < π. In that case ϑ ∈ (max{ϕP , ϕQ + φ}, π − min{ϕP , ϕQ + φ}) exists. First assume that ϕP ≥ ϕQ +φ. This implies ϕP < ϑ < π−(ϕQ +φ) respectively ϕQ + φ < π − ϑ < π − ϕP . If ϕP ≥ π2 then π − ϑ < π − ϕP < ϕP yields arg(λ + P (ξ)) < ϕP for λ ∈ Σπ−ϑ and ξ ∈ Rn \ {0}. Now let ϕP < π2 . If π − ϑ ≤ ϕP then arg(λ + P (ξ)) < ϕP holds for λ ∈ Σπ−ϑ and ξ ∈ Rn \ {0} follows. In case ϕP < π − ϑ taking into account π − ϑ < π − ϕP yields arg(λ + P (ξ)) < π − ϑ for λ ∈ Σπ−ϑ and ξ ∈ Rn \ {0}. Since ϕP , π − ϑ < π − (ϕQ + φ) in both cases λ + P (ξ) ∈ Σπ−φ Q(ξ)

(λ ∈ Σπ−ϑ , ξ ∈ Rn \ {0})

follows. Conversely, assume ϕP < ϕQ + φ, that is, ϕQ + φ < ϑ < π − ϕP . Consequently we have ϕP < π − ϑ < π − (ϕQ + φ) and due to ϕP + ϕQ + φ < π necessarily ϕP < π2 . Thus, ϕP < π − ϑ < π − (ϕQ + φ) < π − ϕP implies arg(λ + P (ξ)) < π − ϑ for λ ∈ Σπ−ϑ and ξ ∈ Rn \ {0} as well as π − ϑ + ϕQ < π − φ, that is, λ + P (ξ) ∈ Σπ−φ Q(ξ)

(λ ∈ Σπ−ϑ , ξ ∈ Rn \ {0}).

Remark 6.12. The assertion of Lemma 6.11 remains true if λ ∈ Σπ−ϑ ∪ {0}. The Lp (Rn , E)-realization of A as given in problem (6.2) is defined as D(A) := {u ∈ W m1 ,p (Rn , E); Q(D)Au ∈ Lp (Rn , E)}, Au := A(D)u

(u ∈ D(A)).

Proposition 6.13. Let 1 < p < ∞, let E be a Banach space of class HT enjoying property (α), and let A ∈ ΨRS(E). For homogeneous polynomials P and Q assume that (i) P is parameter-elliptic with angle ϕP ∈ [0, π), (ii) Q is parameter-elliptic with angle ϕQ ∈ [0, π), (iii) ϕP + ϕQ + φR A < π. p n R Set ϕ0 := max{ϕP , ϕQ + φR A }. Then A ∈ RS(L (R , E)) and φA ≤ ϕ0 . Moreover, for each φ > ϕ0 it holds that |α| 1− (6.3) R λ m1 Dα (λ + A)−1 ; λ ∈ Σπ−φ , α ∈ Nn < ∞. 0 , 0 ≤ |α| ≤ m1

In case Q ≡ c, c = 0 subject to condition (ii), 0 ∈ ρ(A) implies 0 ∈ ρ(A).


63

Proof. We consider the formal representation (λ + A)−1 = Tmλ of the resolvent of A, where Tmλ denotes the operators associated with

−1 mλ (ξ) := λ + P (ξ) + Q(ξ)A . More generally, with α ∈ Nn 0 we consider λ mα λ (ξ) := λ

|α| 1

1− m

|α| 1

1− m

Dα (λ + A)−1 = Tmα , where λ

−1 ξ α λ + P (ξ) + Q(ξ)A .

To justify the representation formulas, we make use of Theorem 3.17 to establish n mα λ as a Fourier multiplier. By uniqueness of the Fourier transform in S (R , E) it follows that λ + A is bijective and that Tmλ defines its inverse operator. R Let φ ∈ (max{ϕP , ϕQ + φR A }, π − min{ϕP , ϕQ + φA }). We apply Lemma 6.1 γ γ α in order to calculate ξ D mλ (ξ). As in the proof of Proposition 6.8 it suffices to show that |α|

−1 1− (6.4) λ m1 ξ ω Dω N (ξ) λ + P (ξ) + Q(ξ)A ; λ ∈ Σπ−φ , ξ ∈ Rn \ {0} for N (ξ) := ξ α and arbitrary ω ≤ γ,

−1 ; λ ∈ Σπ−φ , ξ ∈ Rn \ {0} (6.5) ξ ω Dω P (ξ) λ + P (ξ) + Q(ξ)A for 0 < ω ≤ γ, and

−1 (6.6) ξ ω Dω Q(ξ)A λ + P (ξ) + Q(ξ)A ; λ ∈ Σπ−φ , ξ ∈ Rn \ {0} for 0 < ω ≤ γ are R-bounded. Due to our assumptions on P and Q, the pseudo-R-sectoriality of A, and Lemma 6.11 −1 λ + P (ξ) λ + P (ξ) ; λ ∈ Σπ−φ , ξ ∈ Rn \ {0} +A Q(ξ) Q(ξ) is R-bounded. This readily yields R-boundedness of

−1 (6.7) λ + P (ξ) λ + P (ξ) + Q(ξ)A ; λ ∈ Σπ−φ , ξ ∈ Rn \ {0} and further R-boundedness of

−1 (6.8) Q(ξ)A λ + P (ξ) + Q(ξ)A ; λ ∈ Σπ−φ , ξ ∈ Rn \ {0} . Recall that λ + P (ξ) = 0 for λ ∈ Σπ−φ by our choice of φ > ϕP . Therefore, quasi-homogeneity of (λ, ξ) → λ + P (ξ) allows to apply the contraction principle of Kahane to prove (6.4) and (6.5) (cf. Lemma 6.5). Similarly, ellipticity and homogeneity of Q proves (6.6). Finally, Q(D)A(λ + A)−1 f ∈ Lp (Rn , E) for arbitrary f ∈ Lp (Rn , E) follows from (6.8).

64


We prove injectivity of A. To this end, consider u ∈ D(A) such that Au = 0, that is, (λ + A)u = λu. Hence, for each λ ∈ R+ the unique solution of (6.2) with right-hand side f := λu is given by u. Due to (6.3) for α ∈ Nn 0 subject to |α| = m1 we have Dα u p ≤ λu p . Since λ ∈ R+ is arbitrary, Dα u p = 0 follows. Because of u ∈ W m,p (Rn , E), this yields u = 0. In case Q ≡ c, c = 0 subject to condition (ii), and 0 ∈ ρ(A), what has been proved so far remains valid if A is replaced by A − δ with δ > 0 sufficiently small. This proves 0 ∈ ρ(A).

Figure 6.1: Possible shift in case 0 ∈ ρ(A).

Remark 6.14. By ellipticity of Q it holds that D(A) = {u ∈ W m1 ,p (Rn , E); Dα Au ∈ Lp (Rn , E) (|α| = m2 )}. Consider Aδ (D) := A(D) + δA, δ > 0. Then all steps of the proof above apply unchanged and we end up with R-boundedness of

−1 ; λ ∈ Σπ−φ , ξ ∈ Rn \ {0} Q(ξ) + δ A λ + P (ξ) + (Q(ξ) + δ)A instead of (6.8). Let Aδ denote the corresponding Lp -realization of Aδ (D). Then the assertion of Proposition 6.13 remains true for Aδ and we have D(Aδ ) = {u ∈ W m1 ,p (Rn , E); Au ∈ W m2 ,p (Rn , E)} due to parameter-ellipticity of Q. In the sequel we consider Aδ (D), where δ > 0 in case m2 = 0, and assume it to be even, that is, Aδ (D) = pα D α + qα Dα A + δA, |α|=m1

|α|=m2

where m1 and m2 are even. In case m2 = 0 the shift δA can be dropped.


ñ = Let R problem

n j=1

65

Rj,(+) with Rj,(+) ∈ {R, R+ }. We consider a boundary value λu + Aδ (D)u B1 (D)u B2 (D)Au

(6.9)

= = =

f 0 0

˜ n, in R ˜ n, on ∂ R ˜ n. on ∂ R

˜ n is defined to be the union of all sets Here the boundary ∂ R R1,(+) × Rj−1,(+) × {0} × Rj+1,(+) × Rn,(+)

(j = 1, . . . , n; Rj,(+) = R+ ).

The pairwise intersections of these sets are empty, i.e. the Lebesgue null sets of vertices is neglected. In each direction j ∈ {1, . . . , n} such that Rj,(+) = R+ , the boundary operator B(D) := (B1 (D), B2 (D)) endows problem (6.9) with Dirichlet (i) Dj u|xj =0 = 0 ( = 0, 2, . . . , m1 − 2) and Dj Au|xj =0 = 0 ( = 0, 2, . . . , m2 − 2) or Neumann (ii) Dj u|xj =0 = 0 ( = 1, 3, . . . , m1 − 1) and Dj Au|xj =0 = 0 ( = 1, 3, . . . , m2 − 1) boundary conditions. Note that the types may be different in different directions. ˜ n , E), then B1 (D)u = 0 implies Remark 6.15. If u, Au ∈ W m1 ,p (R Dj Au|xj =0 = 0

( = 1, 3, . . . , m1 − 1)

due to closedness of A. In case m1 < m2 , it is therefore enough to assume Dj Au|xj =0 = 0

( = m1 − 2, m1 , . . . , m2 − 2)

in (i), respectively Dj Au|xj =0 = 0

( = m1 − 1, m1 + 1, . . . , m2 − 1)

in (ii). Accordingly, m2 ≤ m1 renders the boundary condition B2 (D)Au = 0 unnecessary. For the sake of simplicity we assume Rj,(+) = R+ for all j = 1, . . . , n which will ˜ n . The Lp (Kn , E)-realization of problem be indicated by writing Kn instead of R (6.9) is defined as D(Aδ,B ) := {u ∈ W m1 ,p (Kn , E); Au ∈ W m2 ,p (Kn , E), B(D)u = 0}, Aδ,B u := Aδ (D)u

(u ∈ D(Aδ,B )).

66


Proposition 6.16. Given the assumptions of Proposition 6.13, let A(D) be even. Then Aδ,B ∈ RS(Lp (Rn , E)) and φR Aδ,B ≤ ϕ0 . Moreover, for each φ > ϕ0 it holds that |α| 1− (6.10) R λ m1 Dα (λ + Aδ,B )−1 ; λ ∈ Σπ−φ , α ∈ Nn < ∞. 0 , 0 ≤ |α| ≤ m1 Proof. For simplicity of notation we consider the case n = 2 and boundary conditions of type (i) in direction x1 and of type (ii) in direction x2 . Following e.g. [PS93], we carry out a reflection procedure. Let f ∈ Lp (Kn , E) be arbitrary and λ ∈ ρ(−Aδ ). First considering the odd extension of f to R × R+ and afterwards its even extension to R2 , we end up with a function F fulfilling F (x1 , x2 ) = −F (−x1 , x2 ) as well as F (x1 , x2 ) = F (x1 , −x2 ) a.e. in R2 . Now we can apply Proposition 6.13 and Remark 6.14 which yield existence of a unique solution U ∈ {u ∈ W m1 ,p (Rn , E); Au ∈ W m2 ,p (Rn , E)} of λU + Aδ (D)U = F

in R2 .

Symmetry of A(D) now shows that V1 (x1 , x2 ) := −U (−x1 , x2 )

and

V2 (x1 , x2 ) := U (x1 , −x2 )

(x ∈ R2 )

define solutions, too. Thanks to uniqueness the equality V1 = U = V2 follows. Hence, Ux2 := U (·, x2 ) ∈ W m1 ,p (R, E) ⊂ C m1 −1 (R, E) for a.e. x2 ∈ R is odd which yields (0) = 0 ( = 0, 2, . . . , m1 − 2). Ux() 2 Accordingly, for a.e. x1 ∈ R we have that Ux1 is even, hence, Ux() (0) = 0 1

( = 1, 3, . . . , m1 − 1).

The same arguments applied to AU yield

and

(AU )() x2 (0) = 0

( = 0, 2, . . . , m2 − 2)

(AU )() x1 (0) = 0

( = 1, 3, . . . , m2 − 1).

Therefore, u := U |K2 solves λu + Aδ (D)u = f with boundary conditions (i) for j = 1 and (ii) for j = 2. For arbitrary n ∈ N and boundary conditions of Dirichlet or Neumann type the construction of the solution follows the same lines: we choose odd extensions in directions subject to case (i) and even extensions in directions subject to case (ii). On the other hand, let u be a solution of the boundary value problem (6.9). We extend u and f to U and F defined on Rn as described above. From symmetry of Aδ (D) we infer that U ∈ {u ∈ W m1 ,p (Rn , E); Au ∈ W m2 ,p (Rn , E)} solves λU + Aδ (D)U = F

in Rn .


67

Thus, uniqueness of U yields uniqueness of u. Altogether, (λ + Aδ,B )−1 = R(λ + Aδ )−1 E, where E defines the operator of extension to Rn as explained above and R defines the operator of restriction to Kn . Thus, Proposition 6.13 and Remark 6.14 prove the claim. Besides R-sectoriality, we are interested in R-boundedness of the H∞ -calculus of the operators introduced so far. Lemma 6.17. Let E be a Banach space and let A ∈ ΨRH∞ (E). Let further ∞ 0 ≤ ϑ, ζ < π such that ϑ + ζ < π − φR∞ A . Then the H0 -calculus of (z1 + z2 A) in E is uniformly R-bounded with respect to z1 ∈ Σϑ and z2 ∈ Σζ with angle R∞ R∞ φR∞ z1 +z2 A ≤ max{ϑ, ζ + φA }. More precisely, for each σ > max{ϑ, ζ + φA } it holds that R({h(z1 + z2 A); |h|σ∞ ≤ 1, (z1 , z2 ) ∈ Σϑ × Σζ }) < ∞. Proof. The proof is carried out in two steps separately. First we consider z1 ∈ Σϑ and z2 = 1, i.e. ζ = 0. Afterwards we consider z1 = 0 and z2 ∈ Σζ , i.e. ϑ = 0 (see Figure 6.2). Iteratively this proves the assertion. R∞ ∞ Let φ ∈ (max{ϑ, φR∞ A }, π − min{ϑ, φA }). Let σ > φ and h ∈ H0 (Σσ ) with σ R∞ |h|∞ ≤ 1 be arbitrary. Pick ψ ∈ (φ, min{σ, π − min{ϑ, φA }) and set Γ := (∞, 0]eiψ ∪ [0, ∞)e−iψ .

(6.11)

Then λ − z ∈ ρ(A) for all λ ∈ Γ and z ∈ Σϑ by our choice of ψ and ϑ. For arbitrary z ∈ Σϑ choose 0 < δ < dist(z, Γ). With ψ ∈ (φR∞ A , min{σ, π −ϑ}) set Γδ := (∞, δ]eiψ ∪ δei(ψ ,2π−ψ ) ∪ [δ, ∞)e−iψ . Then

h(z + A) = Γ

h(λ)(λ − z − A)−1 dλ =

Γδ +z

h(λ)(λ − z − A)−1 dλ

since both h and R(λ, z + A) are holomorphic in the area between Γ and Γδ + z. By means of the transformation λ → λ + z we end up with h(λ)(λ − z − A)−1 dλ = h(λ + z)(λ − A)−1 dλ h(z + A) = =

Γδ +z

Γδ

Γδ

hz (λ)(λ − A)−1 dλ = hz (A),

where hz (μ) := h(μ + z) fulfills hz ∈ E(Σσ ) for all z ∈ Σϑ . By Remark 4.13 this proves the claim. ∞ σ Now let φ ∈ (ζ + φR∞ A , π), σ > φ, and h ∈ H0 (Σσ ) such that |h|∞ ≤ 1. Pick ψ ∈ (φ, σ) and set (6.12)

Γ := (∞, 0]eiψ ∪ [0, ∞)e−iψ .

68


Then λ ∈ ρ(zA) and λz ∈ ρ(A) for all λ ∈ Γ and z ∈ Σζ by our choice of ψ and ζ. Pick ψ ∈ (φR∞ A , σ − ζ) and set

+arg(z))

∪ [0, ∞)ei(−ψ

Γ := (∞, 0]eiψ ∪ [0, ∞)e−iψ . For arbitrary z ∈ Σζ set Γz := (∞, 0]ei(ψ Then

h(zA) = Γ

h(λ)(λ − zA)−1 dλ =

Γz

+arg(z))

.

h(λ)(λ − zA)−1 dλ

since both h and R(λ, zA) are holomorphic in the area between Γ and Γz . By means of the transformation λ → λz we end up with h(zA) = h(λ)(λ − zA)−1 dλ = h(λz)(λ − A)−1 dλ

Γz

= Γ

Γ

hz (λ)(λ − A)−1 dλ = hz (A),

where hz (μ) := h(μz) fulfills hz ∈ E(Σσ ) for all z ∈ Σζ . In virtue of Remark 4.13 the claim follows. Lemma 6.18. Let A ∈ ΨRH∞ (E), ϑ, ζ, and σ as in Lemma 6.17. Let Γ1 and Γ2 be the paths defined in (6.11) and (6.12), respectively, and let h ∈ H0∞ (Σσ ) with |h|σ∞ ≤ 1 be arbitrary. Set h(λ)(λ − z − A)−1 dλ and H2 (z) := h(λ)(λ − zA)−1 dλ. H1 (z) := Γ1

Γ2

∞

∞

Then H1 ∈ H (Σϑ , L(E)) and H2 ∈ H (Σζ , L(E)). Proof. Let Γ0 denote any closed curve in Σϑ . Then H1 (z)dz = h(λ)(λ − z − A)−1 dλdz = h(λ) (λ − z − A)−1 dzdλ = 0 Γ0

Γ0

Γ

Γ

Γ0

since z → R(λ, z + A) is holomorphic in Σϑ . Here Fubini’s theorem can be applied for we have h(λ)(λ − z − A)−1 dλ < ∞ (z ∈ Γ0 ). L(E) Γ

Due to Morera’s theorem H1 is holomorphic in Σϑ . The assertion on H2 is proved similarly. As a final ingredient, we need Cauchy’s integral formula in the following extended version (cf. [KW04, Remark 9.3]).


69

(a)

(b)

Figure 6.2: Different paths of integration. The path defined by ψ is deformed to a path defined by means of ψ without any change of the integral. In Figure (a) a possible deformation is illustrated for the case that (z + A) is considered. Figure (b) shows a possible deformation for the case that (zA) is considered.

Lemma 6.19. Let ϑ < ψ < ϑ < π, let H ∈ H∞ (Σϑ , L(E)), and k ∈ N. Set

Γψ := (∞, 0]eiψ ∪ [0, ∞)e−iψ . Then z k H (k) (z) =

k! 2πi

Γψ

zk mh (μ)dμ (μ − z)k+1

(z ∈ Σϑ ).

70


Proof. Let z ∈ Σϑ be arbitrary. Pick 0 < r < |z| < R and set

Γψ ,r,R := [R, r]eiψ ∪ rei(ψ

,−ψ )

∪ [r, R]e−iψ ∪ Rei(−ψ

,ψ )

(see Figure 6.3). Then by Cauchy’s formula for closed rectifiable curves we have 1 1 H(μ)dμ H(z) = 2πi Γψ ,r,R μ − z and z k H (k) (z) =

k! 2πi

Γψ ,r,R

zk H(μ)dμ. (μ − z)k+1

k

1 z Since gz (μ) := (μ−z) k+1 fulfills gz ∈ L (Γψ ), for each z ∈ Σϑ the integral on the right-hand side exists. Recall boundedness of H on Σϑ .

Figure 6.3: The path of integration in the extended Cauchy integral formula.

We estimate the integrals over the two arcs in the representation formula for z k H (k) (z) and find |z|k zk H(μ)dμ ≤ C dμ k+1 k+1 rei(ψ ,−ψ ) (μ − z) rei(ψ ,−ψ ) |μ − z| L(E) r|z|k |z|k ≤C dμ ≤ C dμ → 0 (r → 0) k+1 (r − |z|)k+1 rei(ψ ,−ψ ) (r − |z|)


as well as

Rei(−ψ ,ψ )

≤C

n→∞

Γ 1 ψ , ,n

= lim

n→∞

R|z|k |z|k dμ ≤ C dμ → 0 k+1 (R − |z|) (R − |z|)k+1

(R → ∞).

zk H(μ)dμ (μ − z)k+1

n

= lim

n→∞

|z|k zk H(μ)dμ ≤ C dμ k+1 k+1 (μ − z) Rei(−ψ ,ψ ) |μ − z| L(E)

Rei(−ψ ,ψ )

This yields lim

71

Γ 1 ψ , ,n

zk H(μ)dμ − (μ − z)k+1

i(ψ ,−ψ )

ne

n

iψ

1 ]e [n, n

∪[n−1 ,n]e

−iψ

1e ∪n

zk H(μ)dμ = (μ − z)k+1

i(−ψ ,ψ )

Γψ



which proves the claim. With these results at hand, we return to the the investigation of the operator A(D) := P (D) + Q(D)A := pα D α + qα Dα A. |α|≤m1

|α|≤m2

Proposition 6.20. Let 1 < p < ∞, let E be a Banach space of class HT enjoying property (α), and let A ∈ ΨRH∞ (E). For homogeneous polynomials P and Q assume that (i) P is parameter-elliptic with angle ϕP ∈ [0, π), (ii) Q is parameter-elliptic with angle ϕQ ∈ [0, π), (iii) ϕP + ϕQ + φR∞ < π. A ≤ max{ϕP , ϕQ + φR∞ Then A ∈ RH∞ (Lp (Rn , E)) and φR∞ A }. A R∞ Proof. Let φ ∈ (max{ϕP , ϕQ +φR∞ A }, π −min{ϕP , ϕQ +φA }) and σ > φ. Define iψ −iψ where ψ ∈ (φ, min{σ, π − min{ϕP , ϕQ + φR∞ Γ := (∞, 0]e ∪ [0, ∞)e A }}) and ∞ consider an arbitrary h ∈ H0 (Σσ ) such that |h|σ∞ ≤ 1. We formally apply the Fourier transform to the Dunford integral representation of h(A) to the result h(A) = h(λ)(λ − A)−1 dλ Γ = F −1 h(λ)(λ − P (·) − Q(·)A)−1 dλF = F −1 (mh ◦ (P, Q))F. Γ

72


Here

mh (z1 , z2 ) := Γ

h(λ)(λ − z1 − z2 A)−1 dλ

((z1 , z2 ) ∈ Σϑ × Σζ ),

< π. Thus, Lemma 6.17 applies to where ϑ ≥ ϕP , ζ ≥ ϕQ such that ϑ + ζ + φR∞ A the result R({mh (z1 , z2 ); |h|σ∞ ≤ 1, (z1 , z2 ) ∈ Σϑ × Σζ }) < ∞. Now choose ψp ∈ (ϕP , ϑ) and ψq ∈ (ϕQ , ζ) to define Γp := (∞, 0]eiψp ∪ [0, ∞)e−iψp

and

Γq := (∞, 0]eiψq ∪ [0, ∞)e−iψq .

Due to Lemma 6.18 mh is holomorphic in each variable separately. Hence, we can apply Cauchy’s integral formula for polydiscs (see e.g. [Ran86, Theorem 1.3]) which again can be extended along the lines of Lemma 6.19. For α ∈ N20 this yields |α|! zα mh (μ1 , μ2 )dμ2 dμ1 z α ∂ α mh (z1 , z2 ) = 2 α +1 1 (2πi) Γp Γq (μ1 − z1 ) (μ2 − z2 )α2 +1 z1α1 z2α2 |α|! = mh (μ1 , μ2 )dμ2 dμ1 . 2 α +1 α2 +1 (2πi) Γp (μ1 − z1 ) 1 Γq (μ2 − z2 ) For ϑ ∈ (ϕP , ψp ) and ζ ∈ (ϕQ , ψq ) we have sup gz1 L1 (Γp ) ≤ C

and

z1 ∈Σϑ

sup gz2 L1 (Γq ) ≤ C,

z2 ∈Σζ

where we have set gz1 (μ1 ) :=

z1α1 (μ1 − z1 )α1 +1

and

gz2 (μ2 ) :=

z2α2 . (μ2 − z2 )α2 +1

Thus, Lemma 3.3 applies in iteration to the result R({z α ∂ α mh (z1 , z2 ); |h|σ∞ ≤ 1, (z1 , z2 ) ∈ Σϑ × Σζ }) < ∞. In particular, R({P (ξ)α1 Q(ξ)α2 (∂ α mh )(P (ξ), Q(ξ)); |h|σ∞ ≤ 1, ξ ∈ Rn \ {0}}) < ∞ follows due to parameter-ellipticity of P and Q. Employing Lemma 6.1 we see that the weighted derivatives ξ γ Dγ (mh ◦ (P, Q))(ξ) for 0 ≤ γ ≤ 1 are the sum of terms of the form

ξ γ ∂ α mh (P (ξ), Q(ξ)) · Dκ1 P (ξ) · · · Dκr P (ξ) · Dι1 Q (ξ) · · · Dιs Q (ξ),


73

where α1 + α2 ≤ |γ|, r = α1 and s = α2 . For β (1) := ri=1 κi and β (2) := si=1 ιi , (1) (2) = γ. By parameter-ellipticity of P and Q we thus can it holds that β + β estimate

(1) Dκ1 P (ξ) · · · Dκr P (ξ)| ≤ C|P (ξ)r | = C|P (ξ)α1 | |ξ β and |ξ β

(2)

Dι1 Q (ξ) · · · Dιs Q (ξ)| ≤ C|Q(ξ)s | = C|Q(ξ)α2 |.

Hence, the contraction principle of Kahane yields R({ξ γ Dγ (mh ◦ (P, Q))(ξ); |h|σ∞ ≤ 1, ξ ∈ Rn \ {0}, 0 ≤ γ ≤ 1}) < ∞ which shows mh ◦ (P, Q) to be a Fourier multiplier. Thus, A ∈ ΨRH∞ (Lp (Rn , E)) p n ≤ max{ϕP , ϕQ + φR∞ with φR∞ A }. Because of A ∈ RS(L (R , E)) by ProposiA ∞ p n tion 6.13 we have A ∈ RH (L (R , E)). Proposition 6.21. Given the assumptions of Proposition 6.20, let δ > 0 and let Aδ (D) be even. Then the assertion of Proposition 6.20 carries over to Aδ,B . Proof. Observe that Proposition 6.20 also applies to Aδ for δ ≥ 0. Hence, the assertion follows immediately from (λ + Aδ,B )−1 = R(λ + Aδ )−1 E. Recall that E and R define bounded operators. Hence, they commute with the integral sign in the Dunford integral representation. Remark 6.22. Note that related results on sectoriality and a bounded H∞ -calculus of Aδ can still be deduced if E does not enjoy property (α).

7 Fourier series approach to operator-dependent problems This chapter can be seen as the counterpart of the previous chapter. This time operator-dependent partial differential equations in the cube (0, 2π)n are treated. Here generalized periodic and mixed Dirichlet-Neumann boundary conditions are imposed. This chapter further completes the abstract background for the upcoming treatment of cylindrical boundary value problems.

7.1 Preliminaries In view of Theorems 3.19 and 3.24, respectively in view of our intermediate condition (3.12), the following lemma provides useful formulas for discrete derivatives. It can be seen as an analogue to Lemma 6.1. In the discrete case, however, the proof is more tedious since one has to keep track of each shift that comes from one of the single derivations. To do so, we will find it convenient to define ω∗j :=

r

ωl

l=j+1

for W ∈ Zα , r = rW , and W = (ω 1 , . . . , ω r ). Here W and Zα are defined as in (6.1). Also recall the discrete differential operator Δα from (3.5). Lemma 7.1. a) [Leibniz rule] Given m ∈ N, let (Tk )k∈Zn denote a sequence of operators X → Y and (Sk )k∈Zn a sequence of operators Y → Z, such that the composition Sk(1) Tk(2) : X → Z is well-defined for all k(1) , k(2) ∈ Zn . For α ∈ Nn 0 such that |α| ≤ m we then have α (Δα−β S)k−β (Δβ T )k Δ (ST )k = β β≤α

α

(k ∈ Zn ).

n b) Let

((Sj )k )k∈Z for j = 1, . . . , r denote sequences of operators Xj−1 → Xj such that rj=1 (Sj )k(j) : X0 → Xr is well-defined for all k(1) , . . . , k(r) ∈ Zn . Then we have

Δ ei

r j=1

Sj

k

=

r l−1 l=1

j=1

Sj

k−ei

(Δei Sl )k

r

Sj

k

.

j=l+1


76

7 Fourier series approach to operator-dependent problems

c) Given m ∈ N, let (Sk )k∈Zn denote a sequence of operators X → Y such that the inverse (S −1 )k := (Sk )−1 exists for all k ∈ Zn . Then, for α ∈ Nn 0 such that |α| ≤ m we have Δα (S −1 )k =

(−1)rW (S −1 )k−α

W∈Zα

rW

j

(Δω S)S −1

j=1

j

k−ω∗

(k ∈ Zn ).

Proof. a) We proof the assertion by induction on n. First let n = 1, i.e. α = a ∈ N, the case a = 0 being obvious. By definition for Δ = Δ1 we have Δ(ST )k = (ST )k − (ST )k−1 = Sk−1 (ΔT )k + (ΔS)k Tk and by induction on a we obtain a−1 a Δ (ST )k = Δ (Δa−1−b S)k−b (Δb T )k b b≤a−1 a − 1 a−1−b = S)k−b−1 (Δb+1 T )k + (Δa−b S)k−b (Δb T )k . (Δ b b≤a−1 We split the sum to the result a−1 a−1 a−1−b b+1 S)k−b−1 (Δ T )k + (Δ (Δa−b S)k−b (Δb T )k . b b b≤a−1 b≤a−1 The first sum equals a−1 (Δa−b S)k−b (Δb T )k b − 1 0 N and ζk2 = M + 9

j0

10−j + δ,

j=1

where M = ζk2 and 0 < δ < ε. Hence, if we pick k1 := M + 1, we obtain the inequality ζk2 < k1 < ε + ζk2 ⇐⇒ 0 < k1 − ζk2 < ε. In what follows the assumption that (λ + μA)−1 exists for λ, μ ∈ C is meant to imply both (λ + μA)−1 ∈ L(E) and (λ + μA)−1 (E) = D(A). Proposition 7.8. Let A be a closed operator in a Banach space E of class HT and let P, Q : Zn → C denote elliptic polynomials. Let N define an arbitrary polynomial

7.1 Preliminaries

81

−1 subject to deg N ≤ deg P and let P (k) + Q(k)A ∈ L(E) exist for all k ∈ n Z \ {0}. If the set

−1 P (k) P (k) + Q(k)A ; k ∈ Zn is R-bounded, then for each 1 < p < ∞

−1 M : Zn → L(E), k → N (k) P (k) + Q(k)A defines an Lp -multiplier. Proof. Lemma 7.1 yields |k||γ| Δγ M (k) = γ

−1 (−1)rW |k||γ−β| (Δγ−β N )(k − β) P (k − β) + Q(k − β)A β W∈Z β≤γ β

·

rW

|k|

−1 . Δωj P (k − ωj∗ ) + Δωj Q(k − ωj∗ )A P (k − ωj∗ ) + Q(k − ωj∗ )A

|ωj |

j=1

By Remark 7.6 we know that deg(Δγ−β N ) ≤ deg N − |γ − β|. This and the ellipticity of P imply |k||γ−β| |Δγ−β N (k)| ≤ C|P (k)| for k ∈ Zn \ G with a finite set G ⊂ Zn . By Kahane’s contraction principle we obtain the R-boundedness of

−1 ; k ∈ Zn \ G . |k||γ−β| Δγ−β N (k − β) P (k − β) + Q(k − β)A Since

−1

−1 = idE − P (k) P (k) + Q(k)A , Q(k)A P (k) + Q(k)A

in the same way the R-boundedness of

−1 ; k ∈ Zn \ G |k||ωj | Δωj Q(k − ωj∗ )A P (k − ωj∗ ) + Q(k − ωj∗ )A follows from the ellipticity of Q. Now the assertion follows from the fact that G is finite, from Lemma 3.2, and from Theorem 3.19. Proposition 7.8 is closely related to the concept of 1-regularity of complexvalued sequences considered in [KL04] for the one dimensional case n = 1. There the notion of 1-regularity as introduced by Pr¨ uss in [Pr¨ u93] for functions of a complex variable (cf. Definition 5.7) is transferred to sequences (ak )k∈N ⊂ C \ {0} by assuming k(ak+1 − ak ) ak k∈Z to be bounded. It turns out that 1-regularity is the right concept to investigate multipliers of type M (k) = ak (bk − A)−1 with (ak )k∈Z , (bk )k∈Z ⊂ C. Indeed, by

82


1-regularity, R-boundedness of discrete derivatives of an operator family can be deduced from R-boundedness of the family itself. This allows for an easy application of the multiplier theorem due to Arendt and Bu. In fact, it is shown in [KLP09, Proposition 5.3] that R-boundedness of {ak (bk − A)−1 ; k ∈ Z} implies that M defined as above defines a multiplier for 1 < p < ∞, where both (ak )k∈Z and (bk )k∈Z are 1-regular sequences such that (bk /ak )k∈Z is bounded and (bk )k∈Z ⊂ ρ(A). If Q(k) = 0 for all k ∈ Zn , we may write −1 N (k) P (k) M (k) = (k ∈ Zn ). +A Q(k) Q(k) Hence, for n = 1 we enter the framework of 1-regularity, i.e. M (k) = ak (bk − A)−1 with (ak )k∈Z , (bk )k∈Z ⊂ C defined by ak :=

N (k) Q(k)

and

bk :=

P (k) . Q(k)

In the subsequent lines, we will give a generalization of this concept to arbitrary n and briefly indicate the connection to the results above. Definition 7.9. We call a pair of sequences (ak , bk )k∈Zn ⊂ C2 1-regular if for all 0 ≤ γ ≤ 1 there exist a finite set K ⊂ Zn and a constant C > 0 such that (7.1)

|k γ | max{|(Δγ a)k |, |(Δγ b)k |} ≤ C|bk |

(k ∈ Zn \ K).

We say the pair (ak , bk )k∈Zn is strictly 1-regular if |k γ | can be replaced by |k||γ| in (7.1). A sequence (ak )k∈Zn is called (strictly) 1-regular if (ak , ak )k∈Zn has this property. Remark 7.10. a) In the case n = 1 a sequence (ak )k∈Z ⊂ C \ {0} is 1-regular k(ak+1 −ak ) in Z in the sense of Definition 7.9 if and only if the sequence ak k∈Z is bounded. Hence, our definition extends the one from [KL04] for a sequence (ak )k∈Z . b) With γ = 0 the definition especially requests |ak | ≤ C|bk | for k ∈ Zn \ K. c) Strict 1-regularity implies 1-regularity. If n = 1 both concepts are equivalent. (Strict) 1-regularity by definition allows to apply the contraction principle of Kahane to deduce the sufficient conditions of the multiplier theorem for resolvent type multipliers from R-boundedness of the range of the multiplier itself. It is therefore not surprising that the ellipticity condition used in Proposition 7.8 leads to strictly 1-regular sequences. Lemma 7.11. Subject to the assumptions of Proposition 7.8, let Q(k) = 0 for N (k) P (k) n is strictly 1-regular. k ∈ Z . Then the pair Q(k) , Q(k) k∈Zn

Proof. We have to show the existence of C > 0 and a finite set K ⊂ Zn such that N P P |k||γ| max Δγ (k), Δγ (k) ≤ C (k) (k ∈ Zn \ K). Q Q Q

7.1 Preliminaries

83

Due to Lemma 7.1 for any polynomial M we have M (k) = Q γ

|k||γ| Δγ

β≤γ

β

(−1)W |k||γ−β|

W∈Zβ

rW Δγ−β M Δω j Q |k||ωj | (k − β) (k − ωj∗ ). Q Q j=1

Ellipticity of P now implies that there exists a finite set G ⊂ Zn such that P (k) = 0 for all k ∈ Zn \ G. Therefore it suffices to show the existence of C > 0 and a finite set K ⊃ G such that |k|

|γ−β| Δ

γ−β

Q

M

(k − β)

rW

|k|

j=1

|ωj | Δ

ωj

Q

Q

(k −

Q ωj∗ ) (k) P

≤ C.

If deg M ≤ deg P the assertion follows from ellipticity of Q(· − β)

rW

Q(· − ωj∗ )P,

j=1

which is given since P and Q are assumed to be elliptic. With the aid of Lemma 7.1 we deduce the following more abstract version of Proposition 7.8. Here (ak )k∈Zn and (bk )k∈Zn no longer have to be given as quotients of discrete polynomials. Proposition 7.12. Let (ak , bk )k∈Zn be strictly 1-regular, let bk ∈ ρ(A) for all k ∈ Zn , and let R({bk (bk − A)−1 ; k ∈ Zn \ G}) < ∞ for some finite subset G ⊂ Zn . Then M (k) := ak (bk − A)−1 defines a Fourier multiplier. Proof. By Lemma 7.1 we have |k|

|γ|

−1 γ Δ M (k) = (−1)rW |k||γ−β| (Δγ−β a)k−β bk−β − A β W∈Z β≤γ γ

β

·

rW

−1 |k||ωj | (Δωj b)k−ωj∗ bk−ωj∗ − A .

j=1

Now the claim follows at once from the contraction principle of Kahane.

84


7.2 ν-periodic problems in cubical domains Let E be a Banach space of class HT and A be a closed operator in E. With n ∈ N and ν ∈ Cn we consider the boundary value problem in Qn given by (7.2) A(D)u = f (x ∈ Qn ), (j = 1, . . . , n; |β| < m1 ), (Dβ u)|xj =2π − e2πνj (Dβ u)|xj =0 = 0 (j = 1, . . . , n; |β| < m2 ). (Dβ Au)|xj =2π − e2πνj (Dβ Au)|xj =0 = 0 In view of the boundary conditions, we refer to the boundary value problem (7.2) as ν-periodic. Note that νj = 0 corresponds to periodic boundary conditions and νj = 2i to antiperiodic boundary conditions with respect to the j-th coordinate direction. Here pα D α + qα D α A A(D) := P (D) + Q(D)A := |α|≤m1

|α|≤m2

with m1 , m2 ∈ N and pα , qα ∈ C. Note that (D u)|xj =2π = e2πνj (Dβ u)|xj =0 implies (Dβ Au)|xj =2π = e2πνj (Dβ Au)|xj =0 by closedness of A. Thus m2 ≤ m1 renders the second boundary condition unnecessary. With m := max{m1 , m2 } α α in what follows we frequently write A(D) = |α|≤m (pα D + qα D A), where additional coefficients are understood that we define to be equal to zero. Besides the complex polynomials P (z) := |α|≤m1 pα z α and Q(z) := |α|≤m2 qα z α for z ∈ Cn . Definition 7.13. A solution of the ν-periodic boundary value problem (7.2) is a m1 ,p m2 ,p (Qn , E) such that Au ∈ Wν,per (Qn , E) and A(D)u(x) = f (x) function u ∈ Wν,per for almost every x ∈ Qn . Remark 7.14. Since the trace operator with respect to one direction and tangential derivation commute, the ν-periodic boundary conditions as imposed in (7.2) are equivalent to β

(Dj u)|xj =2π − e2πνj (Dj u)|xj =0 (Dj Au)|xj =2π − e2πνj (Dj Au)|xj =0

= =

0 0

(j = 1, . . . , n, 0 ≤ < m1 ), (j = 1, . . . , n, 0 ≤ < m2 ).

Again the assumption that (λ + μA)−1 exists for λ, μ ∈ C is meant to imply both (λ + μA)−1 ∈ L(E) and (λ + μA)−1 (E) = D(A). Theorem 7.15. Let 1 < p < ∞ and assume P and Q to be elliptic. Then the following assertions are equivalent: (i) For each f ∈ Lp (Qn , E) there exists a unique solution of (7.2).

−1 (ii) P (k − iν) + Q(k − iν)A exists for k ∈ Zn and

−1 Mα (k) := kα P (k − iν) + Q(k − iν)A defines a Fourier multiplier for every |α| = m1 .

7.2 ν-periodic problems in cubical domains

85

−1 (iii) P (k − iν) + Q(k − iν)A exists for k ∈ Zn and for all |α| = m1 there exists a finite subset G ⊂ Zn such that the sets {Mα (k); k ∈ Zn \ G} are R-bounded.

−1 (iv) P (k − iν) + Q(k − iν)A exists for k ∈ Zn and there exists a finite subset n G ⊂ Z such that the set

−1 P (k − iν) P (k − iν) + Q(k − iν)A ; k ∈ Zn \ G is R-bounded. Proof. (i) ⇒ (ii): Let f ∈ Lp (Qn , E) be arbitrary and let u be a solution of (7.2) with right-hand side eν· f . Then e−ν· A(D)u = f . In order to compute the Fourier coefficients, we first remark that

−ν· e P (D)u ˆ(k) = P (k − iν)(e−ν· u)ˆ(k) and

e−ν· Q(D)Au ˆ(k) = Q(k − iν)(e−ν· Au)ˆ(k) = Q(k − iν)A(e−ν· u)ˆ(k)

by Lemma 2.16. Writing kν := k − iν for short, we obtain

(7.3) P (kν ) + Q(kν )A e−ν· u ˆ(k) = fˆ(k).

ik· For arbitrary y ∈ E and k ∈ Zn the choice P (kν ) + Q(kν )A to

f := e y shows be surjective. Let z ∈ D(A) such that P (kν ) + Q(kν )A z = 0. For fixed k ∈ Zn set v := eik· z and u := eν· v. Then

P (kν ) e−ν· u ˆ(k) + Q(kν )A e−ν· u ˆ(k) = 0. As (e−ν· u)ˆ() = 0 for all = k, this gives A(D)u = 0, hence v = u = 0 and z = 0. Altogether we have shown bijectivity of P (kν ) + Q(kν )A for k ∈ Zn . The

−1 closedness of A yields P (kν ) + Q(kν )A ∈ L(E). For f ∈ Lp (Qn , E) let u be a solution of (7.2) with right hand side eν· f and m1 ,p m2 ,p (Qn , E), Av ∈ Wper (Qn , E), and (7.3) implies v := e−ν· u. Then v ∈ Wper

−1 vˆ(k) = P (kν ) + Q(kν )A fˆ(k). This shows

−1 M0 : Zn → L(Lp (Qn , E)); k → P (kν ) + Q(kν )A m1 ,p to be a Fourier multiplier such that TM0 maps Lp (Qn , E) into Wper (Qn , E). Due to Lemma 3.11 we have that Mα is a Fourier multiplier for all |α| = m1 . (ii) ⇒ (iii): This follows from Proposition 3.9.

86


(iii) ⇔ (iv): Let ej , j = 1, . . . , n denote the j-th unit vector. For k = 0 on the one hand it holds that n

−1

−1 P (kν ) = km1 ej P (kν ) + Q(kν )A . P (kν ) P (kν ) + Q(kν )A n km1 ej j=1 j=1

On the other hand for |α| = m1 and k ∈ Zn such that P (kν ) = 0 we have

−1

−1 kα = kα P (kν ) + Q(kν )A . P (kν ) P (kν ) + Q(kν )A P (kν ) m 1 ej there exist c, C > 0 such that the estimate By ellipticity of k → n j=1 k c|k α | ≤ |P (kν )| ≤ C|

n

km1 ej |

j=1

is valid for k ∈ Z \ G with suitably chosen finite G ⊂ Zn . Recall that m1 has to be even in case n > 1 due to ellipticity of P . Lemma 3.2 now shows the claim. (iii) ⇒ (i): Since (iii) ⇔ (iv), by Proposition 7.8 it follows that Mα for |α| = m1 as well as P (· − iν)M0 are Fourier multipliers. For arbitrary f ∈ Lp (Qn , E) we m1 ,p (Qn , E). As therefore get v := TM0 (e−ν· f ) ∈ Wper

−1

−1 (7.4) Q(kν )A P (kν ) + Q(kν )A = idE − P (kν ) P (kν ) + Q(kν )A , n

Q(· − iν)AM0 is a Fourier multiplier, too. By ellipticity of Q and Lemma 3.2 again

−1 , |α| ≤ m2 . the same holds for kα A P (kν ) + Q(kν )A ν· ν· −ν· By construction u := e v = e TM0 e f solves (7.2) and Lemma 3.11 yields m1 ,p m2 ,p (Qn , E) and Au ∈ Wν,per (Qn , E). Finally, uniqueness of u follows u ∈ Wν,per immediately from the uniqueness of the representation as a Fourier series. Theorem 7.15 enables us to treat Dirichlet-Neumann type boundary conditions ˜ n := (0, π)n for even operators. More precisely, for all |α| ≤ m we assume on Q α α pα = qα = 0 or α ∈ 2Nn 0 in the representation A(D) = |α|≤m (pα D + qα D A). In particular, m1 and m2 are even. Given an even operator A(D) we consider the boundary value problem (7.5)

A(D)u B1 (D)u B2 (D)Au

= = =

f 0 0

ñ, in Q ñ, on ∂ Q ñ. on ∂ Q

˜ n is defined to be the union of all sets Here the boundary ∂ Q ˜ n−j ˜ j−1 × {0, 2π} × Q Q

(j = 1, . . . , n).

The pairwise intersections of these sets are empty, i.e. the Lebesgue null sets of vertices are neglected. In each direction j ∈ {1, . . . , n}, the boundary operator B(D) := (B1 (D), B2 (D)) represents one of the following boundary conditions:

7.2 ν-periodic problems in cubical domains

87

(i) Dj u|xj =0 = Dj u|xj =π = 0 ( = 0, 2, . . . , m1 − 2) and Dj Au|xj =0 = Dj Au|xj =π = 0 ( = 0, 2, . . . , m2 − 2), (ii) Dj u|xj =0 = Dj u|xj =π = 0 ( = 1, 3, . . . , m1 − 1) and Dj Au|xj =0 = Dj Au|xj =π = 0 ( = 1, 3, . . . , m2 − 1), (iii) Dj u|xj =0 = Dj+1 u|xj =π = 0 ( = 0, 2, . . . , m1 − 2) and Dj Au|xj =0 = Dj+1 Au|xj =π = 0 ( = 0, 2, . . . , m2 − 2), (iv) Dj+1 u|xj =0 = Dj u|xj =π = 0 ( = 0, 2, . . . , m1 − 2) and Dj+1 Au|xj =0 = Dj Au|xj =π = 0 ( = 0, 2, . . . , m2 − 2). Note that for a second-order operator, (i) is of Dirichlet type, (ii) is of Neumann type, and (iii) and (iv) are of mixed type. For instance, in case (iii) we have u|xj =0 = 0 and Dj u|xj =π = 0. In general, the types may be different in different directions. Therefore, we refer to these boundary conditions as conditions of Dirichlet-Neumann type. Also recall that the boundary operator might B1 render some or even all boundary conditions represented by B2 unnecessary. This depends on m1 and m2 (see Remark 6.15). Proposition 7.16. Let A(D) be even, let P and Q be elliptic, and let the boundary conditions in each coordinate direction be of Dirichlet-Neumann type as explained above. Define ν ∈ Cn by setting νj := 0 in cases (i) and (ii) and νj := i/2 in cases (iii) and (iv). If for this ν one of the equivalent conditions of Theo˜ n , E) there exists a unique solution rem 7.15 is fulfilled, then for each f ∈ Lp (Q m1 ,p ˜ m2 ,p ˜ (Qn , E) with Au ∈ W (Qn , E) of problem (7.5). u∈W Proof. Following an idea from [AB02], the solution is constructed by a suitable even or odd extension of the right-hand side from (0, π)n to (0, 2π)n . For simplicity of notation we consider the case n = 2 and boundary conditions of type (ii) in direction x1 and of type (iii) in direction x2 . By definition this leads to ν1 = 0 and ν2 = 2i . ˜ 2 , E) be arbitrary. First considering the even extension of f to Let f ∈ Lp (Q the rectangle (0, 2π) × (0, π) and afterwards its odd extension to (0, 2π) × (0, 2π) we end up with a function F on Q2 fulfilling F (x1 , x2 ) = F (2π − x1 , x2 ) and F (x1 , x2 ) = −F (x1 , 2π − x2 ) a.e. in Q2 . Now we can apply Theorem 7.15 with ν = (ν1 , ν2 )T as above. This yields a unique solution U of

(7.6)

A(D)U D1 U |x1 =0 −D2 U |x2 =0 D1 AU |x1 =0 −D2 AU |x2 =0

= = = = =

F D1 U |x1 =2π D2 U |x2 =2π D1 AU |x1 =2π D2 AU |x2 =2π

in Q2 , ( = 0, . . . , m1 − 1), ( = 0, . . . , m1 − 1), ( = 0, . . . , m2 − 1), ( = 0, . . . , m2 − 1).

88


Symmetry of A(D) now shows that V1 (x1 , x2 ) := U (2π − x1 , x2 )

and

V2 (x1 , x2 ) := −U (x1 , 2π − x2 )

(x ∈ Q2 )

define solutions of the boundary value problem (7.6) as well. Due to uniqueness V1 = U = V2 and Ux2 := U (·, x2 ) ∈ W m1 ,p ((0, 2π), E) ⊂ C m1 −1 ((0, 2π), E) is even for a.e. x2 ∈ (0, 2π). Together with periodicity of Ux2 due to (7.6) this yields (0) = Ux() (π) = 0 Ux() 2 2

( = 1, 3, . . . , m1 − 1).

Accordingly, for a.e. x1 ∈ (0, 2π) we have that Ux1 is odd, and antiperiodicity due to (7.6) gives (0) = Ux(+1) (π) = 0 Ux() 1 1

( = 0, 2, . . . , m1 − 2).

The same arguments applied to AU yield () (AU )() x2 (0) = (AU )x2 (π) = 0

and

(+1) (π) = 0 (AU )() x1 (0) = (AU )x1

( = 1, 3, . . . , m2 − 1) ( = 0, 2, . . . , m2 − 2).

Therefore, u := U |(0,π)2 solves A(D)u = f with boundary conditions (ii) for j = 1 and (iii) for j = 2. For arbitrary n ∈ N and arbitrary boundary conditions of Dirichlet-Neumann type the construction of the solution follows the same lines. Here we choose even extensions in the cases (ii) and (iv) and odd extensions in the cases (i) and (iii). On the other hand, let u be a solution of A(D)u = f satisfying boundary conditions of Dirichlet-Neumann type. We extend u and f to U and F on (0, 2π)n m1 ,p m2 ,p as described above. Then U ∈ Wν,per ((0, 2π)n , E), AU ∈ Wν,per ((0, 2π)n , E) and due to symmetry of A(D) we see that U solves (7.2) with right-hand side F and ν defined as above. Thus, uniqueness of U yields uniqueness of u and the proof is complete. We close this section with a discussion of the previous results for the case of homogeneous polynomials P and Q. Remark 7.17. a) Consider ν = 0 in Theorem 7.15 and homogeneous polynomials P and Q. First assume both polynomials to be non-constant. Then M0 (0) is no longer well-defined. However, we can overcome this problem quickly by changing the underlying space from Lp (Qn , E) to Lp(0) (Qn , E) := f ∈ Lp (Qn , E); f (x)dx = 0 . Qn

Now assume only P to be non-constant. Without loss of generality let Q ≡ 1 which yields M0 (0) = A−1 . Hence, 0 ∈ ρ(A) is necessary for unique solvability in Lp (Rn , E). If this assumption is violated, existence and uniqueness of a solution


89

can still be deduced with the aid of Theorem 7.15, provided that N (A) = {0} and that the right-hand side f belongs to Lp(0) (Qn , E). b) The observations in part a) affect Proposition 7.16 as soon as boundary conditions of type (i) or (ii) are imposed in every coordinate direction since then results from Theorem 7.15 on pure periodic boundary conditions are employed. Here it is important to distinguish. First consider pure Neumann boundary conditions on the whole of ∂Qn . In that case, the extension of the right-hand side f is carried out by means of even reflections throughout all coordinate directions. Given non-constant polynomials P and Q, the operator M (0) is not well-defined and the underlying space has to be ˜ n , E). If Q ≡ 1 a change of the underlying space is unnecessary changed to Lp(0) (Q provided 0 ∈ ρ(A). However, if Dirichlet conditions are imposed with respect to at least one coordinate direction, the extension of the right-hand side in that direction is carried out by means of an odd reflection. As a consequence, the resulting right-hand side F fulfills F ∈ Lp(0) (Qn , E) automatically. By virtue of part a) the operator M (0) does no longer affect the outcome of Theorem 7.15. Consequently, if existence and uniqueness of a solution U of the pure periodic problem subject to U ∈ Lp(0) (Qn , E) can be deduced, this in turn shows existence and uniqueness of the solution u of the Dirichlet-Neumann problem in Lp (Qn , E). Remark 7.18. In case n = 1 ellipticity does no longer force m1 and m2 to be even. Therefore, the same results as in Proposition 7.16 can be achieved if A(D) is odd in the obvious sense, e.g. A(Dt ) := Dt3 + Dt + Dt A. Moreover, we can as well treat a combination of ν-periodic and Dirichlet-Neumann-type boundary conditions on ˜ n2 as long as the structures of P and Q allow for an application of Qn1 × Q ˜ n2 . Proposition 7.16 with respect to Q

7.3 R-sectoriality and RH∞ -calculus Let E be a Banach space of class HT and A be a closed operator in E. We expand the boundary value problems as considered in the previous section by an additional complex parameter λ. In the sequel we allow for a shift δA, where δ ≥ 0 (cf. Remark 6.14). More precisely, we treat the parameter-dependent ν-periodic boundary value problem given by (7.7) (x ∈ Qn ), λu + Aδ (D)u = f (j = 1, . . . , n; |β| < m1 ), (Dβ u)|xj =2π − e2πνj (Dβ u)|xj =0 = 0 (j = 1, . . . , n; |β| < m2 ). (Dβ Au)|xj =2π − e2πνj (Dβ Au)|xj =0 = 0 Here Aδ (D) is defined as in Remark 6.14.

90


We further define the Lp (Qn , E)-realization of the boundary value problem (7.7) to be m1 ,p m2 ,p (Qn , E); Au ∈ Wν,per (Qn , E) , D(Aδ ) := u ∈ Wν,per Aδ u := Aδ (D)u

(u ∈ D(Aδ )).

If δ = 0 we frequently write A instead of Aδ . In order to emphasize the ν under consideration, we also employ the notation Aδ,ν and Aν , respectively. Remark 7.19. If m2 ≤ m1 it holds that m1 ,p m2 ,p (Qn , E) ∩ Wν,per (Qn , D(A)). D(Aδ ) = Wν,per

In the sequel we restrict ourselves to ν = 0 or purely imaginary components of ν. In that case, it is sufficient to consider ν ∈ i(−1, 1)n . Note that periodic as well as antiperiodic boundary conditions are still captured. The reason for this constraint lies in the fact that kν ∈ Rn has real components only. Hence, (parameter-)ellipticity conditions on P and Q need not to be extended to P = P (z) and Q = Q(z) for z out of a whole open sector of the complex plane. Furthermore, we agree to consider homogeneous parameter-elliptic polynomials P and Q, that is, P = P # and Q = Q# , for elliptic estimates involving lower order terms of polynomials are valid up to k ∈ G in finite sets G ⊂ Zn only. In view of (R-)sectoriality, however, the additional parameter λ leads to uncountably many multiplier functions Mλ which in turn come along with the challenge to keep control of uncountably many sets Gλ . Proposition 7.20. Let 1 < p < ∞, let E be a Banach space of class HT enjoying property (α), let ν ∈ i(−1, 1)n , and let A ∈ ΨRS(E). For homogeneous polynomials P and Q assume that (i) P is parameter-elliptic with angle ϕP ∈ [0, π), (ii) Q is parameter-elliptic with angle ϕQ ∈ [0, π), (iii) ϕP + ϕQ + φR A < π. p Set ϕ0 := max{ϕP , ϕQ + φR A }. Then Aδ ∈ ΨRS(L (Qn , E)) for each δ > 0, R φAδ ≤ ϕ0 and for each φ > ϕ0 it holds that

(7.8)

R

λ

|α| 1

1− m

Dα (λ + Aδ )−1 ; λ ∈ Σπ−φ , α ∈ Nn 0 , 0 ≤ |α| ≤ m1

< ∞.

In case ν = 0 moreover Aδ ∈ RS(Lp (Qn , E)) and 0 ∈ ρ(Aδ ) for δ ≥ 0. In case A ∈ RS(E) moreover Aδ ∈ RS(Lp (Qn , E)) for δ > 0. In case ν = 0 and Q ≡ c, c = 0 subject to condition (ii) and A ∈ RS(E) moreover Aδ ∈ RS(Lp (Qn , E)) for δ ≥ 0. In that case, 0 ∈ ρ(A) implies 0 ∈ ρ(Aδ ) for δ ≥ 0.


91

Proof. We consider the following formal representation (λ + Aδ )−1 = eν· TMλ e−ν· of the resolvent of Aδ , where TMλ denotes the operator associated with

−1 . Mλ (k) := λ + P (k − iν) + Qδ (k − iν)A) More generally, for α ∈ Nn 0 , the Leibniz rule shows

α −1 D (λ + Aδ ) f = Dα eν· TMλ e−ν· f = gβ (ν)eν· TM β e−ν· f , β≤α

λ

where gβ is a polynomial depending on β. Here TM β denotes the operator associλ ated with

−1 Mλβ (k) := kβ λ + P (k − iν) + Qδ (k − iν)A where β ≤ α. In case ν = 0 we simply have D α (λ + Aδ )−1 = TMλα . First consider ν = 0 and δ ≥ 0. Given α ∈ Nn 0 with 0 ≤ |α| ≤ m1 let 0 ≤ β ≤ α, 0 ≤ γ ≤ 1, and φ > ϕ0 . To prove (7.8), we apply Lemma 7.1 in order to calculate kγ Δγ Mλβ (k). In what follows we write kν := k − iν and Qδ (kν ) := Q(kν ) + δ for short. Note that kν = 0 for all k ∈ Zn . As in the proof of Theorem 7.15 it suffices to show that |α|

−1 1− (7.9) ; λ ∈ Σπ−φ , k ∈ Zn λ m1 kω Δω N (k) λ + P (kν ) + Qδ (kν )A for N (k) := kβ and arbitrary ω ≤ γ,

−1 (7.10) kω Δω P (kν ) λ + P (kν ) + Qδ (kν )A ; λ ∈ Σπ−φ , k ∈ Zn for 0 < ω ≤ γ, and

−1 ; λ ∈ Σπ−φ , k ∈ Zn (7.11) kω Δω Qδ (kν )A λ + P (kν ) + Qδ (kν )A for 0 < ω ≤ γ are R-bounded. Due to our assumptions on P and Q, the pseudo-R-sectoriality of A, and Lemma 6.11 −1 λ + P (kν ) λ + P (kν ) ; λ ∈ Σπ−φ , k ∈ Zn +A Qδ (kν ) Qδ (kν ) is R-bounded. Recall that kν ∈ Rn . Hence, we can employ our previous results on the continuous extensions of the discrete polynomials under consideration. Consequently, we get R-boundedness of

−1 ; λ ∈ Σπ−φ , k ∈ Zn (7.12) λ + P (kν ) λ + P (kν ) + Qδ (kν )A

92


and by means of the resolvent equation R-boundedness of

−1 (7.13) Qδ (kν )A λ + P (kν ) + Qδ (kν )A ; λ ∈ Σπ−φ , k ∈ Zn . To prove (7.9) and (7.10) we cannot copy the arguments of Proposition 6.13 directly since the continuous extensions of Δω N respectively Δω P do no longer define homogeneous polynomials. However, since ν is supposed to have at least one non-zero component, there exists ε > 0 such that |kν | > ε holds true for all k ∈ Zn . In particular (7.12) remains valid for λ = 0. Moreover, the ideas in part c) of Lemma 6.5 yield the existence of C > 0 such that |k||ω| |Δω N (k)| λ ≤ C, |λ + P (kν )|

|α| 1

1− m

|k||ω| |Δω N (k)| ≤ C, |λ + P (kν )|

and

|k||ω| |Δω P (kν )| ≤C |λ + P (kν )|

for all k ∈ Zn and all λ ∈ Σπ−φ . Note that Lemma 6.11 applies to λ taken from the closed sector Σπ−φ , in particular λ = 0, since |kν | > 0. Again we apply the contraction principle of Kahane to prove (7.9) and (7.10). Similarly, parameterellipticity of Q and part c) of Lemma 6.5 proves (7.11) as well as

−1 ; λ ∈ Σπ−φ , k ∈ Zn kω A λ + P (kν ) + Qδ (kν )A for |ω| ≤ m2 . Since λ = 0 was considered, too, sectoriality of Aδ and 0 ∈ ρ(Aδ ) for δ ≥ 0 follows. Now consider the case ν = 0 and δ > 0. With what has been proved so far, the sufficient intermediate condition (3.12) for the multiplier theorem can be applied successfully. Note that the ideas of the first part of the proof carry over to this situation only if k = 0. Thus, two R-boundedness statements have to be proved. Firstly, the R-boundedness of |α| 1− λ m1 Mλα ; λ ∈ Σπ−φ , k ∈ Zn . This follows immediately due to homogeneity arguments. Recall the structure of Mλα , in particular the fact that we no longer have to consider Mλβ with |β| < |α| and that |α| 0, α = 0, 1− m α 1 λ Mλ (0) =

−1 λ λ + δA , α = 0. Secondly, we have to prove the R-boundedness of (7.9), (7.10) and (7.11). This time, however, it suffices to consider k ∈ Zn \{0} instead of k ∈ Zn . Thus, part one of the proof applies verbatim, except for the statement on λ = 0. In particular, we have shown Aδ ∈ ΨRS(Lp (Qn , E)). Finally let A ∈ RS(E), i.e. we additionally have N (A) = {0}. Assume Aδ u = 0 for some u ∈ D(Aδ ). Calculating Fourier coefficients gives

P (k) + (Q(k) + δ)A u ˆ(k) = 0 (k ∈ Zn ).


93

Due to injectivity of A and parameter-ellipticity of P and Q u ˆ(k) = 0

(k ∈ Zn )

follows. Hence, u = 0 and Aδ ∈ RS(Lp (Qn , E)) for δ > 0. In case Q ≡ c, c = 0 subject to condition (ii) we can obviously allow δ = 0 by the same arguments. The assertion on 0 ∈ ρ(Aδ ) follows as in Proposition 6.13. Remark 7.21. Note that additionally we have proved (7.14)

R({Dα A(λ + Aδ )−1 ; λ ∈ Σπ−φ , α ∈ Nn 0 , 0 ≤ |α| ≤ m2 }) < ∞.

Remark 7.22. a) The shift δ > 0 cannot be neglected in case Q ≡ c and ν = 0. To see this, take a right-hand side f ∈ Lp (Qn , E) which is given as a constant function η ∈ E \ D(A). If λu + A(D)u = f , then λˆ u(0) = fˆ(0) = η by parameterellipticity of P and Q. Hence, u ∈ / D(A). On the other hand, δ = 0 and ν = 0 can simultaneously be handled if the underlying space is changed to Lp(0) (Qn , E). b) Likewise we cannot omit the assumption N (A) = {0} in case Q ≡ c and δ = 0 in order to deduce A ∈ RS(Lp (Qn , E)). To see this, let η ∈ N (A) \ {0}. Then u defined by u(x) := η for x ∈ Qn fulfills u ∈ D(A) \ {0} and Au = 0. For what follows assume Aδ (D) to be even. Let Aδ,B denote the Lp -realization of the parameter-dependent boundary value problem λu + Aδ (D)u B1 (D)u B2 (D)Au

(7.15)

= = =

f 0 0

ñ, in Q ñ, on ∂ Q ñ, on ∂ Q

where the boundary operator B(D) := (B1 (D), B2 (D)) represents boundary conditions as in problem (7.5), cf. Remark 6.15. Since λ + Aδ (D) is even, too, we have the following immediate consequence of Proposition 7.20. Proposition 7.23. Let the assumptions of Proposition 7.20 on Aδ (D) be fulfilled ˜ n , E)) for each δ > 0, φR and let Aδ (D) be even. Then Aδ,B ∈ ΨRS(Lp (Q Aδ,B ≤ ϕ0 and for each φ > ϕ0 it holds that (7.16) R

λ

|α| 1

1− m

Dα (λ + Aδ,B )−1 ; λ ∈ Σπ−φ , α ∈ Nn 0 , 0 ≤ |α| ≤ m1

< ∞.

In case B includes non-pure Neumann boundary conditions in at least one direction ˜ n , E)) and 0 ∈ ρ(Aδ,B ) for δ ≥ 0. moreover Aδ,B ∈ RS(Lp (Q ˜ n , E)) for δ > 0. In case A ∈ RS(E) moreover Aδ,B ∈ RS(Lp (Q In case ν = 0 and Q ≡ c, c = 0 subject to condition (ii) of Proposition 7.20 and ˜ n , E)) for δ ≥ 0. A ∈ RS(E) moreover Aδ,B ∈ RS(Lp (Q

94


Proof. Let E denote the operator of extension and R the operator of restriction as described in the proof of Proposition 7.16. Let further Aδ = Aδ,ν with ν as described in Proposition 7.16. Then the assertion on pseudo-R-sectoriality follows due to the representation (λ + Aδ,B )−1 = R(λ + Aδ,ν )−1 E and Proposition 7.20. The results on R-sectoriality of Aδ,B , i.e. N (Aδ,B ) = {0}, can also be deduced from Proposition 7.20. To see this, observe that u ∈ D(Aδ,B ) subject to Aδ,B u = 0 implies U := Eu ∈ D(Aδ,ν ) and Aδ,ν U = 0 with ν as described in Proposition 7.16. Clearly U = 0 implies u = 0. See further Remark 7.17. It remains to prove R-boundedness of the H∞ -calculus for the operators under consideration. To do so, we more or less adopt the proofs of the continuous setting. First recall the holomorphic, operator-valued functions H1 ∈ H∞ (Σϑ , L(E)) and H2 ∈ H∞ (Σζ , L(E)) from Lemma 6.18. In the continuous setting a representation (k) of e.g. z k H1 (z) based on Cauchy’s integral formula in an extended

version en tered. It was employed to deduce R-bounds of derivatives Dγ H1 ◦ P (ξ), whereas in the present situation R-bounds of discrete derivatives Δγ H1 ◦ P (k) have to be inferred. However, shifts of H1 itself can not be carried out since the sector Σσ is not invariant under shifting. Consequently, we have to treat mh ◦ P at once. This is done in the subsequent lines. Lemma 7.24. Let ν ∈ i(−1, 1)n , 0 = γ ≤ 1, and let 0 < ϑ < ψ < ϑ < π. Let H ∈ H∞ (Σϑ , L(E)) and let P : Zn → C be homogeneous and parameter-elliptic with angle ϕP < ϑ . Set

Γψ := (∞, 0]eiψ ∪ [0, ∞)e−iψ . Then there exist gk ∈ L1 (Γψ ) such that sup k∈Zn \{−1,0,1}n

and Δγ (H ◦ P )(kν ) =

1 2πi

kγ gk L1 (Γψ ) < ∞

gk (μ)H(μ)dμ Γψ

(k ∈ Zn \ {−1, 0, 1}n ).

Proof. Let k ∈ Zn \ {−1, 0, 1}n be arbitrary and set z := P (kν ) ∈ Σϑ . Due to homogeneity and parameter-ellipticity of P and our choice of k and ν there exist 0 < r < R such that 0 < r < |P (kν − ω)| < R for all 0 ≤ ω ≤ 1. Set

Γψ ,r,R := [R, r]eiψ ∪ rei(ψ

,−ψ )

∪ [r, R]e−iψ ∪ Rei(−ψ

Then Cauchy’s integral formula for closed rectifiable curves yields 1 1 H(z) = H(μ)dμ. 2πi Γψ ,r,R μ − z

,ψ )

.


95

With gk (μ) := Δγ (μ − P )−1 (kν ) we further have 1 Δγ (H ◦ P )(kν ) = gk (μ)H(μ)dμ 2πi Γψ ,r,R and from Lemma 7.1 we infer that each value gk (μ) equals

W

−1

−1 ωj (−1)rW μ − P (kν − γ) . Δ P (kν − ω∗j ) μ − P (kν − ω∗j )

r

W∈Zγ

j=1

/ {−1, 0, 1}n . Recall that 0 < r < |P (kν − ω)| < R for all 0 ≤ ω ≤ 1 thanks to k ∈ Since γ = 0, we have rW ≥ 1 in this representation formula of gk (μ). Therefore, gk ∈ L1 (Γψ ) and Γ gk (μ)H(μ)dμ exists due to boundedness of H on Σϑ . As ψ

in Lemma 6.19 we estimate the integrals over the two arcs in the representation formula for Δγ (H ◦ P )(kν ) to find 1 Δγ (H ◦ P )(kν ) = gk (μ)H(μ)dμ. 2πi Γψ

Figure 7.1: The path of integration, discrete case. All values P (kν − ω), where 0 ≤ ω ≤ 1 and k ∈ Zn \ {−1, 0, 1}n stay in the exterior of the red half circle.

In order to establish the uniform L1 -bound for kγ gk , consider δ > 0 such that |kν − ω| > δ for all 0 ≤ ω ≤ 1 and all k ∈ Zn \ {−1, 0, 1}n . By homogeneity of P

96


we observe existence of a δ > 0 such that |P (kν − ω)| > δ for all 0 ≤ ω ≤ 1 and all k ∈ Zn \ {−1, 0, 1}n . Moreover, there exists C > 0 such that we can estimate

|ξ ω Δω P (ξ)| ≤ C|P (ξ)| (ξ ∈ Rn , |ξ| ≥ δ, 0 ≤ ω ≤ 1) (γ)

z

|γ|

1 it for the continuous extensions of P and Δω P . For gz1 ,z2 (μ) := (μ−z )|γ| (μ−z2 ) 1 therefore holds that (γ) gz ,z 1 kγ gk L1 (Γϑ ) ≤ sup < ∞. 1 2 L (Γ )

z1 ,z2 ∈Σϑ ,|z1 |,|z2 |>δ

ϑ

As already indicated by the previous lemma, our intermediate condition (3.12) plays a crucial role in the upcoming result. Proposition 7.25. Let 1 < p < ∞, let E be a Banach space of class HT enjoying property (α), let ν ∈ i(−1, 1)n , and let A ∈ ΨRH∞ (E). For the homogeneous polynomials P and Q assume that (i) P is parameter-elliptic with angle ϕP ∈ [0, π), (ii) Q is parameter-elliptic with angle ϕQ ∈ [0, π), (iii) ϕP + ϕQ + φR∞ < π. A Then Aδ ∈ ΨRH∞ (Lp (Qn , E)) and φR∞ ≤ max{ϕP , ϕQ + φR∞ A } for each δ > 0. Aδ ∞ p In case ν = 0 moreover Aδ ∈ RH (L (Qn , E)) for δ ≥ 0. In case A ∈ RH∞ (E) moreover Aδ ∈ RH∞ (Lp (Qn , E)) for δ > 0. In case ν = 0 and Q ≡ c, c = 0 subject to condition (ii) and A ∈ RH∞ (E) moreover Aδ ∈ RH∞ (Lp (Qn , E)) for δ ≥ 0. R∞ Proof. Let φ ∈ (max{ϕP , ϕQ +φR∞ A }, π −min{ϕP , ϕQ +φA }) and σ > φ. Define iψ −iψ where ψ ∈ (φ, min{σ, π − min{ϕP , ϕQ + φR∞ Γ := (∞, 0]e ∪ [0, ∞)e A }}) and ∞ consider an arbitrary h ∈ H0 (Σσ ) such that |h|σ∞ ≤ 1. We formally have h(Aδ ) = Tmh ◦(P,Qδ ) . Here h(λ)(λ − z1 − z2 A)−1 dλ ((z1 , z2 ) ∈ Σϑ × Σζ ), mh (z1 , z2 ) := Γ

< π. From Lemma 6.17 we deduce where ϑ ≥ ϕP , ζ ≥ ϕQ such that ϑ + ζ + φR∞ A R({mh (z1 , z2 ); |h|σ∞ ≤ 1, (z1 , z2 ) ∈ Σϑ × Σζ }) < ∞. In particular, R({(mh ◦ (P, Qδ ))(kν ); |h|σ∞ ≤ 1, k ∈ Zn }) < ∞.


97

Now choose ψp ∈ (ϕP , ϑ) and ψq ∈ (ϕQ , ζ) to define Γp := (∞, 0]eiψp ∪ [0, ∞)e−iψp

and

Γq := (∞, 0]eiψq ∪ [0, ∞)e−iψq .

Due to Lemma 6.18 mh is holomorphic in each variable separately. By Cauchy’s integral formula for polydiscs (see e.g. [Ran86, Theorem 1.3]) extended along the lines of Lemma 6.19, we infer 1 1 mh (z1 , z2 ) = mh (μ1 , μ2 )dμ2 dμ1 (2πi)2 Γp Γq (μ1 − z1 )(μ2 − z2 ) 1 1 1 = mh (μ1 , μ2 )dμ2 dμ1 . (2πi)2 Γp (μ1 − z1 ) Γq (μ2 − z2 )

Let 0 = γ ≤ 1 and set gk (μ1 , μ2 ) := Δγ (μ1 − P )−1 (μ2 − Qδ )−1 (kν ). Employing Lemma 7.1 yields γ γ k gk (μ1 , μ2 ) = kγ−θ (Δγ−θ (μ1 − P )−1 )(kν − θ)kθ (Δθ (μ2 − Qδ )−1 )(kν ) θ θ≤γ for k ∈ Zn \ {−1, 0, 1}n . Applying Δγ we get Δγ (mh ◦ (P, Qδ ))(kν ) 1 γ (1),θ (2),θ = h (μ ) hk (μ2 )mh (μ1 , μ2 )dμ2 dμ1 . 1 k (2πi)2 p q θ Γ Γ θ≤γ Here (1),θ

hk

(μ1 ) := (Δγ−θ (μ1 − P )−1 )(kν − θ)

fulfill sup k∈Zn \{−1,0,1}n

γ−θ (1),θ hk k

L1 (Γp )

and

≤ C and

(2),θ

hk

(μ2 ) := (Δθ (μ2 − Qδ )−1 )(kν )

sup k∈Zn \{−1,0,1}n

θ (2),θ k hk

L1 (Γq )

≤ C.

Thus, Lemma 3.3 applies in iteration to the result R({kγ Δγ (mh ◦ (P, Qδ ))(k); |h|σ∞ ≤ 1, k ∈ Zn \ {−1, 0, 1}n , 0 ≤ γ ≤ 1}) < ∞ which shows mh ◦ (P, Qδ ) to be a Fourier multiplier. Proposition 7.26. Given the assumptions of Proposition 7.25, let Aδ (D) be R∞ ˜ n , E)) and φR∞ even. Then Aδ,B ∈ ΨRH∞ (Lp (Q Aδ,B ≤ max{ϕP , ϕQ + φA } for each δ > 0. In case B includes non-pure Neumann boundary conditions in at least one direction ˜ n , E)) for δ ≥ 0. moreover Aδ,B ∈ RH∞ (Lp (Q ˜ n , E)) for δ > 0. In case A ∈ RH∞ (E) moreover Aδ,B ∈ RH∞ (Lp (Q In case ν = 0 and Q ≡ c, c = 0 subject to condition (ii) of Proposition 7.25 and ˜ n , E)) for δ ≥ 0. A ∈ RH∞ (E) moreover Aδ,B ∈ RH∞ (Lp (Q Remark 7.27. Note that related results on sectoriality and a bounded H∞ -calculus of Aδ,ν respectively Aδ,B can still be deduced if E does not enjoy property (α).

8 Application to cylindrical boundary value problems In this chapter the Fourier transform approach and the Fourier series approach from the previous chapters are employed to investigate cylindrical boundary value problems. With their aid, a model problem for cylindrical boundary value problems containing partially non-constant coefficients can be treated. Thanks to Rbounds derived for the solution operator of the model problem a localization procedure as known for problems in the whole space can be carried out to deal with fully non-constant coefficients. The crucial assumption is that the cylindrical boundary value problem is parameter-elliptic. As a main result, we prove pseudoR-sectoriality of the according Lp -realizations. Due to the results from Chapter 5 this implies results on the associated parabolic problems. At the end of the chapter we focus on the Laplacian subject to mixed periodic and Dirichlet-Neumann boundary conditions in cylindrical domains.

8.1 Known results for bounded and exterior domains We start this section with known results on parameter-elliptic boundary value problems in standard domains. Definition 8.1. Let m, n ∈ N. The domain V ⊂ Rn is called a standard domain in n Rn if it is given as the whole space Rn , the half space Rn + or as a domain in R with compact boundary, that is, a bounded or an exterior domain. If a standard domain V is of class C m , it is also called a C m standard domain. Let F be a Banach space, m, n ∈ N, and let V ⊂ Rn be a C 2m standard domain. Set aα (x)Dα u, A(x, D)u := |α|≤2m

where α ∈

Nn 0

and aα : V → L(F ). Furthermore, let

Bj (x, D)u :=

bj,β (x)(Dβ u)|∂V ,

|β|≤mj

where mj < 2m, β ∈ Nn 0 , and bj,β : ∂V → L(F ) for j = 1, . . . , m. We consider the boundary value problem (A, B) given through (8.1)

λu + A(x, D)u Bj (x, D)u

= =

f in V, 0 on ∂V

(j = 1, . . . , m).


100

8 Application to cylindrical boundary value problems

The Lp (V )-realization of (A, B) will be denoted by D(A) := u ∈ W 2m,p (V ); Bj (·, D)u = 0 (j = 1, . . . , m) , Au := A(·, D)u

(u ∈ D(A)).

The differential operator A(x, D) is assumed to be parameter-elliptic in V with angle of parameter-ellipticity ϕ ∈ [0, π), i.e. ϕ is the infimum over all φ ∈ (0, π) such that the principal part of its symbol aα (x)ξ α A# (x, ξ) := |α|=2m

for each x ∈ V is parameter-elliptic with angle of parameter-ellipticity ϕx < φ. Here, an L(F )-valued homogeneous polynomial aα ξ α (ξ ∈ Rn ) a(ξ) := |α|=2m

is called parameter-elliptic if there exists an angle φ ∈ (0, π) such that the spectrum σ(a(ξ)) of a(ξ) in L(F ) satisfies (8.2)

σ(a(ξ)) ⊂ Σφ

(ξ ∈ Rn , |ξ| = 1).

Then ϕ := inf{φ; (8.2) holds} is called angle of parameter-ellipticity of a (see [DHP03]). Definition 8.2. The boundary value problem (A, B) given through (8.1) is called parameter-elliptic in V with angle of parameter-ellipticity ϕ ∈ [0, π) if A(·, D) is parameter-elliptic in V with angle of parameter-ellipticity ϕ ∈ [0, π) and if for each φ > ϕ the Lopatinskii-Shapiro condition holds. In order to indicate that ϕ is the angle of parameter-ellipticity of the boundary value problem (A, B), we use the subscript notation ϕ(A,B) . We refer to [Wlo87] for an introduction to the Lopatinskii-Shapiro condition for scalar-valued boundary value problems and to [DHP03] for an extensive treatment of the F -valued case. In the sequel let the boundary value problem (8.1) be parameter-elliptic. Furthermore, assume the following assumptions on the coefficients to hold true: ⎧ aα ∈ C(V , L(F )) and aα (∞) := lim aα (x) exists (|α| = 2m), ⎪ ⎪ |x|→∞ ⎪ ⎪ ⎨ 2m − k 1 ∞ rk (8.3) aα ∈ [L + L ](V, L(F )), rk ≥ p, (|α| = k < 2m), > ⎪ ⎪ n rk ⎪ ⎪ ⎩ bj,β ∈ C 2m−mj (∂V, L(F )) (j = 1, . . . , m; |β| ≤ mj ). The condition on aα (∞) can be neglected if V is bounded. By employing finite open coverings of V in [DHP03] the following result is proved.

8.1 Known results for bounded and exterior domains

101

Proposition 8.3. Let V ⊂ Rn be a C 2m standard domain. Let the boundary value problem (A, B) be parameter-elliptic and let the assumptions (8.3) on the coefficients be given. Then for each φ > ϕ(A,B) there exists a δ = δ(φ) ≥ 0 such that A + δ ∈ RS(Lp (V, F )) and φRS A+δ ≤ φ. Moreover, we have |γ| < ∞. (8.4) R λ1− 2m Dγ (λ + A + δ)−1 ; λ ∈ Σπ−φ , 0 ≤ |γ| ≤ 2m As a stronger condition on the coefficients assume ⎧ aα ∈ BU C γ (V , L(F )) for some γ ∈ (0, 1), ⎪ ⎪ ⎪ ⎪ ⎪ aα (∞) := lim aα (x) exists, and ⎪ ⎪ |x|→∞ ⎪ ⎪ ⎨ aα (x) − aα (∞) ≤ C|x|−γ for x ∈ V, |x| ≥ 1 (|α| = 2m), (8.5) ⎪ ⎪ ⎪ 2m − k 1 ⎪ ⎪ aα ∈ [L∞ + Lrk ](V, L(F )), rk ≥ p, (|α| = k < 2m), > ⎪ ⎪ n r k ⎪ ⎪ ⎩ 2m−mj bj,β ∈ C (∂V, L(F )) (j = 1, . . . , m; |β| ≤ mj ). Again by finite open coverings of V , in [DDH+ 04] the following result is proved. Proposition 8.4. Let V ⊂ Rn be a C 2m standard domain. Let the boundary value problem (A, B) be parameter-elliptic and let the assumptions (8.5) on the coefficients be given. Then for each φ > ϕ(A,B) there exists a δ = δ(φ) ≥ 0 such that A + δ ∈ RH∞ (Lp (V, F )) and φR∞ A+δ ≤ φ. Remark 8.5. In [DHP03] and [DDH+ 04] formally the case n ≥ 2 is treated, whereas the case n = 1 can be deduced by simplified methods. For the Laplacian according results for domains rougher than the standard domains as defined in Definition 8.1 are known. More precisely, smoothness of the boundary of G can be reduced to be of graph Lipschitz type. Definition 8.6. A domain G ⊂ Rn is called a Lipschitz domain if there exists an M > 0 so that every point x = (x1 , . . . , xn ) ∈ ∂G has a neighborhood U such that, eventually after an affine change of coordinates, ∂G∩U is described by the equation xn = ϕ(x1 , . . . , xn−1 ), where ϕ is a Lipschitz continuous function with Lipschitz constant bounded by M and where G ∩ U = {x ∈ U ; xn > ϕ(x1 , . . . , nn−1 )}. This definition is also used in [JK95], for instance. Domains G of the above type are usually termed graph Lipschitz domains or strongly Lipschitz or domains with Lipschitz boundary. Recall that on such domains standard function space theory, trace results, definition of an outer unit normal, etc., are still available (see [Gri85] and [JK95]). For more general classes of Lipschitz domains see also [Gri85] and [HDR09]. In what follows we distinguish between the weak and the strong Dirichlet LaplaD p cian ΔD p,w and Δp,s on L (V ) defined by D D(Δp,w ) := u ∈ W01,p (V ); Δu ∈ Lp (V ) , ΔD p,w u := Δu

(u ∈ D(ΔD p,w ))

102


and 2,p (V ) ∩ W01,p (V ), D(ΔD p,s ) := W

(u ∈ D(ΔD p,s )).

ΔD p,s u := Δu

N The weak and the strong Neumann Laplacian ΔN p,w and Δp,s are defined by 1,p p 1,p ) := u ∈ W (V ); ∃v ∈ L (V ) ∀ϕ ∈ W (V ) : − ∇u∇ϕ = vϕ , D(ΔN p,w V

ΔN p,w u := v

V

(u ∈ D(ΔN p,w ))

and 2,p (V ); ∂n u = 0 on ∂V , D(ΔN p,s ) := u ∈ W ΔN p,s u := Δu

(u ∈ D(ΔN p,s )).

The following proposition gathers results on these operators suitable for our purposes. Recall that A ∈ (Ψ)RH∞ (Lp (Ω)) implies A ∈ (Ψ)RS(Lp (Ω)) with R∞ φR A ≤ φA . Proposition 8.7. a) Let V ⊂ Rn be a C 2 standard domain and let 1 < p < ∞. Then ∞ p R∞ (i) −ΔD p,s ∈ RH (L (Ω)) and φ−ΔD < p,s

π , 2

∞ p R∞ (ii) −ΔN p,s ∈ ΨRH (L (Ω)) and φ−ΔN < p,s

π , 2

∞ p R∞ (iii) −ΔN p,s + δ ∈ RH (L (Ω)) and φ−ΔN

0.

b) Let V ⊂ Rn , n ≥ 3, be a bounded Lipschitz domain. Then there exists ε > 0 depending only on the Lipschitz character of V such that for all (3+ε) < p < 3+ε ∞ p R∞ (i) −ΔD p,w ∈ RH (L (Ω)) and φ−ΔD

p,w

2mi ,

aiαi (xi ),

else.

The notion of parameter-ellipticity for the entire cylindrical boundary value problem from Definition 8.2 (see also [DHP03]) is no longer appropriate for our intentions since any part Ai (xi , D) of higher order rules out others of lower order. Instead, the differential operators A1 (x1 , D) and A2 (x2 , D) on the one hand and the induced boundary value problem on the other hand can be used to define parameter-ellipticity of a cylindrical boundary value problem efficiently. Definition 8.9. A cylindrical boundary value problem is called parameter-elliptic in Ω if (i) A1 (x1 , D) is parameter-elliptic in Rn1 with angle ϕ1 := ϕA1 ∈ [0, π), (ii) A2 (x2 , D) is parameter-elliptic in Qn2 with angle ϕ2 := ϕA2 ∈ [0, π), (iii) the induced boundary value problem (A3 , B3 ) on the cross-section V is parameter-elliptic with angle ϕ3 := ϕ(A3 ,B3 ) ∈ [0, π), and (iv) it holds that ϕi + ϕj < π for i, j = 1, 2, 3 and i = j.

8.2 Cylindrical boundary value problems

105

We call ϕ(A,Bν ) := max{ϕi , i = 1, 2, 3} the angle of parameter-ellipticity of the cylindrical boundary value problem. In case m1 = m2 = m3 at first sight we could have defined parameter-ellipticity of a cylindrical boundary value problem in Ω simply by subjecting the boundary value problem (8.7) to the common definition of parameter-ellipticity as presented in Definition 8.2. As long as the corresponding angle is less than π2 this works out equivalently well since parameter-ellipticity in the sense of Definition 8.9 is implied. Things change considerably in case this constraint is violated, as then we lose track of the Dore-Venni-type condition in (iv) of Definition 8.9. As we have already seen in the previous chapters, however, this condition is crucial in order to carry out a multiplier approach successfully. As Lp (Ω)-realization of (8.7) we define D(A) := Lp (Rn1 , Lp (Qn2 , D(A3 ))) ∩ 2 ,p W 1 ,p (Rn1 , Wν,per (Qn2 , W 3 ,p (V ))), 1 + 2 + 3 ≤1 2m1 2m2 2m3

Au := A(·, D)u

(u ∈ D(A)).

Given equality of orders, m := m1 = m2 = m3 , the domain of definition D(A) equals Lp (Rn1 , Lp (Qn2 , D(A3 ))) ∩

2m =1

W 1 ,p (Rn1 ,

2m−

κ,p Wν,per (Qn2 , W 2m−−κ,p (V ))),

κ=1

in particular, D(A) ⊂ W 2m,p (Ω). Finally, we require the following smoothness assumptions: ⎧ 1 a 1 ∈ C(Rn1 ) and a1α1 (∞) := lim a1α1 (x1 ) exists (|α1 | = 2m1 ), ⎪ ⎪ ⎪ α |x1 |→∞ ⎪ ⎪ ⎪ ⎪ 2 2 ⎪ ⎪ aα2 ∈ Cper (Qn2 ) (|α | = 2m2 ), ⎪ ⎪ ⎪ ⎪ ⎪ a3α3 ∈ C(V ) and a3α3 (∞) := lim a3α3 (x3 )exists (|α3 | = 2m3 ), ⎪ ⎪ |x3 |→∞ ⎪ ⎪ ⎪ ⎪ ⎨ 1 2m1 − μ 1 aα1 ∈ [L∞ + Lrμ ](Rn1 ) rμ ≥ p, > (|α1 | = μ < 2m1 ), (8.10) n r 1 μ ⎪ ⎪ ⎪ ⎪ ⎪ a2 2 ∈ Lrμ (Q ) r ≥ p, 2m2 − μ > 1 (|α2 | = μ < 2m ), ⎪ n2 μ 2 ⎪ α ⎪ n2 rμ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ a3 3 ∈ [L∞ + Lrμ ](V ) r ≥ p, 2m3 − μ > 1 (|α3 | = μ < 2m ), ⎪ μ 3 ⎪ α ⎪ n3 rμ ⎪ ⎪ ⎪ ⎩ 3 bj,β 3 ∈ C 2m3 −m3,j (∂V ) (j = 1, . . . , m3 ; |β 3 | ≤ m3,j ). Here Cper (Qn2 ) := {f ∈ C([0, 2π]n2 ); f |xj =0 = f |xj =2π (j = 1, . . . , n2 )}. As a matter of course, the limit condition on a3α3 for |α3 | = 2m3 drops out in case V

106


is non-exterior. Note that this limit behavior of the top order coefficients ensures parameter-ellipticity of the limit operators A1 (∞, D) and A2 (∞, D) in case V is an exterior domain to be well-defined. Thus, we can extend the notion of parameterellipticity of a cylindrical boundary value problem to

Ω := Rn ∪ {∞} × Qn2 × V , where we agree on {∞} ⊂ V in case that V is unbounded. Theorem 8.10. Let 1 < p < ∞ and ν ∈ i(−1, 1)n2 . Set Ω := Rn1 × Qn2 × V ⊂ Rn , where V is a standard domain of class C 2m3 in Rn3 . For the boundary value problem (A, Bν ) given through (8.7) on Ω we assume that (i) it is cylindrical, (ii) the coefficients of A(·, D) and B3,j (·, D), j = 1, . . . , m3 , satisfy (8.10), (iii) it is (cylindrically) parameter-elliptic in Ω of angle ϕ(A,Bν ) ∈ [0, π). Then for each φ > ϕ(A,Bν ) there exists δ = δ(φ) ≥ 0 such that A + δ ∈ RS(Lp (Ω)) and φRS A+δ ≤ φ. Moreover, we have (8.11) R

λρ Dα (λ + A + δ)−1 ;

λ ∈ Σπ−φ , ρ ∈ [0, 1], α ∈ Nn 0,0 ≤ ρ +

|α1 | 2m1

+

|α2 | 2m2

+

|α3 | 2m3

≤1

< ∞.

Theorem 8.10 is proved in three steps. First we assume n2 = 0, that is, we consider merely Ω1 := Rn1 × V . In a second step, we assume n1 = 0, that is, we consider merely Ω2 := Qn2 × V . Once we have proved Theorem 8.10 for these special cases, in a third step the assertion of Theorem 8.10 without any restriction follows for we can write Ω = Rn1 × Ω2 . Note that all relevant properties are transferred from A3 to A in each step.

8.2.1 Step 1: Let 1 < p < ∞. Since in this step we consider the Cartesian product of two domains only, for the sake of convenience, we shift the index 3 appearing in the boundary value problem (8.7) to 2. That is, from now on we consider (8.12)

λu + A1 (x, D)u + A2 (y, D)u B2,j (y, D)u

= =

f in Ω, 0 on ∂Ω.

Here (x, y) ∈ Ω := Rn1 × V . Accordingly, the induced boundary value problem (A2 , B2 ) := (A2 (·, D), B2,1 (·, D), . . . , B2,m2 (·, D)).


107

reads as λu + A2 (y, D)u B2,j (y, D)u

(8.13)

= =

f in V, 0 on ∂V

(j = 1, . . . , m2 ).

Clearly, the cross-section V is now assumed to be a C 2m2 standard domain. The Lp (V )-realization of (A2 , B2 ) will be denoted by D(A2 ) := u ∈ W 2m2 ,p (V ); B2,j (·, D)u = 0 (j = 1, . . . , m2 ) , A2 u := A2 (·, D)u

(u ∈ D(A2 )).

Moreover, since ν is no longer involved in (8.7), we write (A, B) instead of (A, Bν ). As the Lp (Ω)-realization of the cylindrical boundary value problem (A, B) we accordingly have D(A) := u ∈ Lp (Ω); Dα u ∈ Lp (Ω) |α1 | |α2 | for 2m + ≤ 1 and B (·, D)u = 0 (j = 1, . . . , m ) , j 2 2m2 1 Au := A(·, D)u

(u ∈ D(A)).

Note that

D(A) = Lp (Rn1 , D(A2 )) ∩

W 1 ,p (Rn1 , W 2 ,p (V )).

1 + 2 ≤1 2m1 2m2

First we consider the model problem for the cylindrical boundary value problem (8.12), i.e. we assume A1 (x, D) on Rn1 to be given as homogeneous differential operator 1 α A1 (D) := aα D |α|=2m

with constant coefficients

a1α

∈ C. We set

A0 (·, D) := A0 (y, D) := A1 (D) + A2 (y, D) and A0 u := A0 (·, D)u

(u ∈ D(A0 ) := D(A)).

Observe that A2 (y, D) remains unchanged in this model problem and that δ2 ≥ 0 exists such that the assertions of Proposition 8.3 are valid for A2 + δ2 . Let φ > ϕ(A0 ,B) , λ ∈ Σπ−φ , and u ∈ S(Rn1 , D(A2 )) ⊂ D(A0 ). For the sake of convenience in the sequel we write E := Lp (V ) and X := Lp (Rn1 , E) ∼ = Lp (Ω). Applying E-valued Fourier transform F to f := (λ + A1 (D) + A2 + δ2 )u gives us (λ + a1 (·) + A2 + δ2 )F u = Ff.

108


Hence, we formally have

u = F −1 m0λ F f,

where m0λ is given by the operator-valued symbol m0λ (ξ) := (λ + a1 (ξ) + A2 + δ2 )−1

(ξ ∈ Rn1 ).

Note that m0λ ∈ C ∞ (Rn1 , L(E)) is well-defined if −(λ + a1 (ξ)) ∈ ρ(A2 + δ2 )

(ξ ∈ Rn1 ).

In view of our choice of λ ∈ Σπ−φ , Lemma 6.11, and Proposition 8.3 this is satisfied. The latter and Proposition 6.13 therefore yield R

λ

|α1 | 1

1− 2m

D(α

1

,0)

˜ 0 +δ2 )−1 ; λ ∈ Σπ−φ , α1 ∈ Nn1 , 0 ≤ |α1 | ≤ 2m1 (λ+ A 0

δ2 . Proposition 8.11. For each φ > ϕ(A0 ,B) and δ2 = δ2 (φ) as in Proposition 8.3 we have A0 + δ2 ∈ RS(X) and φRS A0 +δ2 ≤ φ. Moreover, it holds that

|α1 | |α2 | 1− 2m + 2m α −1 1 2 D (λ + A0 + δ2 ) R λ ; (8.14) |α1 | |α2 | < ∞. λ ∈ Σπ−φ , α ∈ Nn 0 , 0 ≤ 2m1 + 2m2 ≤ 1 Proof. Let φ > ϕ(A0 ,B) . For 0 ≤ γ ≤ 1 we apply Lemma 6.1 to the result ξ γ Dγ mλ (ξ) = λ ·

|α |

|α |

1−( 2m1 + 2m2 ) α1 α2 1 2 ξ D (λ

γ ≤γ W∈Zγ−γ

Cγ ,α1 ,W

Wr j=1

+ a1 (ξ) + A2 + δ2 )−1

j j ξ ω (Dω a1 )(ξ)(λ + a1 (ξ) + A2 + δ2 )−1 ,


109 1

2

|α | |α | for all λ ∈ Σπ−φ , ξ ∈ Rn1 and all α ∈ Nn 0 such that 0 ≤ 2m1 + 2m2 ≤ 1. Here 1 Cγ ,α1 ,W ∈ Z denotes a constant depending on γ , α and W. We want to employ Lemma 3.2 to establish mλ as a Fourier multiplier uniformly in λ ∈ Σπ−φ . Thanks to (8.4) this is possible if we can show that both |α |

κ1 (λ, ξ) :=

|α |

1−( 2m1 + 2m2 ) α1 1 2

λ

ξ

(λ + a1 (ξ)) and κ2 (λ, ξ) :=

|α2 | 2

1− 2m

ξ ω Dω a1 (ξ) λ + a1 (ξ)

are uniformly bounded for (λ, ξ) ∈ Σπ−φ ×Rn1 . Obviously κ2 meets the conditions in the proof of Lemma 6.5 perfectly and uniform boundedness follows. While the structure of κ1 includes fractional powers, quasi-(2m1 ,1)-homogeneity of degree zero is preserved. We therefore can apply arguments very similar to those in Lemma 6.5 to show that κ1 is uniformly bounded as well. Hence, Rλ := F −1 mλ F ∈ L(X) exists, where

R(Rλ ) ⊂

W 1 ,p (Rn1 , W 2 ,p (V )).

1 + 2 ≤1 2m1 2m2

Therefore, the equality Rλ = (λ + A0 + δ2 )−1 and the validity of (8.14) follow if B2,j Rλ f = 0

(f ∈ Lp (Ω); j = 1, . . . , m2 ).

To see this, we represent the resolvent applied to f ∈ S(Rn1 , E) as a Bochner integral via 1 (λ + A0 + δ2 )−1 f (x) = eixξ (λ + a1 (ξ) + A2 + δ2 )−1 Ff (ξ)dξ. (2π)n1 /2 Rn1 Since taking the trace acts as a bounded operator on E, it commutes with the integral sign. This yields B2,j (λ + A0 + δ2 )−1 f = 0

(f ∈ S(Rn1 , E); j = 1, . . . , m2 ).

Employing density of S(Rn1 , E) in Lp (Rn1 , E) we conclude that R(Rλ ) = Lp (Rn1 , D(A2 )) ∩

1 + 2 ≤1 2m1 2m2

W 1 ,p (Rn1 , W 2 ,p (V )) = D(A0 ).

110


Note again that the appearing shift δ2 does not have to be enlarged while carrying over R-sectoriality from A2 to A0 . By a perturbation argument we generalize the R-sectoriality for constant coefficients to the case of slightly varying coefficients of A1 . To this end, we will employ the following perturbation result which is based on a standard Neumann series argument. Lemma 8.12. Let R be a linear operator in X such that D(A0 ) ⊂ D(R) and let δ2 be given as in Proposition 8.11. Assume that there are η > 0 and δ > δ2 such that Ru X ≤ η (A0 + δ)u X (u ∈ D(A0 )). RS Then A0 + R + δ ∈ RS(X), φRS A0 +R+δ ≤ φA0 +δ2 , and for every φ > ϕ(A0 ,B) we have R λρ Dα (λ + A0 + R + δ)−1 ; (8.15) |α1 | |α2 | < ∞, λ ∈ Σπ−φ , α ∈ Nn 0 , ρ ∈ [0, 1], 0 ≤ ρ + 2m1 + 2m2 ≤ 1

whenever η < R({(A0 + δ)(λ + A0 + δ)−1 })−1 . Proof. As R(λ + A0 + δ)−1 L(X) ≤ η (A0 + δ)(λ + A0 + δ)−1 L(X) ≤ ηR({(A0 + δ)(λ + A0 + δ)−1 }) < 1 by assumption, we see that

λ + A0 + R + δ = 1 + R(λ + A0 + δ)−1 (λ + A0 + δ)

is invertible. This implies λρ Dα (λ + A0 + R + δ)−1 = λρ Dα (λ + A0 + δ)−1

∞

(−R(λ + A0 + δ)−1 )j .

j=0

By assumption we have δ0 := δ − δ2 > 0. The fact that |λ + δ0 | ≥ cφ δ0

(λ ∈ Σπ−φ )

for some cφ > 0 yields the existence of a Mφ > 0 such that |λρ | ≤ Mφ |α1 | |α2 | (λ + δ0 )1−( 2m1 + 2m2 )

(λ ∈ Σπ−φ ).

Thanks to the contraction principle of Kahane and Proposition 8.11 we deduce R({λρ Dα (λ + A0 + δ)−1 }) |α1 | |α2 | 1− 2m + 2m α −1 1 2 D ((λ + δ0 ) + A0 + δ2 ) ≤ CR (λ + δ0 ) ≤ C.


111

Lemma 3.2a) then yields R({λρ Dα (λ + A0 + δ)−1 (−R(λ + A0 + δ)−1 )j }) ≤ R({λρ Dα (λ + A0 + δ)−1 })R({(R(λ + A0 + δ)−1 )j }) ≤ Cη j R({(A0 + δ)(λ + A0 + δ)−1 )})j ≤ Cν j

(j ∈ N0 )

with ν := ηR({(A0 + δ)(λ + A0 + δ)−1 }) < 1. Employing again Lemma 3.2a), in particular the fact that the R-bound is preserved when taking the closure in the strong operator topology, the assertion follows. |α1 | |α2 | For the particular choice ρ := 1 − 2m + 2m , Lemma 8.12 does not require 1 2 any enlargement of δ2 . However, for the localization procedure in order it is of importance to be aware of the R-boundedness condition (8.15) for each α ∈ Nn 0 and all ∈ N0 subject to 0 ≤ + 2m2 |α1 | + 2m1 |α2 | ≤ 4m1 m2 . α1 Corollary 8.13. Let R(x, D) := be given such that the |α1 |=2m1 rα1 (x)D rα1 ∞ < η is satisfied. Set condition |α1 |=2m1

(8.16)

Ava (·, D) := Ava (x, y, D) := A0 (y, D) + R(x, D)

((x, y) ∈ Ω)

and denote its X-realization by Ava u := Ava (·, D)u

(u ∈ D(Ava ) := D(A0 )).

RS Then there exists a δ > 0 such that Ava + δ ∈ RS(X) and φRS Ava +δ ≤ φA0 +δ2 provided that η is sufficiently small. In this case, for φ > ϕ(A0 ,B) we have R λρ Dα (λ + Ava + δ)−1 ; (8.17) |α1 | |α2 | < ∞. λ ∈ Σπ−φ , α ∈ Nn 0 , ρ ∈ [0, 1], 0 ≤ ρ + 2m1 + 2m2 ≤ 1

Proof. By Proposition 8.11, in particular by relation (8.14), there exists a C > 0 such that 1 1 1 Dα (A0 + δ)−1 L(X) ≤ C (α1 ∈ Nn 0 , |α | = 2m1 ) for each δ > δ2 . For a fixed δ > δ2 this implies 1 Ru p ≤ rα1 ∞ Dα (A0 + δ)−1 (A0 + δ)u p |α1 |=2m1

≤

Cη (A0 + δ)u p

(u ∈ D(A0 )).

Thus, if we assume that η < 1/CR({(A0 +δ)(λ+A0 +δ)−1 }), the assertion follows from Lemma 8.12. In the next lemma we establish estimates that will turn out to be crucial for the localization procedure.

112


1 Lemma 8.14. Let 1 nrμ1 . Let b ∈ [L∞ + Lrμ ](Rn1 ), Ava be the operator as defined in (8.16), and assume that φ > ϕ(A,B) . a) For every ε > 0 there exists C(ε) > 0 such that

bDβ u p ≤ ε u p,2m1 + C(ε) u p

(u ∈ W 2m1 ,p (Rn1 , E)).

b) For every ε > 0 there exists a δ = δ(ε) > 0 such that R({bDβ (λ + Ava + δ)−1 ; λ ∈ Σπ−φ }) ≤ ε. Proof. a) Let ε > 0 be arbitrary. For b ∈ L∞ (Rn1 ) we obtain by H¨ older’s inequality and vector-valued complex interpolation (see e.g. [Ama95]) that μ 2m

μ 1− 2m

1 u p bD β u p ≤ b ∞ u p,μ ≤ C b ∞ u p,2m 1

1

(u ∈ W 2m1 ,p (Rn1 , E)).

With the help of Young’s inequality we further deduce bDβ u p ≤ ε u p,2m1 + C(ε) u p

(u ∈ W 2m1 ,p (Rn1 , E)).

Now let b ∈ Lrμ (Rn1 ), r := pμ , and r1 + r1 = 1. Then H¨ older’s inequality and the vector-valued version of the Gagliardo-Nirenberg inequality (see [SS05]) imply r

bDβ u p ≤ C b pr Dβ u pr ≤ C b rμ u τp,2m1 u 1−τ , p n1 where τ = rμ (2m ∈ (0, 1) by our assumption on rμ . Again an application of 1 −μ) Young’s inequality yields

bDβ u p ≤ ε u p,2m1 + C(ε) u p

(u ∈ W 2m1 ,p (Rn1 , E)).

b) Let (εj )j∈N be a family of independent symmetric {−1, 1}-valued random variables on a probability space ([0, 1], M, P ), λj ∈ Σπ−φ , and fj ∈ X. For any b ∈ L∞ (Rn1 ), δ0 > 0, and arbitrary t ∈ [0, 1] we have

N

εj (t)bDβ (λj + δ0 + Ava + δ)−1 fj p

j=1

≤ b ∞

N

εj (t)Dβ (λj + δ0 + Ava + δ)−1 fj p .

j=1

Note that there is a cφ > 0 such that |λ + δ0 | ≥ cφ δ0

(λ ∈ Σπ−φ , δ0 > 0).


113

Taking Lp -norm with respect to t and applying the contraction principle of Kahane therefore yields N εj (·)bDβ (λj + δ0 + Ava + δ)−1 fj j=1

Lp ([0,1],X)

N 1− |β| 2m1 λ +δ ≤ C b ∞ εj (·) jδ0 0 Dβ (λj + δ0 + Ava + δ)−1 fj j=1

Lp ([0,1],X)

.

Thanks to (8.17) this implies N εj (·)bDβ (λj + δ0 + Ava + δ)−1 fj j=1 |β| N −(1− 2m )

≤ C b ∞ δ0

1

εj (·)fj

j=1

Lp ([0,1],X)

Lp ([0,1],X)

.

Thus, for δ0 > (C b ∞ /ε)1/(1−|β|/2m1 ) the assertion follows. older’s inequality and the Gagliardo-Nirenberg In case that b ∈ Lrμ (Rn1 ) H¨ inequality imply for τ (2m1 − μ) = nrμ1 and arbitrary t ∈ [0, 1] that N εj (t)bDβ (λj + δ0 + Ava + δ)−1 fj j=1

p

N ≤ b pr εj (t)Dβ (λj + δ0 + Ava + δ)−1 fj j=1

≤ C b rμ

pr

N p τ /p εj (t)Dα (λj + δ0 + Ava + δ)−1 fj

|α|=2m1

p

j=1

N 1−τ · εj (t)(λj + δ0 + Ava + δ)−1 fj . p

j=1

Taking Lp -norm with respect to t and applying once more H¨ older’s inequality we deduce N εj (·)bDβ (λj + δ0 + Ava + δ)−1 fj p L ([0,1],X)

j=1

≤ C b rμ

|α|=2m1

N p εj (·)Dα (λj + δ0 + Ava + δ)−1 fj p

L ([0,1],X)

j=1

N 1−τ · εj (·)(λj + δ0 + Ava + δ)−1 fj j=1

Lp ([0,1],X)

.

τ /p

114


The contraction principle of Kahane then gives us N εj (·)bDβ (λj + δ0 + Ava + δ)−1 fj j=1

≤ C b rμ

|α|=2m1

Lp ([0,1],X)

N p εj (·)Dα (λj + δ0 + Ava + δ)−1 fj p

τ /p

L ([0,1],X)

j=1

N 1−τ λ +δ · εj (·) jδ0 0 (λj + δ0 + Ava + δ)−1 fj

Lp ([0,1],X)

j=1

.

Taking into account (8.17) we arrive at N εj (·)bDβ (λj + δ0 + Ava + δ)−1 fj j=1 N ≤ C b rμ δ0τ −1 εj (·)fj j=1

Lp ([0,1],X)

Lp ([0,1],X)

.

Choosing δ0 > (C b rμ /ε)1/(1−τ ) proves the claim. Now we are in the position to perform a successful localization procedure. We denote by 1 aα (x)Dα A# 1 (x, D) := |α|=2m

the principal part of A1 (x, D). Freezing the coefficients at x0 ∈ Rn1 ∪ {∞}, Proposition 8.11 applies to A1 (D) := A# 1 (x0 , D). So, we first choose a large ball Br0 (0) ⊂ Rn1 with a fixed radius r0 > 0 such that |a1α1 (x) − a1α1 (∞)| ≤ η/Mα (|x| ≥ r0 , |α1 | = 2m1 ), 1 1 1 were Mα = {α ∈ Nn 0 ; |α | = 2m1 , aα1 = 0} and η = η(∞) is the constant 1 α given in Corollary 8.13 applied to A# (∞, D) = 1 |α|=2m1 aα (∞)D . For every fixed x0 ∈ B r0 (0) let η = η(x0 ) be the constant given in Corollary 8.13 applied to A# 1 (x0 , D). Due to the continuity assumptions on the coefficients there exists a radius r = r(x0 ) such that |a1α1 (x) − a1α1 (x0 )| ≤ η(x0 )/Mα ,

(|x − x0 | ≤ r(x0 ), |α1 | = 2m1 ).

Obviously the collection {Br(x0 ) (x0 ); x0 ∈ B r0 (0)} represents an open covering of B r0 (0). Thus, by compactness we have B r0 (0) ⊆

N j=1

Br(xj ) (xj )


115

for a certain finite set (xj )N j=1 . For simplicity we set rj := r(xj ), and Uj := Brj (xj ) for j = 1, . . . , N , as well as U0 := Rn1 \B r0 (0) and x0 := ∞. For each j = 0, . . . , N we define coefficients of A# 1 (x, D)-localizations 1 (x, D) := aj,α (x)Dα A1,loc j |α|=2m

by reflection, that is, we set a10,α (x) = and a1j,α (x) =

⎧ ⎨a1 (x), α

⎩a1α (

⎧ ⎨a1α (x), ⎩a1α (xj

Then by definition we have

+

2 r0 x), |x|2

rj2

x∈ / B r0 (0), x ∈ B r0 (0) x ∈ B rj (xj ),

|x−xj |2

(x − xj )),

x∈ / B rj (xj ).

|a1j,α1 (x) − a1α1 (xj )| ≤ η(xj )

|α1 |=2m1

for x ∈ Rn1 and j = 0, . . . , N , that is, 1,loc (x, D) + A2 (y, D) Aloc j (x, y, D) := Aj

is a small variation of Aj,# (y, D) := A# 1 (xj , D) + A2 (y, D) in the sense of (8.16). Hence, Corollary 8.13 applies to loc Aloc j u := Aj (·, D)u

(u ∈ D(Aloc j ) := D(A)).

For j = 0, . . . , N and each φ > ϕ(A,B) this yields the existence of δ = δ(φ) > 0 such that Aloc j + δ ∈ RS(X) and −1 ; R λρ Dα (λ + Aloc j + δ) (8.18) |α1 | |α2 | < ∞. λ ∈ Σπ−φ , α ∈ Nn 0 , ρ ∈ [0, 1], 0 ≤ ρ + 2m1 + 2m2 ≤ 1 ∞ n1 ) of Rn1 subordinate to Next we choose a partition of unity (ϕj )N j=0 ⊂ C (R N the open covering (Uj )j=0 such that 0 ≤ ϕj ≤ 1. In addition, we fix ψj ∈ C ∞ (Rn1 ) such that ψj ≡ 1 on supp ϕj and supp ψj ⊂ Uj . We further set

Alow (x, D) := A(x, y, D) − A# (x, y, D)

116


which in fact acts on Rn1 only. Pick λ ∈ Σπ−φ . Then λu + A(·, D)u = f holds if and only if λu + A# (·, D)u = f − Alow (·, D)u. Multiplying the line above by ϕj we obtain λϕj u + A# (·, D)ϕj u = ϕj f + [A# (·, D), ϕj ]u − ϕj Alow (·, D)u, where the commutators [A# (·, D), ϕj ] := A# (·, D)ϕj − ϕj A# (·, D) = [A# 1 (·, D), ϕj ] loc in fact do only depend on A# to the localized 1 (·, D). Applying the resolvent of Aj equations we deduce −1 −1 ϕj f + (λ + Aloc ([A# ϕj u = (λ + Aloc j + δ) j + δ) 1 (·, D), ϕj ]u − ϕj Alow (·, D)u).

By multiplying with ψj and by summing up over j we gain the representation u=

N

−1 ψj (λ + Aloc ϕj f j + δ)

j=0

+

N

−1 ψj (λ + Aloc ([A# j + δ) 1 (·, D), ϕj ]u − ϕj Alow (·, D))u.

j=0

Hence, we obtain (I −

N

−1 ψj (λ + Aloc Cj (·, D))u = j + δ)

j=0

where

N

−1 ψj (λ + Aloc ϕj f, j + δ)

j=0

Cj (·, D) := [A# 1 (·, D), ϕj ] − ϕj Alow (·, D)

is a differential operator of lower order, acting on Rn1 only, whose coefficients fulfill the assumptions of Lemma 8.14. We set (8.19)

R0 (λ, δ) :=

N

−1 ψj (λ + Aloc ϕj j + δ)

j=0

and R1 (λ, δ) :=

N j=0

−1 ψj (λ + Aloc Cj (·, D). j + δ)


117

Both R0 (λ, δ) and R1 (λ, δ) map to E := Lp (Rn1 , D(A2 )) ∩

W 1 ,p (Rn1 , W 2 ,p (V )).

1 + 2 ≤1 2m1 2m2

Given u ∈ E, relation (8.18) and Lemma 8.14a) further imply that R1 (λ, δ + δ0 )u E + δ0 R1 (λ, δ + δ0 )u p

≤ C R1 (λ + δ0 , δ )u E + |λ + δ0 | R1 (λ + δ0 , δ )u p ≤ C Cj (·, D)u p ≤ C (ε u 2m1 ,p,Rn1 ,E + C(ε) u p ) ≤

1 2

( u 2m1 ,p,Rn1 ,E + δ0 u p ) ≤

1 2

( u E + δ0 u p )

(λ ∈ Σπ−φ )

for some δ > 0 and provided that δ0 > 0 is sufficiently large. Setting δ := δ + δ0 we see that then Lλ := (I − R1 (λ, δ))−1 R0 (λ, δ) : Lp (Rn1 , E) → D(A) is a left inverse of λ + A + δ which admits an estimate λLλ f p ≤ C f p

(λ ∈ Σπ−φ ).

Therefore, if we can show that there exists a right inverse as well we have proved A + δ ∈ S(X) and φA+δ ≤ φ. To this end, let f ∈ X be arbitrary and consider (λ + A(·, D) + δ)R0 (λ, δ)f = (λ + A# (·, D) + δ)R0 (λ, δ)f + Alow (·, D)R0 (λ, δ)f. The sum on the right can be rewritten as N

−1 ψj (λ + A# (·, D) + δ)(λ + Aloc ϕj f + j + δ)

j=0

where

N

−1 Dj (·, D)(λ + Aloc ϕj f, j + δ)

j=0

Dj (·, D) := [A# 1 (·, D), ψj ] + Alow (·, D)ψj

is again a differential operator of lower order, acting on Rn1 only, whose coefficients fulfill the assumptions of Lemma 8.14. Since supp ψj ⊂ Uj and ψ ≡ 1 on supp ϕj , we obtain (λ + A(·, D) + δ)R0 (λ, δ)f = f + R2 (λ, δ)f with R2 (λ, δ) :=

N j=0

−1 Dj (·, D)(λ + Aloc ϕj . j + δ)

118


Lemma 8.14b) implies R2 (λ, δ) L(X) ≤ 12 and thus existence of (I + R2 (λ, δ))−1 for large enough δ > 0. Consequently, Rλ := R0 (λ, δ)(I + R2 (λ, δ))−1 is a right inverse of λ + A + δ. With the help of the Leibniz rule and the contraction principle of Kahane, from representation (8.19) and relation (8.18) we obtain that R({λρ Dα R0 (λ, δ)}) ≤ C(N + 1). Note that commutators [Dα , ψj ] are involved in this calculation. It is here that we make use of the more general assertion that the R-boundedness condition (8.18) is 1 2 valid for each α ∈ Nn 0 and all ∈ N0 subject to 0 ≤ +2m2 |α |+2m1 |α | ≤ 4m1 m2 |α1 | |α2 | and ρ := 4m1 m2 instead of for ρ := 1 − 2m1 + 2m2 only. In view of Lemma 8.14b) and Lemma 3.2 the representation (λ + A + δ)−1 = R0 (λ, δ)

∞

R2 (λ, δ)i

i=0

as a Neumann series finally gives us R({λρ Dα (λ + A + δ)−1 ; λ ∈ Σπ−φ , 0 ≤ + |α| ≤ 2m}) ∞ ≤ R({λρ Dα R0 (λ, δ)})R R2 (λ, δ)i i=0

≤ (N + 1)C

∞

(N + 1)i (Cε)i =

i=0

(N + 1)C < ∞. 1 − (N + 1)Cε

Hence, the proof of Theorem 8.10 within step 1 is complete.

8.2.2 Step 2: As in the previous step we take into account that Ω is given as the Cartesian product of two domains only. More precisely, we shift the indices 2 and 3 to 1 and 2, respectively. In particular, we consider Ω := Qn1 × V ⊂ Rn , where V is a standard domain in Rn2 . Within this step, we investigate the ν-periodic boundary value problem (A, Bν ) given through

(8.20)

λu + A1 (x, D)u + A2 (y, D)u Bj (y, D)u (Dβ u)|xj =2π − e2πνj (Dβ u)|xj =0

= = =

f 0 0

in Ω, on Qn1 × ∂V, j = 1, . . . , m2 , (j = 1, . . . , n1 , |β| < m1 ).

The induced boundary value problem on V and its Lp (V )-realization remain unchanged. As the Lp (Ω)-realization of the cylindrical ν-periodic boundary value


119

problem (8.20) we define

D(Aν ) := Lp (Qn1 , D(A2 )) ∩

1 ,p Wν,per (Qn1 , W 2 ,p (V )),

1 + 2 ≤1 2m1 2m2

Aν u := A(·, D)u

(u ∈ D(Aν )).

Similar to the preceding step, we first consider the model problem for the cylindrical boundary value problem (8.20), that is, we assume A1 (x, D) on Qn1 to be given as homogeneous differential operator 1 α aα D A1 (D) := |α|=2m

with constant coefficients

a1α

∈ C. We set

A0 (·, D) := A0 (y, D) := A1 (D) + A2 (y, D) and Aν,0 u := A0 (·, D)u

(u ∈ D(Aν,0 ) := D(Aν )).

Again no restrictions on A2 (y, D) have to be assumed. Proposition 8.15. For each φ > ϕ(A0 ,B,ν) we have Aν,0 + δ2 ∈ RS(X) with δ2 = δ2 (φ) as in Proposition 8.3. Moreover, φRS Aν,0 +δ2 ≤ φ and it holds that

(8.21)

R

|α1 |

|α2 |

1− 2m + 2m 1 2

λ

Dα (λ + Aν,0 + δ2 )−1 ; λ ∈ Σπ−φ , α ∈ Nn 0, 0 ≤

|α1 | 2m1

+

|α2 | 2m2

≤1

< ∞.

Proof. Let φ > ϕ(A0 ,B) and consider |α1 |

Mλ (k) := λ

|α2 |

1−( 2m + 2m ) α1 α2 1 2 k D (λ

+ a1 (kν ) + A2 + δ2 )−1 1

2

|α | |α | for λ ∈ Σπ−φ , k ∈ Zn1 and α ∈ Nn 0 such that 0 ≤ 2m1 + 2m2 ≤ 1. Distinguishing the cases ν = 0 and ν = 0, we combine the proofs of Proposition 7.20 and Proposition 8.11 to prove the assertion. For β ≤ α observe the existence of C > 0 such that |α1 | 1

|α2 | 2

1−( 2m + 2m )

λ

|k||ω| |Δω N (k)|

|λ + P (kν )| 1

|β 2 | 2

1− 2m

≤C

holds true for N (k) := kβ , 0 < ω ≤ γ, all k ∈ Zn1 \ {0} and all λ ∈ Σπ−φ due to parameter-ellipticity of P . The validity of Bj (y, D)u = 0 follows from density of T(Qn1 , Lp (V )) in Lp (Ω) (see Proposition 2.4).

120


From Lemma 8.12 we obtain as in the previous step the following result on small variations. 1 1 α1 Corollary 8.16. Let R(x , D) := |α1 |=2m1 rα1 (x )D be given such that the rα1 ∞ < η is satisfied. Set condition |α1 |=2m1

(8.22)

Ava (·, D) := Ava (x, y, D) := A0 (y, D) + R(x, D)

((x, y) ∈ Ω)

and denote its X-realization by va Ava ν u := A (·, D)u

(u ∈ D(Ava ν ) := D(A0 )).

RS Then there exists a δ > 0 such that Ava ≤ φRS ν + δ ∈ RS(X) and φAva A0 +δ2 ν +δ provided that η is sufficiently small. In this case, for φ > ϕ(A0 ,B,ν) we have −1 ; R λρ Dα (λ + Ava ν + δ) (8.23) |α1 | |α2 | < ∞. λ ∈ Σπ−φ , α ∈ Nn 0 , ρ ∈ [0, 1], 0 ≤ ρ + 2m1 + 2m2 ≤ 1

Within this step we use the same strategy of localization as employed in step 1. Again local operators are defined by reflection in order to achieve smallness of variation for all top order coefficients. However, there is the boundary ∂Qn1 which has to be taken into account. We denote by 1 aα (x)Dα A# 1 (x, D) := |α|=2m

the principal part of A1 (x, D). Freezing the coefficients at some arbitrary point x0 ∈ Qn1 , the operator A# 1 (x0 , D) fulfills the assumptions of Proposition 8.15 and we denote by η = η(x0 ) the constant from Corollary 8.16 applied to A# 1 (x0 , D). For the given non-constant top order coefficients a1α1 we consider the periodic extensions a1α1 ,per to Rn1 . Note that we have a1α1 ,per ∈ BU C(Rn1 ) for all α1 such that |α1 | = 2m1 due to the periodicity assumptions. In particular, there exists a length d = d(x0 ) such that |a1α1 (x) − a1α1 (x0 )| ≤ η(x0 )/Mα

(|x − x0 |∞ ≤ d(x0 ), |α1 | = 2m1 ).

Without loss of generality, we may assume d(x0 )
0 such that Aloc ν,j + δ ∈ RS(X) and we have −1 ; R λρ Dα (λ + Aloc ν,j + δ) (8.24) |α1 | |α2 | < ∞. λ ∈ Σπ−φ , α ∈ Nn 0 , ρ ∈ [0, 1], 0 ≤ ρ + 2m1 + 2m2 ≤ 1 ∞ n1 We choose a partition of unity (ϕj )N ) subordinate to the open j=1 ⊂ C (R such that 0 ≤ ϕ ≤ 1. In addition, we fix ψj ∈ C ∞ (Rn1 ) such covering (Uj )N j j=1


123

that ψj ≡ 1 on supp ϕj and supp ψj ⊂ Uj . Again we think of ϕj and ψj as continued on the opposite side of Qn1 in case Uj intersects with the boundary of Qn 1 .

(a)

(b)

(c)

Figure 8.2: Reflections depending on intersections with the boundary.

With Cj (·, D) and Dj (·, D) defined similar to the preceding step we set (8.25)

R0 (λ, δ) :=

N

−1 ψj (λ + Aloc ϕj , ν,j + δ)

j=0

R1 (λ, δ) :=

N

−1 ψj (λ + Aloc Cj (·, D), ν,j + δ)

j=0

and R2 (λ, δ) :=

N j=0

−1 Dj (·, D)(λ + Aloc ϕj . ν,j + δ)

124


Thanks to the definition of ψj for the case that Uj intersects with the boundary of Qn1 , both R0 (λ, δ) and R1 (λ, δ) map to E := Lp (Qn1 , D(A2 )) ∩

2m

,p Wν,per (Qn1 , W 2m−,p (V )).

=1

Just as in step 1 we deduce existence of δ > 0, such that (I − R1 (λ, δ))−1 R0 (λ, δ) defines a left and R0 (λ, δ)(I + R2 (λ, δ))−1 a right inverse of λ + Aν + δ. Again a Neumann series argument shows (8.11). This proves the assertion within step 2.

8.2.3 Step 3: We can now combine the results from step 1 and step 2 to prove Theorem 8.10. Proof of Theorem 8.10. We apply step 2 to the ν-periodic boundary value problem (A, Bν ) given through A23 (x2 , x3 , D) := A2 (x2 , D) + A3 (x3 , D) supplemented with Bj (x3 , D), j = 1, . . . , m3 and observe validity of the R-boundedness condition (8.11) for its Lp (Qn2 × V, F )-realization Aν . Hence, the claim follows from step 1 replacing A2 on V there by A23 on Qn2 × V . In the subsequent lines we comment on generalizations of Theorem 8.10. Remark 8.17. We point out that these proofs easily generalize to the case of different p-integrability with respect to Rn1 , Qn2 and V (cf. Remark 10.7). Remark 8.18. As frequently mentioned in the proof of Theorem 8.10, it is a strength of the Fourier multiplier approach that the induced boundary value problem on V does not have to be investigated during the localization process performed above. Indeed, it can be treated separately for our proof only relies on the R-boundedness estimates (8.4) established in [DHP03]. Therefore, with a Banach space F of class HT that enjoys property (α), we can also treat F valued boundary value problems with L(F )-valued coefficients in A3 and B3,j , j = 1, . . . , m3 . Remark 8.19. Using our results from Chapters 6 and 7 on the transference of an RH∞ -calculus and the results from [DDH+ 04] stated in Proposition 8.4, we can similarly prove an RH∞ -calculus for the model problem of our partially ν-periodic cylindrical boundary value problem (8.7). Then, the coefficients of A3 and B3,j , j = 1, . . . , m3 , have to match the assumptions of Proposition 8.4. In that case, non-constant coefficients of A1 and A2 can be treated by means of a localization procedure again. However, perturbation arguments for the H∞ -calculus are known to be much more involved (see e.g. [DDH+ 04] or [KKW06]). We thus restrict ourselves to what has been presented so far and refer to Chapter 10 for results on the RH∞ -calculus for cylindrical boundary value problems.


125

Remark 8.20. Also observe that to some extent we can weaken the assumption that (A, Bν ) is cylindrical. Indeed, consider homogeneous differential operators with constant coefficients A1 (Dx1 ), Q1 (Dx1 ), A2 (Dx2 ), and Q2 (Dx2 ). Furthermore, let (A3 , B3 ) define a boundary value problem as described in Remark 8.18. Then with the aid of the results from Chapter 6 and Chapter 7, we can treat differential operators

A(x3 , D) := A1 (Dx1 ) + Q1 (Dx1 ) A2 (Dx2 ) + Q2 (Dx2 )A3 (x3 , Dx3 ) . Remark 8.21. Assume A(x, D) to have constant coefficients with respect to one of its parts Ai , i = 1, 2. As in the previous chapters, additional results by means of reflection techniques can be obtained provided Ai (D) is even. To some extent, the reflection techniques can be applied successfully even in case of non-constant coefficients. For instance, consider a second order parameter-elliptic differential operator in divergence form 1+n2

A(x, D) := −

∂i aij (x)∂j

i,j=1

in Ω := R+ × V , V ⊂ Rn2 . On the lateral surface we impose Dirichlet or Neumann boundary conditions. On {0} × V in what follows we impose Neumann boundary conditions which are obtained by an even extension of the data. Dirichlet boundary conditions with respect to {0} × V can be obtained similarly using an odd extension instead. Set a(x) := aij (x) i,j=1,...,n , where n := 1 + n2 . In order to enter the context of cylindrical boundary value problems we assume that ⎛ ⎞ 0 ... 0 a11 (x1 ) ⎜ 0 a22 (x2 ) . . . a2n (x2 ) ⎟ ⎜ ⎟ a11 (x1 ) 0 ⎜ . ⎟. a(x) = := . . 2 ⎜ ⎟ .. .. 0 a ˜(x ) ⎝ .. ⎠ 0 an2 (x2 ) . . . ann (x2 ) Let aij ∈ BU C 1 (Ω) and assume that a11 (∞) := limx1 →∞ a11 (x1 ) exists. This yields A(x, D) := −a11 (x1 )Δ1 −

n

n

aij (x2 )∂i ∂j − ∂1 a11 (x1 )∂1 − ∂i aij (x2 )∂j

i,j=2

and so

i,j=2

A(x, D) := −a11 (x1 )Δ1 − ∂1 a11 (x1 )∂1 + A2 (x2 , D2 ),

where A2 (x2 , D), acting on V , is defined by A2 (x2 , D) := −

n2 i,j=1

a ˜ij (x2 )∂i ∂j −

n2

n2

˜ij (x2 )∂i = − ∂i a ˜ij (x2 )∂j . ∂j a

j=1

i,j=1

126


Note that A2 is of divergence form again. The operator

A1 (x1 , D1 ) := −a11 (x1 )Δ1 − ∂1 a11 (x1 )∂1 = −∂1 a11 (x1 )∂1 is of even structure in the following sense: consider the Dirichlet-Neumann type boundary value problem for A(x, D) on Ω as explained above and let f ∈ Lp (Ω) be any given right-hand side. We extend the boundary value problem to the whole of R × V by means of even extensions of f to F as well as of a11 to A11 . This yields A11 ∈ C(R) as well as limx1 →±∞ A11 (x1 ) = a11 (∞). Moreover, ∂1 A11 exists classically on R \ {0}, is odd with respect to x1 , and belongs to L∞ (R × V ). Let U define the unique solution to the (possibly shifted) extended problem on R × V and set V (x) := U (−x1 , x2 ). Then A11 (x1 )(Δ1 V )(x) = A11 (−x1 )(Δ1 U )(−x1 , x2 ) and

∂1 A11 (x1 )∂1 V (x) = − ∂1 A11 (−x1 ) − ∂1 U )(−x1 , x2 )

= ∂1 A11 (−x1 ) ∂1 U )(−x1 , x2 ).

Therefore, Aδ (x, D)V (x) = Aδ (x, D)U (−x1 , x2 ) = F (−x1 , x2 ) = F (x) and by uniqueness U = V , i.e. U ∈ W 2,p (R × V ) is even with respect to x1 . Hence, u := U |Ω is a solution to the original (possibly shifted) boundary value problem subject to Neumann boundary conditions with respect to {0} × V . Along the same lines we can treat the according boundary value problem in Ω := (0, π) × V subject to mixed Dirichlet-Neumann boundary conditions. Here we make use of our previous results on periodic respectively antiperiodic boundary value problems applied to the extended problem in (0, 2π) × V . We do not specify all possible boundary value problems which are covered by the results obtained by reflection so far but rather focus on the Laplacian at the end of this chapter.

8.3 A focus on the Laplacian In what follows let ni ∈ N0 , i = 1, . . . , 5, n := Ω := R Given ν ∈ i(−1, 1)

n2

(8.26)

n1

× Q n2 × K

n3

5 i=1

ni , V ⊂ Rn5 , and

˜ n4 × V. ×Q

, we consider the boundary value problem for the Laplacian

λu − Δu B3 u B4 u B5 u 2πνj u|xj =0 u|xj =2π − e ∂j u|xj =2π − e2πνj ∂j u|xj =0

= = = = = =

f 0 0 0 0 0

in Ω, ˜ n4 × V, on Rn1 × Qn2 × ∂Kn3 × Q ˜ n4 × V, on Rn1 × Qn2 × Kn3 × ∂ Q ˜ n4 × ∂V, on Rn1 × Qn2 × Kn3 × Q j = n1 + 1, . . . , n1 + n2 , j = n1 + 1, . . . , n1 + n2 .

8.3 A focus on the Laplacian

127

Here the boundary operator B := {Bi ; i = 3, 4, 5} endows the boundary value problem for the Laplacian with mixed Dirichlet-Neumann boundary conditions on ˜ n4 × V ). Rn1 × Qn2 × ∂(Kn3 × Q We point out that on each part of this boundary the types of boundary conditions might be different. On ˜ n4 ) × V, Rn1 × Qn2 × Kn3 × ∂(Q in particular, we can even distinguish between opposite sides of the boundary. The Laplacians with respect to each of the five components of Ω are denoted by Δi , i = 1, . . . , 5. Here we consider different Lp -realizations of these operators. For i = 1, . . . , 4 the domains of definition are defined as ˜ n4 × V )), D(Δp,1 ) := W 2,p (Rn1 , Lp (Qn2 × Kn3 × Q 2,p ˜ n4 × V )), D(Δp,2 ) := Wν,per (Qn2 , Lp (Kn3 × Q ˜ n4 × V )); B3 u = 0 , D(Δp,3 ) := u ∈ W 2,p (Kn3 , Lp (Q ˜ n4 , Lp (V )); B4 u = 0 . D(Δp,4 ) := u ∈ W 2,p (Q

For i = 5 we consider either the strong Dirichlet or Neumann Laplacian Δp,s,5 or the weak Dirichlet or Neumann Laplacian Δp,w,5 as defined in Section 8.1. For all operators the same notation will be used for their canonical extensions to Lp (Ω). Without further explanations we also employ the notation Δu = Δ1 u + . . . + Δ5 u, where Δi acts on the according component of Ω. We define the strong Laplacian subject to the boundary conditions (B, ν) to be D(Δp,s,B,ν ) :=

4

D(Δp,i ) ∩ D(Δp,s,5 )

i=1

∩

2

2−m

,p ˜ n4 × V ) , W m,p Rn1 , Wν,per Qn2 , W 2−m−,p (Kn3 × Q

m=0

=0

(u ∈ D(Δp,s,B,ν )).

Δp,s,B,ν u := Δu

Additionally, we will consider the weak Laplacian subject to the boundary conditions (B, ν) defined as D(Δp,w,B,ν ) :=

4

D(Δp,i ) ∩ D(Δp,w,5 )

i=1

∩

2

W

m,p

m=0

Δp,w,B,ν u := Δu

R

n1

,

2−m

,p Wν,per Qn2 , W 2−m−,p (Kn3

=0

(u ∈ D(Δp,w,B,ν )).

˜ n4 , Lp (V )) ×Q

,

128


˜ n4 , Lp (V )) ∩ W 1,p (Ω) only, Note that D(Δp,w,B,ν ) ⊂ W 2,p (Rn1 × Qn2 × Kn3 × Q whereas D(Δp,s,B,ν ) ⊂ W 2,p (Ω). Theorem 8.22. a) Let 1 < p < ∞ and let V ⊂ Rn5 be a C 2 standard domain. Then the strong Laplacian subject to the boundary conditions (B, ν) fulfills −Δp,s,B,ν ∈ ΨRH∞ (Lp (Ω))

and

π . 2

φR∞ −Δp,s,B,ν
0 depending only on the Lipschitz character of V such that the weak Laplacian subject to the boundary conditions (B, ν) fulfills −Δp,w,B,ν ∈ ΨRH∞ (Lp (Ω))

and

φR∞ −Δp,w,B,ν
0 such that {kν ; k ∈ Zn \ {0}} ⊂ Rn \ Bε (0). Hence, there exists C > 0 such that |α|+|β| 1− |α|+|β| 2 kω Δω Sβ (k) ≤ C|λ + T (k)|1− 2 λ

(k ∈ Zn \ {0}, λ ∈ Σπ−φ )

due to parameter-ellipticity of T . By Kahane’s contraction principle the claim follows. We agree to simply write E for the set Const(Qn , E) of constant, E-valued func1,p 1,p (Qn , E) := Wper (Qn , E) ∩ Lp(0) (Qn , E) tions defined on Qn . Let further W(0),per denote the subset of functions of mean value zero, that is, the set of functions 1,p (Qn , E) such that fˆ(0) = 0. f ∈ Wper Theorem 9.5. For the Stokes resolvent problem (9.2) there exists a unique solution u

∈

2

,p Wν,per (Qn , W 2−,p (Rm ))n+m ,

=0

p

∈

1,p (Qn , Lp (Rm )), ν = 0 or fˆ(0) = 0, W 1,p (Ω) ∩ Wν,per

1,p 1,p ˆ 1,p (Rm ), else. (Qn , Lp (Rm )) + W W (Ω) ∩ W(0),per

We have ∇2 u p +

√

λ ∇u p + λ u p + ∇p p ≤ C f p

and additionally p p ≤ C f p if ν = 0 or fˆ(0) = 0. Moreover, for each φ > 0 the family of solution operators Rλ f := u fulfills |α| < ∞. R λ1− 2 ∂ α Rλ ; λ ∈ Σπ−φ , 0 ≤ |α| ≤ 2

9 Application to the Stokes equation

139

Proof. Lemma 9.4 and Theorem 3.24 prove Mλα,β to define a Fourier multiplier for λ ∈ Σπ−φ and 0 ≤ |α| + |β| ≤ 2. Let ν = 0. Then by the last assertion of Lemma 9.4 we see that

−1 T ∇kν Q∗ : Zn → L Lp (Rm )n+m , Lp (Rm ) ; k → Q∗kν := |kν |2 − Δ and

−1 T ∇kν Q : Zn → L Lp (Rm )n+m ; k → ∇kν Q∗kν = ∇kν |kν |2 − Δ define Fourier multipliers. Hence,

1,p (Qn , Lp (Rm )) TQ∗ ∈ L Lp (Ω)m+n , Lp (Qn , W 1,p (Rm )) ∩ Wν,per

and u := eν· Tλ−1 M 0,0 e−ν· f and p := −eν· TQ∗ e−ν· f define the solution of λ

(9.2) for fixed λ ∈ Σπ−φ . Recall that we dropped the factor e−ν· right after (9.3). If ν = 0, we still have

(k ∈ Zn ) Q(k) = ∇k Q∗k ∈ L Lp (Rm )n+m and

Q∗ (k) = Q∗k ∈ L Lp (Rm )n+m , Lp (Rm )

However,

(k ∈ Zn \ {0}).

ˆ 1,p (Rm )). Q∗ (0) = Q∗0 ∈ L(Lp (Rm )n+m , W

We therefore define Q∗I (k) :=

0,

k = 0,

Q∗k ,

k = 0

and

Q∗II (k) :=

Q∗0 ,

k = 0,

0,

k = 0.

Then ˆ 1,p (Rm )) Q∗I : Zn → L(Lp (Rm )n+m , Lp (Rm )) and Q∗II : Zn → L(Lp (Rm )n+m , W define Fourier multipliers. Note that by their structure TQ∗I maps to functions of mean value zero, whereas TQ∗II maps to constant functions which of course are periodic. Thus, we have

1,p (Qn , Lp (Rm ) TQ∗I ∈ L Lp (Ω)m+n , Lp (Qn , W 1,p (Rm )) ∩ W(0),per as well as

ˆ 1,p (Rm ) . TQ∗II ∈ L Lp (Ω)m+n , W

Consequently, for each λ ∈ Σπ−φ we find as above that u := eν· Tλ−1 M 0,0 e−ν· f λ

and p := −eν· TQ∗I f −TQ∗II e−ν· f solve problem (9.2). Uniqueness finally follows from the uniqueness theorem for Fourier series.

140


In what follows we aim for the definition of a ν-periodic Helmholtz decomposition. To this end, we set 1,p (Qn , Lp (Rm )) , ν = 0, ∇p; p ∈ W 1,p (Ω) ∩ Wν,per p Gν,per (Ω) :=

1,p ˆ 1,p (Rm ) , else. (Qn , Lp (Rm )) + W ∇p; p ∈ W 1,p (Ω) ∩ W(0),per (Rn+m ) of partially ν-periodic distributions and its propRecall the class Dν,per,n (Rn+m ) if T ∈ Dν,per (Rn+m ). erties from Lemma 2.21. In particular, div T ∈ Dν,per For the sake of convenience in this chapter we agree to write Dν,per (Ω) instead of Dν,per,n (Rn+m ). Furthermore, we drop the argument k and write fˆ only instead ˆ of f (k) if an assertion is valid for all k ∈ Zn .

Lemma 9.6. For ν ∈ i(−1, 1)n define PΩ,ν := eν· TI+Qkν e−ν· . Then PΩ,ν ∈ L(Lp (Ω)n+m ) defines a projection in Lp (Ω)n+m such that N (PΩ,ν ) = Gpν,per (Ω) and R(PΩ,ν ) = {f ∈ Lp (Ω)n+m ; divf = 0 in Dν,per (Ω)}.

Proof. Due to Lemma 9.4 I + Qkν defines a discrete Fourier multiplier. In particular, PΩ,ν ∈ L(Lp (Ω)n+m ) is well-defined. Since 2 −1 ∇T = −I, kν ∇kν |kν | − Δ we have (I + Qkν )(I + Qkν ) = I + 2Qkν + Qkν Qkν = I + Qkν and PΩ,ν defines a projection. In what follows we consider partial Fourier coefficients to prove the equalities for the kernel and range of PΩ,ν . Due to Lemma 2.21 we have −ν·

e div(eν· TI+Qkν e−ν· f ) ˆ = ikνT TI+Qkν e−ν· f ˆ + ∇T TI+Qkν e−ν· f ˆ −ν· f ˆ = 0, = ∇T kν (I + Qkν ) e where we have used (9.6) in the last line. Hence, div(eν· TI+Qkν e−ν· f ) = 0 in −ν· (Ω). On the other hand Dν,per −ν·let divf = 0 in Dν,per (Ω). Then e divf = 0 in T Dper (Ω) and therefore ∇kν e f ˆ = 0 which implies

Qkν e−ν· f ˆ = 0. Thus,

TI+Qkν e−ν· f ˆ = (I + Qkν ) e−ν· f ˆ = e−ν· f ˆ

and eν· TI+Qkν e−ν· f = f


141

by means of the representation formula from Theorem 2.19. Altogether we have (Ω)}. shown R(PΩ,ν ) = {f ∈ Lp (Ω); divf = 0 in Dν,per p p Now let f ∈ Gν,per (Ω), i.e. f = ∇p ∈ L (Ω). This implies

TI+Qkν e−ν· f ˆ = TI+Qkν e−ν· ∇p ˆ = (I + Qkν ) e−ν· ∇p ˆ

−1 T −ν· ∇kν ∇kν e p ˆ = 0. = (I + Qkν )∇kν e−ν· p ˆ = I + ∇kν |kν |2 − Δ

In turn consider f ∈ Lp (Ω)n+m such that PΩ,ν f = 0. Then (I + Qkν ) e−ν· f ˆ = 0 and

e−ν· f ˆ = −Qkν e−ν· f ˆ

−1 T = ∇kν |kν |2 − Δ ∇kν − e−ν· f ˆ = − e−ν· ∇TUkν ,0 ∇T f ˆ.

Hence, f = ∇p, where p := −TQ∗k f ∈ Gpν,per (Ω). Here just as in the proof of ν Theorem 9.5 we decompose Q∗ = Q∗I + Q∗II in case ν = 0. Thus, the kernel of p PΩ,ν is equal to Gν,per (Ω) which finishes the proof. We refer to the projection from Lemma 9.6 as the ν-periodic Helmholtz projection. It implies the decomposition Lp (Ω) = Lpσ,ν,per (Ω) ⊕ Gpν,per , where Lpσ,ν,per (Ω) := R(PΩ,ν ).

(9.8)

With the aid of this ν-periodic Helmholtz decomposition, we are now in the position to define the ν-periodic Stokes operator in Lpσ,ν,per (Ω). Definition 9.7. For ν ∈ i(−1, 1)n we define the ν-periodic Stokes operator AS,ν in Lpσ,ν,per (Ω) to be D(AS,ν )

:=

AS,ν u

:=

2

,p Wν,per (Qn , W 2−,p (Rm ))n+m ∩ Lpσ,ν,per (Ω)

=0

−PΩ,ν Δu

(u ∈ D(AS,ν )).

As a consequence of the following lemma, just as in Rm , the ν-periodic Stokes Lp (Ω)n+m -realization of the ν-periodic operator AS,ν is given as a part of the & ,p (Qn , W 2−,p (Rm ))n+m . negative Laplacian −Δν with D(Δν ) := 2=0 Wν,per & ,p (Qn , W 2−,p (Rm ))n+m . Then Lemma 9.8. Let ν ∈ i(−1, 1)n and u ∈ 2=0 Wν,per ∂ α PΩ,ν u = PΩ,ν ∂ α u

(α ∈ Nn+m , |α| ≤ 2). 0

142


Proof. Let α = (α, α ) ∈ Nn+m . In what follows we write e−νx instead of e−ν· to 0 indicate the dependency on x ∈ Rn only. Then Lemma 2.21 shows

e−νx ∂ α PΩ,ν u ˆ = e−νx ∂ α eνx TI+Qkν e−νx u ˆ = (ikν )α ∂ α (I + Qkν ) e−νx u ˆ

as well as −νx

e PΩ,ν ∂ α u ˆ = (I + Qkν ) e−νx ∂ α u ˆ = (I + Qkν )(ikν )α ∂ α e−νx u ˆ.

Since ∂ α and Qkν commute, the assertion follows. For the ν-periodic Stokes operator AS,ν we therefore have (9.9)

AS,ν = −Δν |D(AS,ν )

and

(λ − AS,ν )−1 = (λ + Δν )−1 |Lpσ,ν,per (Ω) .

From that we deduce the following result on the RH∞ -calculus of the ν-periodic Stokes operator. Theorem 9.9. Let ν ∈ i(−1, 1)n . Then we have AS,ν ∈ RH∞ (Lpσ,ν,per (Ω)) and φR∞ AS,ν = 0. Proof. Thanks to the observation in (9.9) this follows from −Δν ∈ RH∞ (Lp (Ω)) with φR∞ −Δν = 0 by Theorem 8.22 and the subsequent remarks. In the sequel we prove existence of the standard Helmholtz decomposition ˜ For technical reasons we choose Ω ˜ := Q ˜ n × Rm , where for the space Lp (Ω). n ˜ Qn := (0, π) . To this end, we make use of the periodic Helmholtz projection ˜ PΩ,per ∈ L(Lp (Ω)) to prove existence of the Helmholtz projection PΩ˜ ∈ L(Lp (Ω)). Moreover, with suitable extension and restriction operators E and R we establish the representation formula PΩ˜ = RPΩ,per E.

˜ (blue) and Ω (gray). Figure 9.1: The domains Ω


143

First we give an equivalent representation of Lpσ,per (Ω). Given f ∈ Lp (Ω) such that div f ∈ Lp (Ω), the expression

γn·f (ω) := f · ∇ϕ + ϕ divf ϕ ∈ W 1,p (Ω) Ω

Ω

is well-defined. Here n denotes the outer normal, ω = γ(ϕ) denotes the trace of older conjugate of p, i.e. ϕ on ∂Ω (see e.g. [Gal94, Section III.2]), and p is the H¨ 1 1 ˆ 1,p (Ω) as the + = 1. If div f = 0, we can extend γ (ω) formally to ϕ ∈ W n·f p p integral Ω f · ∇ϕ is still well-defined. We abbreviate

1,p 1,p Wper (Ω) := W 1,p (Ω) ∩ Wper (Qn , Lp (Rm )) ⊂ W 1,p (Ω) and 1,p

1,p 1,p ˆ 1,p (Rm ) ⊂ W ˆ 1,p (Ω). (Ω) ∩ W(0),per (Qn , Lp (Rm )) + W Wper, ˆ (Ω) := W + For f subject to div f = 0 in Ω we define periodic traces by means of

1,p (9.10) γper f = 0 :⇐⇒ γn·f (ω) = 0 ϕ ∈ Wper, ˆ (Ω) . +

1,p Note that the trace ω = γ(ϕ) of ϕ ∈ Wper, ˆ (Ω) is a well-defined element in the +

1−1/p ˆ 1,p (∂Ω). This extends the notion of periodic boundary space Wp (∂Ω) + W conditions as known from smooth functions. Precisely, consider the following class

Dσ,per (Ω) := {ϕ ∈ C ∞ (Ω); div ϕ = 0, ∞ ∀y ∈ Rm : Dyα ϕ(·, y)per ∈ Cper (Qn )

(α ∈ Nm 0 ),

∃U = Uϕ ⊂ Rm bounded : ∀x ∈ Qn : supp ϕ(x, ·) ⊂ U } of smooth, divergence-free, partially periodic functions with tangentially bounded support. Recall the 2π-periodic extension gper of a function g ∈ Lp (Qn ) from Chapter 2. Let f ∈ Dσ,per (Ω), U = Uf ⊂ Rm and ΩU = Qn × U . Then γn·f (ω) = f · ∇ϕ + ϕ divf = f · ∇ϕ + ϕ divf = ωf · n Ω

Ω

ΩU

ΩU

∂ΩU

by the Gauß identity for bounded Lipschitz domains (see [Gal94, Section II.3]). We decompose the boundary of ΩU into ∂ΩU = ∂Qn × U ∪ Qn × ∂U . By our choice of f and U it follows that ωf · n vanishes on Qn × ∂U . The remaining part can be decomposed further using ∂Qn =

n

− Q+ n,j ∪ Qn,j ,

j=1

where Q+ n,j := Qj−1 × {2π} × Qn−j

and

Q− n,j := Qj−1 × {0} × Qn−j .

144


Then

n

ωf · n =

∂ΩU

=

n j=1

and so

ωf · n =

∂Qn ×U

Q+ n,j ×U

ωf · ej −

ωf · n = ∂ΩU

j=1

Qn−1 ×U

ωf · ej =

n,j ×U

Q− n,j ×U

n j=1

Q+ n,j ×U

ω|Q−

n,j ×U

f j | Q+

n,j ×U

ωf · n

ωfj −

− f j | Q−

Q− n,j ×U

ωfj

,

n,j ×U

= ω|Q−

where we have used ω|Q+

Q+ n,j ×U

Q− n,j ×U

n j=1

ωf · n +

n,j ×U

1,p for ω = γ(ϕ), ϕ ∈ Wper (Ω). Since

1,p (Ω) can be chosen arbitrarily, we conclude ϕ ∈ Wper

⇐⇒

γper f = 0

f j | Q+

n,j ×U

= f j | Q−

(j = 1, . . . , n).

n,j ×U

In the sequel we frequently denote the periodic Helmholtz projection on Lp (Ω) by PΩ,per,p = TI+Qk,p . This is done in order to emphasize the underlying Lp -space. Lemma 9.10. The periodic Helmholtz projection fulfills PΩ,per,p = PΩ,per,p .

Proof. Let u ∈ T(Qn , Lp (Rm )) andv ∈ T(Qn , Lp (Rm )) denote two arbitrary trigonometric polynomials, say u = k∈[−α,α] ek ⊗ uk and v = k∈[−α,α] ek ⊗ vk

with uk ∈ Lp (Rm ) and vk ∈ Lp (Rm ), respectively. Then u(x, y) := k∈[−α,α] eikx uk (y) and v(x, y) := k∈[−α,α] eikx vk (y) define

elements of Lp (Ω) and Lp (Ω), respectively. Moreover, we have ek ⊗(I +Qk,p )uk and PΩ,per,p v = ek ⊗(I +Qk,p )vk . PΩ,per,p u = k∈[−α,α]

k∈[−α,α]

This allows for the calculation

PΩ,per,p u, v Ω := (PΩ,per,p u)(x, y)v(x, y)d(x, y) Ω

= eikx (I + Qk,p )uk (y) =

Ω k∈[−α,α]

k∈[−α,α]

eikx

Ω k∈[−α,α]

=

e

k,j∈[−α,α]

= (2π)n

i(k+j)x

Qn

k∈[−α,α]

e

j∈[−α,α]

eikx vk (y)d(x, y)

ijx

(I + Qk,p )uk (y)vj (y)d(x, y)

(I + Qk,p )uk (y)vj (y)dy

dx Rm

(I + Qk,p )uk , v−k

Rm

.


145

Since Qk,p = Q−k,p holds true for all k ∈ Zn , we deduce

(I + Qk,p )uk , v−k Rm PΩ,per,p u, v Ω = (2π)n = (2π)

n

k∈[−α,α]

uk , (I + Q−k,p )v−k

Rm

= u, PΩ,per,p v Ω .

k∈[−α,α]

Thanks to density of trigonometric polynomials (see Proposition 2.4) the claim follows for arbitrary u ∈ Lp (Ω) and v ∈ Lp (Ω). Proposition 9.11. For Lpσ,per (Ω) defined as in (9.8) we have Lpσ,per (Ω) = {f ∈ Lp (Ω); divf = 0 in Ω, γper f = 0}. Proof. First let f ∈ Lpσ,per (Ω). Trivially div f = 0 in Ω. Moreover, for arbitrary 1,p ϕ ∈ Wper, ˆ (Ω) we have +

f, ∇ϕ

Ω

= PΩ,per,p f, ∇ϕ Ω = f, PΩ,per,p ∇ϕ Ω = 0.

Thus, γper f = 0 by (9.10). Now let f ∈ Lp (Ω) fulfill divf = 0 in Ω and γper f = 0. Then

1,p 0= f · ∇ϕ + ϕ divf = f · ∇ϕ = f, ∇ϕ Ω ϕ ∈ Wper, ˆ (Ω) . + Ω

Ω

Ω

Since N (PΩ,per,p ) = R(I − PΩ,per,p ), for each v ∈ Lp (Ω) it holds that

0 = f, (I − PΩ,per,p )v Ω = (I − PΩ,per,p )f, v Ω . Thus, f = PΩ,per,p f ∈ R(PΩ,per,p ) = Lpσ,per (Ω). Recall the standard Helmholtz decomposition in Lp (Ω) given by Lp (Ω) = Lpσ (Ω) ⊕ Gp (Ω), where Lpσ (Ω) = {f ∈ Lp (Ω)n+m ; divf = 0 in Ω, γf ·n = 0} and

Gp (Ω) = {∇p; p ∈ L1loc (Ω), ∇p ∈ Lp (Ω)n+m },

as well as the standard Helmholtz projection PΩ ∈ L(Lp (Ω)) which fulfills PΩ (Lp (Ω)) = Lpσ (Ω)

and

N (PΩ ) = Gp (Ω).

˜ from the Our aim is to derive the standard Helmholtz projection in L(Lp (Ω)) periodic Helmholtz projection in L(Lp (Ω)). As mentioned, we scale the underlying ˜ n := (0, π)n . ˜ := Q ˜ n × Rm with Q rectangular domain and consider Ω

146


A few preparations are in order. Consider u ∈ Lp ((0, π) × G)m+1 with an arbitrary domain G ⊂ Rm . Define U ∈ Lp ((0, 2π) × G)m+1 by ⎧ ⎪ u(x, y), (x, y) ∈ (0, π) × G, ⎪ ⎨

U (x, y) := E1 u (x, y) := −u1 ⎪ (2π − x, y), (x, y) ∈ (π, 2π) × G. ⎪ ⎩ u If u ∈ C 1 ((0, π) × G)m+1 , then div U (x, y) = div U (2π − x, y)

((x, y) ∈ (0, π) × G).

Iteratively, for u ∈ Lp ((0, π)n × G)m+n we define U ∈ Lp ((0, 2π)n × G)m+n by U := Eu := En · · · E1 u, where the Ej are defined to extend the j-th variable accordingly. ˜ m+n . Then we have u ∈ Lpσ (Ω) ˜ if and only if Lemma 9.12. Let u ∈ Lp (Ω) p U := Eu ∈ Lσ,per (Ω). Proof. It is sufficient to consider the case n = 1 as the assertion for arbitrary n ∈ N follows with minor adjustments by iteration. ˜ We show div Uper = 0 in Dper (Ω). To this end, let ϕ ∈ First let u ∈ Lpσ (Ω). 1+m ) be arbitrary and ∈ N such that supp ϕ ⊂ (−2π, 2π) × Rm . Then D(R

m+1

Uper (x, y)∇ϕ(x, y)d(x, y) = Ω

Rm

=

−1 i=−

Rm

m+1

Uper,j (x, y)∂j ϕ(x, y)dxdy

(−2π,2π) j=1

Uj (x, y) ∂j ϕ (x + 2iπ, y)dxdy.

(0,2π) j=1

It therefore suffices to prove

Rm

m+1

Uj (x, y)∂j ϕ(x, y)dxdy = 0

(0,2π) j=1

for arbitrary ϕ ∈ W 1,p (Ω). Obviously

m+1

Uj (x, y)∂j ϕ(x, y)dx =

(0,2π) j=1

m+1

uj (x, y)∂j ϕ(x, y)dx

(0,π) j=1

+ (π,2π)

−u1 (2π − x, y)∂1 ϕ(x, y) +

m+1 j=2

uj (2π − x, y)∂j ϕ(x, y)dx


147

holds true for U = E1 u. A substitution of variables and the relation

∂1 ϕ (2π − x, y) = −∂x ϕ(2π − x, y)

(9.11) yields

m+1

m+1

Uj (x, y)∂j ϕ(x, y)dx =

(0,2π) j=1


(0,π) j=1

+ (0,π)

m+1

−u1 (x, y) ∂1 ϕ (2π − x, y) + uj (x, y)∂j ϕ(2π − x, y)dx. j=2

˜ by We define ψ ∈ W 1,p (Ω)

ψ(x, y) := ϕ(x, y) + ϕ(2π − x, y). Then

Rm

m+1

Uj (x, y)∂j ϕ(x, y)dxdy =

(0,2π) j=1

˜ Ω

u(x, y)∇ψ(x, y)d(x, y) = 0

˜ due to u ∈ Lpσ (Ω). ˜ Given ϕ ∈ W 1,p (Ω) ˜ we In turn let U ∈ Lpσ,per (Ω) which implies div u = 0 in Ω. set ϕ(x, y), (x, y) ∈ (0, π) × Rm , Φ(x, y) := ϕ(2π − x, y), (x, y) ∈ (π, 2π) × Rm .

1,p Due to the fact that this extension is even it holds that Φ ∈ Wper (Ω). Hence,

m+1

U (x, y)∇Φ(x, y)d(x, y) =

0= Ω

Rm

Uj (x, y)∂j Φ(x, y)dxdy

(0,2π) j=1

by the fact that U ∈ Lpσ,per (Ω), Proposition 9.11, and the definition of γper in (9.10). The structures of U and Φ further allow for the calculation

m+1

Uj (x, y)∂j Φ(x, y)dx =

(0,2π) j=1

m+1

+ (π,2π)


(0,π) j=1

−u1 (2π − x, y)∂x ϕ(2π − x, y) +

m+1 j=2

uj (2π − x, y)∂j ϕ(2π − x, y)dx.

148


In the second integral on the right-hand side we employ relation (9.11) to arrive at m+1 m+1 Uj (x, y)∂j Φ(x, y)dx = uj (x, y)∂j ϕ(x, y)dx (0,2π) j=1

+ (π,2π)

(0,π) j=1 m+1

u1 (2π − x, y) ∂1 ϕ (2π − x, y) + uj (2π − x, y)∂j ϕ(2π − x, y)dx. j=2

Altogether this yields m+1 uj (x, y)∂j ϕ(x, y)dxdy 0= Rm

(0,π) j=1

u1 (x, y)∂1 ϕ(x, y) +

+ Rm

=2

˜ Ω

(0,π)

m+1

uj (x, y)∂j ϕ(x, y)dxdy

j=2

u(x, y)∇ϕ(x, y)d(x, y)

and the proof is complete. ˜ m+n . Then u ∈ Gp (Ω) ˜ if and only if Eu ∈ Gpper (Ω). Lemma 9.13. Let u ∈ Lp (Ω) 1,p Proof. First let Eu ∈ Gpper (Ω), that is, Eu = ∇p with p ∈ Wper, ˆ (Ω). Thus, +

˜ u = ∇p|Ω˜ with p|Ω˜ ∈ L1loc (Ω). ˜ that is, u = ∇p with p ∈ L1loc (Ω). ˜ Again On the other hand, let u ∈ Gp (Ω), it is sufficient to consider the case n = 1. Then Eu = E1 ∇p, where the first component (E1 ∇p)1 is odd and all other components (E1 ∇p) for = 2, . . . , m + 1 are even with respect to x1,0 = π. Let pe denote the even extension of p to Ω. From ∂1 pe = (E1 ∇p)1 , Proposition 2.15, and E1 ∇p ∈ Lp (Ω) we deduce pe = p(0) + p+ 1,p ˆ 1,p (Rm ). with p(0) ∈ W 1,p (Ω) ∩ W(0),per (Qn , Lp (Rm )) and p+ ∈ W The following lemma shows that PΩ,per preserves the structure endowed by E. ˜ m+n and U := Eu. Then there exists v ∈ Lpσ (Ω) ˜ m+n Lemma 9.14. Let u ∈ Lp (Ω) such that Ev = PΩ,per U . Proof. By definition of E for each j = 1, . . . , n the j-th component Uj is odd with respect to xj,0 := π, whereas the components U are even with respect to xj,0 if = j. For the sake of convenience in the sequel we regard U as a vector-valued function defined on Qn , i.e. U ∈ Lp (Qn , Lp (Rm ))m+n . The calculation of Fourier coefficients then yields ˆ ˆ (kj , k ) = −U (−kj , k ), = j (k ∈ Zn , = 1, . . . n + m, j = 1, . . . , n). U ˆ (−kj , k ), = j U


149

Let V := PΩ,per U , that is, V = TI+Qk U . Hence, ˆ (k) + ik Q∗k U ˆ (k), = 1, . . . , n U Vˆ (k) = ˆ (k), = n + 1, . . . , n + m ˆ (k) + ∂ Q∗k U U

(k ∈ Zn ),

where ˆ (k) = i Q∗k U

n

|k|2 − Δ

−1

n+m

ˆ (k) + k U

=1

|k|2 − Δ

−1

ˆ (k) ∂ U

=n+1

fulfills ˆ (kj , k ) = Q∗(−k ,k ) U ˆ (−kj , k ) Q∗(kj ,k ) U j

(k ∈ Zn , j = 1, . . . , n).

Thus,

Vˆ (kj , k ) =

−Vˆ (−kj , k ), Vˆ (−kj , k ),

=j = j

(k ∈ Zn , = 1, . . . n + m, j = 1, . . . , n).

˜ m+n such that Ev = V = PΩ,per U . Since This shows existence of v ∈ Lp (Ω) p m+n p ˜ m+n implies v ∈ Lσ (Ω) by Lemma 9.12 the assertion is proved. V ∈ Lσ (Ω) We are now in the position to prove a representation formula for the Helmholtz ˜ To this end, let R define the restriction operator from projection PΩ˜ in L(Lp (Ω)). ˜ In particular, Ω to Ω. ˜ RE = I ∈ L(Lp (Ω)). ˜ and the peTheorem 9.15. The standard Helmholtz projection PΩ˜ ∈ L(Lp (Ω)) riodic Helmholtz projection PΩ,per ∈ L(Lp (Ω)) fulfill PΩ˜ = RPΩ,per E. Proof. Since PΩ,per defines a bounded projection in Lp (Ω), it follows immediately ˜ that RPΩ,per E defines a bounded projection in Lp (Ω). ˜ be arbitrary. We show the range of both projections to coincide. Let u ∈ Lpσ (Ω) yields Then U := Eu ∈ Lpσ,per (Ω) by Lemma 9.12 which immediately

PΩ,per U = U . This shows u = RPΩ,per Eu. On the other hand let v ∈ R RPΩ,per E , that is, there ˜ such that v = RPΩ,per Eu. Then V := PΩ,per Eu ∈ Lpσ,per (Ω) and exists u ∈ Lp (Ω) ˜ V = Ev due to Lemma 9.14 which thanks to Lemma 9.12 yields v ∈ Lpσ (Ω). ˜ Finally assume u ∈ N (RPΩ,per E). Due to Lemma 9.14 there exists v ∈ Lp (Ω) such that PΩ,per Eu = Ev. By assumption we have v = RPΩ,per Eu = 0 which implies PΩ,per Eu = 0. Hence, Eu ∈ N (PΩ,per ) and Lemma 9.13 yields u ∈ N (PΩ˜ ). In turn u ∈ N (PΩ˜ ) implies Eu ∈ N (PΩ,per ) by Lemma 9.13, thus, u ∈ N (RPΩ,per E). Altogether we have proved R(PΩ˜ ) = R(RPΩ,per E)

and

from which we conclude PΩ˜ = RPΩ,per E.

N (RPΩ,per E) = N (PΩ˜ )

150


With the standard Helmholtz projection at hand, we can now easily transfer the result of Theorem 9.9 on the ν-periodic Stokes operator to the Stokes operator subject to pure-slip boundary conditions. To this end, we first establish useful ˜ subject to pure-slip boundary conditions results on the resolvent problem in Ω given through λu − Δu + ∇p div u un ∂ n uτ

(9.12)

= = = =

f 0 0 0

˜ in Ω, ˜ in Ω, ˜ on ∂ Ω, ˜ on ∂ Ω.

˜ is understood as The boundary ∂ Ω ˜ := ∂Ω

n

˜± Q n,j ,

where

˜± ˜ ˜ Q n,j := Qj−1 × {0, π} × Qn−j

(j = 1, . . . , n).

j=1

˜ n × Rm is neglected. Here n denotes Note that the Lebesgue null set of edges of Q the outer normal, un the normal component of u, uτ the tangential component of u, and ∂n the normal derivative with respect to the boundary. Hence,

T un = uj and ∂n uτ = ∂j u1 , . . . , ∂j uj−1 , ∂j uj+1 , . . . , ∂j un ˜ j−1 × {0, π} × Q ˜ n−j . on Q Theorem 9.16. For the Stokes resolvent problem (9.12) there exists a unique solution (u, p) It admits the estimate ∇2 u p +

√

∈

˜ n+m × W ˆ 1,p (Ω). ˜ W 2,p (Ω)

λ ∇u p + λ u p + ∇p p ≤ C f p .

˜ λ := RRλ E with Rλ as defined in ˜ λ f := u is given by R The solution operator R Theorem 9.5. Moreover, for each φ > 0 it holds that |α| ˜ λ ; λ ∈ Σπ−φ , 0 ≤ |α| ≤ 2 < ∞. R λ1− 2 ∂ α R Proof. We extend the right-hand side f with the extension operator E to F defined on the whole of Ω. Due to Theorem 9.5 there exists a unique solution U

∈

P

∈

2,p (Qn , Lp (Rm ))n+m , W 2,p (Ω)n+m ∩ Wper 1,p

1,p ˆ 1,p (Rm ) W (Ω) ∩ W(0),per (Qn , Lp (Rm )) + W

of the periodic Stokes resolvent problem with right-hand side F . Set Uj (x , −x , y), j= () ( = 1, . . . , n) V (x, y) := −Uj (x , −x , y), j =


and

Q() (x, y) := P (x , −x , y) ()

151

(j = 1, . . . , n + m, = 1, . . . , n).

()

Then (V , Q ) define solutions of the periodic Stokes resolvent problem with right-hand side F , too. By uniqueness V () = U

and

Q() = P

( = 1, . . . , n)

follows. Taking into account periodicity of U , these symmetry properties yield RUn

=

∂n RUτ

=

˜ 0 on ∂ Ω, ˜ 0 on ∂ Ω.

Hence, u := RU and p := RP solve (9.12). An indirect argument along the same steps proves uniqueness. We define the Stokes operator AS,ps subject to pure-slip boundary conditions in ˜ to be Lpσ (Ω) ˜ m+n ∩ Lpσ (Ω); ˜ un = ∂n uτ = 0 on ∂ Ω ˜ , u ∈ W 2,p (Ω) D(AS,ps ) := AS,ps u

:=

−PΩ˜ Δu = −ΔPΩ˜ u

(u ∈ D(AS,ps )).

˜ m+n which fulfill the Note that ΔPΩ˜ u = PΩ˜ Δu holds true for all u ∈ W 2,p (Ω) ˜ This follows immediately from the boundary condition un = ∂n uτ = 0 on ∂ Ω. representation PΩ˜ = RPΩ,per E and Lemma 9.8. Consequently, we obtain the following result on the RH∞ -calculus of AS,ps by Theorem 9.9. ˜ and φR∞ Theorem 9.17. We have AS,ps ∈ RH∞ (Lpσ (Ω)) AS,ps = 0.

10 The functional calculus approach This chapter extends the results from Chapter 8 in two directions. Firstly, domains given as the Cartesian product of finitely many standard domains are considered. Secondly, with a Banach space F , the cylindrical boundary value problems are F valued and contain L(F )-valued coefficients. Here we employ the operator-valued Dunford calculus and the Kalton-Weis-Theorem. Again we first consider rather arbitrary cylindrical boundary value problems and focus on the Laplacian at the end. The results of this chapter also appear in [NS11a].

10.1 Operator-valued Dunford calculus In this section the Dunford calculus from Chapter 4 is extended to the operatorvalued context. Let A ⊂ L(X) denote the subalgebra of bounded operators on X which commute with the resolvent (μ − A)−1 . For σ ∈ (0, π] we denote by H∞ (Σσ , A) the commutative algebra of bounded, A-valued, holomorphic functions on Σσ , that is, H∞ (Σσ , A) := {f : Σσ → A; f is holomorphic, |f |σ∞ < ∞} , where |f |σ∞ := sup{ f (z) L(X) ; z ∈ Σσ }. Using ρ(z) :=

z (1+z)2

we define the subalgebra

H0∞ (Σσ , A) := {f ∈ H∞ (Σσ , A); there are C, ε > 0 such that f (z) L(X) ≤ C|ρ(z)|ε for all z ∈ Σσ }. Consider a sectorial operator A in X with spectral angle φA ∈ [0, π). We pick σ ∈ (φA , π] and ψ ∈ (φA , σ) and set Γ := (∞, 0]eiψ ∪ [0, ∞)e−iψ . Similar to the scalar valued case, by Cauchy’s integral formula and the sectoriality of A, the Bochner integral 1 f (μ)(μ − A)−1 dμ f (A) := 2πi Γ represents a well-defined element in L(X) for every f ∈ H0∞ (Σσ , A). As f is supposed to take values in A, the above formula defines an algebra homomorphism (10.1)

ΦA : H0∞ (Σσ , A) → L(X);

f → f (A),

known as operator-valued Dunford calculus. For arbitrary f ∈ H∞ (Σσ , A) we set f (A) := ρ(A)−1 (ρf )(A). T. Nau, Lp-Theory of Cylindrical Boundary Value Problems, DOI 10.1007/978-3-8348-2505-6_10, © Vieweg+Teubner Verlag | Springer Fachmedien Wiesbaden 2012

154

10 The functional calculus approach

As in the scalar-valued setting, this definition gives rise to a closed, densely defined operator in X. Moreover, by Cauchy’s theorem it is consistent with the former one for f ∈ H0∞ (Σσ , A). Again convergence properties are important in order to make the functional calculus a useful tool. The following most important result in this direction is a special operator-valued version of the convergence lemma for sectorial operators (see e.g. [DV05, Theorem 4.7] or [HD03]). Recall the approximation sequence ρn ∈ H0∞ (Σσ ) from Lemma 4.7 defined by ρn (z) :=

1 1 n2 z − z − = . 1 + z/n 1 + zn (1 + nz)(n + z)

Lemma 10.1. Let f ∈ H∞ (Σσ , A) and A ∈ S(X). Then f (A) ∈ L(X) if and only if supn∈N (ρn f )(A) < ∞. In this case, f (A)x = limn→∞ (ρn f )(A)x for all x ∈ X. Given an operator admitting an R-bounded H∞ -calculus, the question arises, whether R-boundedness can be extended to some class of A-valued functions. An affirmative answer to this question is given in [KW01, Corollary 5.4]. This result nowadays is known as the Kalton-Weis-Theorem. Theorem 10.2. Let X be a Banach space that has property (α) and let A ∈ S(X). Given an R-bounded subset T ⊂ L(X) set H∞ (Σσ , T ) := {f ∈ H∞ (Σσ , A); f (z) ∈ T (z ∈ Σσ )} . If A admits a bounded H∞ -calculus, then for σ > φ∞ A we have R({f (A); f ∈ H∞ (Σσ , T )}) < ∞.

10.2 Applications to cylindrical boundary value problems n In what follows we consider a domain Ω ⊂ R given as product

of finitely many n = n, that is Ω = N domains Vi ⊂ Rni with ni ∈ N and N i=1 i i=1 Vi . For x ∈ Ω we write x = (x1 , . . . , xN ) with xi ∈ Vi and i = 1, . . . , N , whenever we want to refer to the cylindrical geometry of Ω. Accordingly, for a multi-index α ∈ Nn we

N ni 1 N write α = (α , . . . , α ) ∈ i=1 N0 . Finally we set

(10.2)

∂Vi := V1 × . . . × Vi−1 × ∂Vi × Vi+1 × . . . × VN

' and ∂Ω := N i=1 ∂Vi . As ∂Vi ∩ ∂Vj = ∅ for i = j, all points belonging to the Lebesgue null set of edges of Ω are neglected in this definition of ∂Ω.

10.2 Applications to cylindrical boundary value problems

155

Let F be a Banach space. In this section we particularly deal with F -valued boundary value problems λu + A(x, D)u B(x, D)u

(10.3)

= =

f in Ω, 0 on ∂Ω

with operators A(x, D) and B(x, D) of cylindrical form. Hence, we first extend our previous definition of cylindrical boundary value problems defined for the Cartesian product of three sets to the present situation of finitely many sets. Definition 10.3. Let mi ∈ N for i = 1, . . . , N . The boundary value problem (10.3) is called cylindrical if Ω is a cylindrical domain as introduced above and if the operator A(·, D) is represented as A(x, D) =

N

Ai (xi , D) =

N

i

aiαi (xi )Dxαi

i=1 |αi |≤2mi

i=1

and if the boundary operator on ∂Ω is given as B := (Bi ; i = 1, . . . , N ), where Bi , acting on ∂Vi only, is defined as Bi (x, D) := Bi,j (xi , D); j = 1, . . . , mi with Bi,j (xi , D)u =

i

bij,β i (xi )(Dxβi u)|∂Vi

(i = 1, . . . , N )

(mi,j < mi , j = 1, . . . , mi ).

|β i |≤mi,j

In other words, the differential operators A(x, D) and B(x, D) resolve completely into parts Ai (xi , D) and Bi,j (xi , D) of which each one acts merely on Vi . The boundary operator B acting on the whole of ∂Ω can as well be represented by means of B(x, D)u =

N

χ∂Vi (x)Bi (x, D)u,

i=1

where χ∂Vi denotes the characteristic function of the set ∂Vi . With the aid of the definition m := max{mi ; i = 1, . . . , N } and Bi,j (·, D) := 0 for j > mi we can further write B(x, D) := {Bj (x, D); j = 1, . . . , m}, where Bj (x, D)u =

N i=1

χ∂Vi (x)Bi,j (xi , D)u.

156


For 1 < p < ∞ the Lp (Ω, F )-realization of the boundary value problem (A, B) given through (10.3) is defined by D(A) :=

u ∈ Lp (Ω); Dα u ∈ Lp (Ω) for

N

|αi | 2mi

i=1

(j = 1, . . . , m) ,

Bj (·, D)u = 0 Au := A(·, D)u

≤ 1,

(u ∈ D(A)).

In case that mi = m for all i = 1, . . . , N we obviously have D(A) := u ∈ W 2m,p (Ω, F ); Bj (·, D)u = 0

(j = 1, . . . , m)

which coincides with the domains of definition of the operators considered in Proposition 8.3 and Proposition 8.4. For i = 1, . . . , N we consider the induced boundary value problems (Ai , Bi ) := (Ai (·, D), Bi,1 (·, D), . . . , Bi,mi (·, D)) given through

(10.4)

λu + Ai (x, D)u Bi,j (x, D)u

= =

f in Vi , 0 on ∂Vi

(j = 1, . . . , mi )

which arise by cylindrical decomposition of (A, B). From now on we assume every Vi ⊂ Rni to be given as a C 2mi standard domain (cf. Definition 8.1). Furthermore, we consider the following cylindrical parameterellipticity. Definition 10.4. A cylindrical boundary value problem in the sense of Definition 10.3 is called parameter-elliptic in Ω if (i) each induced boundary value problem (Ai , Bi ) is parameter-elliptic in Vi with angle ϕi := ϕ(Ai ,Bi ) ∈ [0, π) in the sense of Definition 8.2, and (ii) it holds that ϕi + ϕj < π for i, j = 1, . . . , N , i = j. We call ϕ(A,B) := max{ϕi ; i = 1, . . . , N } the angle of parameter-ellipticity of the cylindrical boundary value problem.


157

Finally, with arbitrary γi ∈ (0, 1), i = 1, . . . , N , we impose the following smoothness assumptions on the coefficients of (Ai , Bi ): ⎧ i ⎪ aαi ∈ BU C γi (Vi , L(F )) for all |αi | = 2mi , ⎪ ⎪ ⎪ ⎪ ⎪ aiαi (∞) := lim aiαi (xi ) exists if Vi is unbounded, and ⎪ ⎪ ⎪ |xi |→∞ ⎪ ⎪ ⎪ ⎪ i i ⎨ aαi (x ) − aiαi (∞) ≤ C|xi |−γi (xi ∈ Vi , |xi | ≥ 1)), (10.5) i ∞ rν i ⎪ ⎪ ⎪ aαi ∈ [L + L ](Vi , L(F )) for all |α | = ν < 2mi , ⎪ ⎪ ⎪ 1 2mi − ν ⎪ ⎪ where rν ≥ p, > , ⎪ ⎪ n r ⎪ i ν ⎪ ⎪ ⎩ i bj,β i ∈ C 2mi −mi,j (∂Vi , L(F )) (j = 1, . . . , mi ; |β i | ≤ mi,j ). Note that the limit behavior of the top order coefficients ensures parameterellipticity of the limit operators Ai (∞, D) in case Vi is an unbounded domain to be well-defined. Thus, we can extend the notion of parameter-ellipticity of a cylindrical boundary value problem to Ω :=

N

V i,

i=1

where we again agree on {∞} ⊂ V i in case Vi is unbounded. We are now in the position to present the main theorem of this section. Theorem 10.5. Let 1

< p < ∞ and let F be a Banach space of class HT enjoying 2mi standard domain in Rni , property (α). Let Ω := N i , where every Vi is a C i=1 V N ni ∈ N, i = 1, . . . , N , and let i=1 ni = n. Furthermore, we assume that (i) the boundary value problem (A, B) is cylindrical, (ii) the coefficients of (Ai , Bi ) satisfy (10.5), (iii) (A, B) is parameter-elliptic with angle ϕ(A,B) ∈ [0, π) in Ω, (iv) aiαi (xi )ajαj (xj ) = ajαj (xj )aiαi (xi ) in L(F ) for i, j = 1, . . . , N , i = j and a.e. x ∈ Ω. Then for every φ > ϕ(A,B) , there exists δ = δ(φ) > 0 such that the realization A of (A, B) fulfills A + δ ∈ RH∞ (Lp (Ω, F )) and φR,∞ A+δ ≤ φ. Moreover, we have (10.6) R

λ

1−

N

|αi | i=1 2mi

Dα (λ + A + δ)−1 ; λ ∈ Σπ−φ , 0 ≤

N |αi | ≤1 < ∞. 2mi i=1

Proof. Step 1: First we perform the cylindrical decomposition.

158


We define Lp (Vi , F )-realizations of the boundary value problems (Ai , Bi ) by D(Ai ) := u ∈ W 2mi ,p (Vi , F ); Bi,j (·, D)u = 0 (j = 1, . . . , mi ) , Ai u := Ai (·, D)u

(u ∈ D(Ai )).

Assumptions (ii) and (iii) allow for an application of Propositions 8.3 and 8.4 to each boundary value problem (Ai , Bi ). Thus, for each i = 1, . . . , N and for every φ > ϕi there exists δi = δi (φ) ≥ 0 such that Ai + δi ∈ RH∞ (Lp (Vi , F )) and φR∞ Ai +δi ≤ φ. Moreover, |α| 1− (10.7) < ∞. R λ 2mi Dα (λ + Ai + δi )−1 ; λ ∈ Σπ−φ , 0 ≤ |α| ≤ 2mi These statements remain true for the canonical extension of Ai to Lp (Ω, F ) which for simplicity will be denoted by Ai again. Note that the domain of Ai in Lp (Ω, F ) reads as

D(Ai ) := u ∈ Lp V1 × · · · × Vi−1 , W 2mi ,p (Vi , Lp (Vi+1 × · · · × VN , F )) ; (10.8) Bi,j (·, D)u = 0 (j = 1, . . . , mi ) . Step 2: Within this step, we restrict ourselves to the case N = 2. a) We first show that resolvents of the extensions A1 and A2 commute. To this end, we will frequently make use of the following observation. If T ∈ L(E1 , E2 ) and u ∈ W k,p (G, E1 ) for Banach spaces E1 and E2 and an open set G ⊂ R, then (10.9)

Dα T u = T Dα u

(|α| ≤ k)

p

in L (G, E2 ). This follows easily by the fact that a derivative ∂xj u represents the limit of a convergent sequence in Lp (G, E1 ) and by the continuity of T . For the following argumentation it will be convenient to introduce the notation D(A1 , X) := u ∈ W 2m1 ,p (V1 , X); B1,j (·, D)u = 0, j = 1, . . . , m1 for the domain of A1 in the X-valued space Lp (V1 , X). Here we are particularly interested in the case X = D(A2 , F ). According to step 1, λ + A1 : D(A1 , D(A2 , F )) → Lp (V1 , D(A2 , F )) is an isomorphism for λ ∈ ρ(−A1 ). Fubini’s theorem yields 2m ,p 2m ,p D(A1 , D(A2 , F )) → W 2m1 ,p (V1 , W 2m2 ,p (V2 , F )) ∼ = W 2 (V2 , W 1 (V1 , F )). 1−1/p

(∂V1 , F ) and consider First we set E1 := W 2m1 ,p (V1 , F ) and E2 := Wp 1−1/p T = B1,j , where j ∈ {1, . . . , m1 } is arbitrary. Here Wp (∂V1 , F ) defines the Slobodeckij space, see [Tri78, Section 4.2.1] for the precise definition. Then boundedness of T : E1 → E2 (cf. [ADF97]) and relation (10.9) implies Dα2 B1,j u = B1,j Dα2 u

(u ∈ W 2m2 ,p (V2 , E1 ), |α2 | ≤ 2m2 ).


159

This shows that D(A1 , D(A2 , F )) → W 2m2 ,p (V2 , D(A1 , F )). Since B2,j u = 0 for u ∈ D(A1 , D(A2 , F )) and j ∈ {1, . . . , m2 }, we even have D(A1 , D(A2 , F )) → D(A2 , D(A1 , F )). Interchanging the roles of A1 and A2 , we obtain the converse embedding. Hence, we have D(A1 , D(A2 , F )) ∼ = D(A2 , D(A1 , F )) with equivalent norms. The above arguments also include that p Lp (V1 , D(A2 , F )) ∼ = D(A2 , L (V1 , F )).

From this we conclude that (10.10)

λ + A1 : D(A2 , D(A1 , F )) → D(A2 , Lp (V1 , F ))

defines an isomorphism. Setting E1 = D(A1 , F ), E2 = Lp (V1 , F ), and T = λ + A1 , relation (10.9) yields Dα2 (λ + A1 )u = (λ + A1 )Dα2 u

(u ∈ D(A2 , D(A1 , F ))).

a1α1 ,

in view of (10.9) we also see that Dα2 and the Setting E1 = E2 = F and T = 1 coefficients aα1 commute. By our assumption (iv) on the coefficients this shows (10.11)

(μ + A2 )(λ + A1 )u = (μ + A1 )(λ + A2 )u

(u ∈ D(A2 , D(A1 , F ))).

For f ∈ Lp (V2 , Lp (V1 , F )) we have (μ + A2 )−1 f ∈ D(A2 , Lp (V1 , F )), provided μ ∈ ρ(−A2 ). Since (10.10) is an isomorphism, we obtain (λ + A1 )−1 (μ + A2 )−1 f ∈ D(A2 , D(A1 , F )). Hence, the application of (λ + A1 )(μ + A2 ) to this expression makes sense and we obtain by virtue of (10.11) that (λ + A1 )−1 (μ + A2 )−1 f = (λ + A2 )−1 (μ + A1 )−1 f. b) Let φ > max{ϕ1 , ϕ2 }. Due to assumption (iii) and step 1 of the proof for i = 1, 2 there exist φ > φi > ϕi with φ1 + φ2 < π and δi = δi (φi ) ≥ 0 such that R∞ R∞ Ai + δi ∈ RH∞ (Lp (Ω, F )) and φR∞ Ai +δi < φi . Since φA1 +δ1 + φA2 +δ2 < π we can ˜ := D(A1 )∩D(A2 ) and A ˜ := A1 +A2 , this employ Proposition 4.19a). Setting D(A) ˜ + δ ∈ RH∞ (Lp (Ω, F )) and φR,∞ < max{φ1 , φ2 } < φ, where δ := δ1 + δ2 . yields A ˜ A+δ c) It remains to show the R-boundedness statement (10.6) which implies that 1 2 ˜ ⊂ D(A). For |α | + |α | ≤ 1 we consider the family of operators D(A) 2m1 2m2 |α1 | |α2 | 1− 2m + 2m (α1 ,α2 ) ˜ −1 ; λ ∈ Σπ−φ . 1 2 D (λ + A) λ

160


By the fact that A1 + δ1 ∈ RH∞ (Lp (Ω, F )) has bounded imaginary powers we obtain D(Aν1 ) = [Lp (Ω, F ), D(A1 )]ν for ν ∈ (0, 1) (see Proposition 4.10), where [Lp (Ω, F ), D(A1 )]ν denotes the complex interpolation space between Lp (Ω, F ) and D(A1 ) of order ν. From this we deduce D(Aν1 ) = [Lp (Ω, F ), D(A1 )]ν → Hp2m1 ν (V1 , Lp (V2 , F )), where Hp2m1 ν (V1 , Lp (V2 , F )) denotes the Bessel-potential space of order 2m1 ν (cf. 1 |α1 | [Tri78, Section 4.3.1]). This shows that D(α ,0) (A1 + δ1 )− /2m1 is bounded in p p 1 L (V1 , L (V2 , F )) for |α | ≤ 2m1 , provided δ1 is suitably large. Thus, thanks to Lemma 3.2a) it suffices to show that the family |α1 | |α2 | |α1 | 1− 2m + 2m (0,α2 ) ˜ −1 ; λ ∈ Σπ−φ , 1 2 (A1 + δ1 ) 2m1 D (λ + A) λ

|α1 | 2m1

+

|α2 | 2m2

≤1

is R-bounded. To this end, pick σ ∈ (φ1 , min{φ, π − φ2 }). For λ ∈ Σπ−φ we define the holomorphic functions |α1 | |α2 | |α1 | 1− 2m + 2m (0,α2 ) 1 2 z 2m1 D Gλ (z) := λ (λ + z + A2 + δ2 )−1 (z ∈ Σσ ). A homogeneity argument yields the existence of C = C(φ, σ) > 0 such that |α1 | |α2 | |α1 | |α2 | + 2m 1− 2m 1− 2m 1 2 z 2m1 ≤ C|λ + z| 2 . λ By virtue of Lemma 3.2b) and relation (10.7) we conclude R({Gλ (z); z ∈ Σσ , λ ∈ Σπ−φ }) < ∞. From step 2b) we also know that 2

D(0,α ) (λ + z + A2 + δ2 )−1 (μ − A1 )−1 2

= (μ − A1 )−1 D(0,α ) (λ + z + A2 + δ2 )−1 . Hence, we may apply Theorem 10.2 to the result that R({Gλ (A1 ); λ ∈ Σπ−φ }) < ∞. By an approximation argument and Lemma 10.1 we therefore see that |α1 | |α2 | |α1 | 1− 2m + 2m (0,α2 ) ˜ −1 . 1 2 (A1 + δ1 ) 2m1 D (λ + A) Gλ (A1 ) = λ ˜ ⊂ D(A), hence A = A. ˜ Consequently, relation (10.6) follows. This yields D(A) Step 3: In this step, we prove the assertion for N > 2.


161

Given f ∈ Lp (Ω, F ), Lemma 10.1 and step 2a) of the proof imply (ζ − Al )−1 (λ − (Ai + Aj ))−1 f 1 = lim ρn (μ)(ζ − Al )−1 (λ − μ − Ai )−1 (μ − Aj )−1 dμ f n→∞ 2πi Γ 1 = lim ρn (μ)(λ − μ − Ai )−1 (μ − Aj )−1 (ζ − Al )−1 dμ f n→∞ 2πi Γ = (λ − (Ai + Aj ))−1 (ζ − Al )−1 f. Hence, the resolvent of an extension commutes with the resolvent of finite sums of extensions. As the bounded H∞ -calculus as well as the estimate (10.6) are preserved in each iteration step, the claim for arbitrary N follows by induction. Remark 10.6. Note that no continuity of the boundary conditions at the edges of Ω has to be assumed. Remark 10.7. It is worthwhile to mention that another advantage of this approach lies in the fact that it easily generalizes to the case of different p-integrability in the single cross-sections Vi . In fact, if p = (p1 , . . . , pk ) ∈ (1, ∞)k we set Lp (Ω, F ) := Lp1 (V1 , Lp2 (V2 , . . . Lpk (Vk , F ) . . .)). In the smoothness assumptions (10.5) then we have to replace p by pi . The remaining definitions, such as the domain of A, remain exactly the same. Also the statement of Theorem 10.5 holds without any change. Observe that the operators (λ + A1 )−1 (μ + A2 )−1 and (μ + A2 )−1 (λ + A1 )−1 are well-defined elements of L(Lp1 (V1 , Lp2 (V2 ))) for λ ∈ ρ(−A1 ) and μ ∈ ρ(−A2 ) due to step 1 of the proof. In step 2a) it is further shown that both operators coincide in L(Lp (V1 , Lp (V2 ))), i.e., for p1 = p2 . This implies equality of these operators being restricted to C0∞ (Ω). Since this space is dense in Lp (Ω) for p = (p1 , . . . , pk ), A1 and A2 are resolvent commuting in the case of different pi as well. The remaining parts of the proof then copy verbatim. We close this section with the treatment of a situation where operators on crosssections do not necessarily commute. We emphasize that we do not aim for the greatest generality. The purpose is just to demonstrate that the approach is not restricted to the commuting situation. Improvements and generalizations in one or the other direction are certainly possible. In particular, we restrict ourselves to the case of two domains and of two operators. Hence, let Ω = V1 × V2 and differential operators A1 (x1 , D) and A2 (x2 , D) such as in Theorem 10.5 be given. Then, for φi > ϕi there exist δi ≥ 0 such that the canonical extensions of Ai fulfill Ai + δi ∈ RH∞ (Lp (Ω, F )) and φR,∞ Ai +δi ≤ φi . For the sake of simplicity we will assume δi = 0 and the operators Ai to be subject to generalized Dirichlet boundary conditions for i = 1, 2.

162


We assume the cylindrical structure to be disturbed in the following way: given a function r on V1 , we consider the differential operator A1 (x1 , D) + r(x1 )A2 (x2 , D) in Ω subject to Dirichlet boundary conditions. Associated to r we define an operator of pointwise multiplication in Lp (Ω, F ) by D(Mr ) := {u ∈ Lp (Ω, F ); ru ∈ Lp (Ω, F )}, Mr u := ru

(u ∈ D(Mr )).

In the sequel we will investigate the operator D(Ar ) := D(A1 ) ∩ D(Mr A2 ), Ar := A1 + Mr A2

(u ∈ D(Ar )).

The main difference to previous boundary value problems is that the operators A1 and Mr A2 are no longer resolvent commuting on Lp (Ω, F ). Therefore, we have to impose conditions on r that allow for an application of Proposition 4.21. Theorem 10.8. Let 1 0 with ϕ1 + ϕ2 + ϑ < π and assume that (i) r ∈ [W 2m1 ,p + W 2m1 ,∞ ](V1 ) if 2m1 p > n1 and r ∈ W 2m1 ,∞ (V1 ) else, (ii) r(x1 ) ∈ Σϑ for all x1 ∈ V1 , and (iii) r−1 Dη r ∈ L∞ (V1 ) for all |η| ≤ 2m1 , (iv) r−1 ∈ L∞ (V1 ) or 0 ∈ ρ(A2 ). Then for every φ > max{ϕ1 , ϕ2 + ϑ} there exists δ = δ(φ) ≥ 0 such that Ar fulfills Ar + δ ∈ RH∞ (Lp (Ω, F )) and φR,∞ Ar +δ ≤ φ. Proof. For r subject to assumption (i) we have Mr ∈ L(Lp (Ω, F )), in particular D(Mr ) = Lp (Ω, F ). This implies D(Mr A2 ) := {u ∈ D(A2 ) : A2 u ∈ D(Mr )} = D(A2 ), hence D(Ar ) = D(A1 ) ∩ D(A2 ). In addition, Mr ∈ H∞ (Lp (Ω, F )) with φ∞ Mr ≤ ϑ by assumption (ii) and from assumption (iv) we deduce 0 ∈ ρ(Mr ) or 0 ∈ ρ(A2 ). Since Mr A2 = A2 Mr due to the special form of the two operators in each case we can apply Proposition 4.19(b) to the result that Mr A2 ∈ H∞ (Lp (Ω, F )) and ∞ φ∞ M r A 2 ≤ ϕ2 + φ M r . We show that A1 and Mr A2 satisfy the Labbas-Terreni condition (4.8). To this end, we may assume 0 ∈ ρ(Mr A2 ) since this can always be derived by a shift which we can compensate at the end by choosing δ ≥ 0 a bit larger if necessary. By the


163

fact that we assume Dirichlet boundary conditions and in view of assumption (i)

we obtain Mr D(A1 ) ⊂ D(A1 ). This implies D(Mr A2 A1 ) = D(A2 A1 ) = D(A1 A2 ) ⊂ D(A1 Mr A2 ).

(10.12)

For u ∈ D(A1 A2 ) therefore the equality Mr A2 (μ + A1 )u = (μ + A1 − R)Mr A2 u

(10.13)

with R := [A1 , Mr ]Mr−1 makes sense in Lp (Ω, F ). Thanks to (10.12) we may also identify R as Ru = [A1 (x1 , D), r(x1 )]r(x1 )−1 u =: R(x1 , D)u for all u ∈ Mr A2 D(A1 A2 ) ⊂ D(A1 ). Due to assumptions on r the differential operator R(x1 , D), and hence also R, is well-defined on all of D(A1 ) and represented as a linear combination of differential operators of the form −1 η lη 1 γ r D r (x )D , Rγ (x1 , D) = aα1 (x1 ) η∈Mγ 1

{η ∈ η = 0, 0 ≤ ηj ≤ αj1 for j = 1, . . . , n1 } and with γ < α , some Mγ ⊂ integers lη ∈ N such that η∈Mγ lη η = α1 − γ. This shows that R(x1 , D) is of lower order with respect to A1 . In view of assumption (iii) we also see that the coefficients of R(x1 , D) satisfy condition (10.5). Hence, there is a δ1 ≥ 0 such that A1 − R + δ1 ∈ S(Lp (Ω, F )) and φA1 −R+δ1 ≤ φA1 . Let φ1 > φA1 , μ ∈ Σπ−φ1 , and v ∈ D(A2 ). With u = (μ + A1 )−1 v ∈ D(A1 A2 ) inserted into (10.13) and (μ + A1 − R + δ1 )−1 applied to the resulting equation, we deduce Mr A2 (μ + A1 )−1 v = (μ + A1 − R + δ1 )−1 Mr A2 v. 1 Nn 0 ;

From this for v ∈ D(A2 ) and μ ∈ Σπ−φ1 we infer that [Mr A2 , (μ + A1 )−1 ]v = (μ + A1 )−1 (R + δ1 )(μ + A1 − R + δ1 )−1 Mr A2 v. Let φ2 > φMr A2 and λ ∈ Σπ−φ2 . With the relation given above, the expression appearing in the Labbas-Terreni commutator condition turns into Mr A2 (λ + Mr A2 )−1 [(Mr A2 )−1 , (μ + A1 )−1 ] = −(λ + Mr A2 )−1 (μ + A1 )−1 (R + δ1 )(μ + A1 − R + δ1 )−1 . This formula can easily be estimated to the result Mr A2 (λ + Mr A2 )−1 [(Mr A2 )−1 , (μ + A1 )−1 ] ≤

C (1 + |λ|)|μ|

1 1+ 2m

(μ ∈ Σπ−φ1 , λ ∈ Σπ−φ2 ), 2

where we employed 0 ∈ ρ(Mr A2 ) and the fact that R is relatively bounded by A1 . The assertion now follows from Proposition 4.21.

164



In this section we consider the Laplacian on domains Ω := N i=1 Vi , however, ni with the difference that Vi ⊂ R now each may be a bounded Lipschitz domain (cf. Definition 8.6). More precisely, we consider the resolvent problem for the Laplacian with mixed Dirichlet-Neumann boundary conditions λu − Δu u ∂n u

(10.14)

= = =

f in Ω, 0 on Γ0 , 0 on Γ1 ,

' where ' ∂ν u denotes the outer normal derivative of u. Here Γ0 := i∈N0 ∂Vi and Γ1 := i∈N1 ∂Vi with N0 ∪ N1 ⊂ {1, . . . , N }, and N0 ∩ N1 = ∅. The sets ∂Vi are / N0 ∪ N1 . The boundary value defined as in (10.2) and ∂Vi = ∅ if and only if i ∈ problem (10.14) decomposes into λu − Δu Bi u

(10.15)

= =

f in Vi , 0 on ∂Vi ,

where Bi u := u for i ∈ N0 and Bi u := ∂n u for i ∈ N1 . In case ∂Vi = ∅, of course, the boundary conditions drop out. For our purposes weak versions of the Dirichlet and Neumann Laplacian have to be considered in Banach space-valued spaces. Their definitions read essentially the same as in scalar-valued context: given a Banach space E, the Lp -realizations are defined as 1,p p D(ΔD p,w,i , E) := u ∈ W0 (Vi , E); Δu ∈ L (Vi , E) , (10.16) ΔD (u ∈ D(ΔD p,w,i u := Δu p,w,i , E)) for the E-valued Dirichlet Laplacian on Vi , i.e. in case i ∈ N0 , and u ∈ W 1,p (Vi , E); ∃v ∈ Lp (Vi , E) ∀ϕ ∈ W 1,p (Vi ) : D(ΔN p,w,i , E) := − ∇u∇ϕ = vϕ , Vi

ΔN p,w,i u

:= v

(u ∈

Vi

D(ΔN p,w,i , E))

for the E-valued Neumann Laplacian on Vi , i.e. in case i ∈ N1 . If E = C these definitions coincide with the former ones. From now on we just write D(Δp,w,i ) if it is clear from the context or the subindex i which boundary conditions are assumed or if investigations in progress do not require to distinguish with respect to the boundary conditions. On the other hand, to emphasize the boundary conditions N in special situations we occasionally come back to the notation ΔD p,w,i or Δp,w,i , respectively. As before, we use the same symbol for the canonical extensions of


165

Δp,w,i to Lp (Ω). We define the Lp -realization of the weak Laplacian with mixed boundary conditions on Ω by D(Δp,w ) :=

N

D(Δp,w,i ),

i=1

(10.17) Δp,w u :=

N

Δp,w,i u

(u ∈ D(Δp,w )).

i=1

Theorem 10.9. For i = 1, . . . , N let Vi be a C 2 standard domain in Rni , ni ∈ N, or a bounded Lipschitz domain in Rni , ni ≥ 2. On two-dimensional Lipschitz cross-sections Vi we assume Δp,w,i to be the Dirichlet Laplacian. Then there exists ε > 0 depending only on the Lipschitz character of the different Vi such that for all (3 + ε) 0 we have −Δp,w + δ ∈ RH∞ (Lp (Ω)) and π φR,∞ −Δp,w +δ < 2 . Proof. Step 1: As in Theorem 10.5, we decompose the problem first. Observe that the assumptions imposed on the Laplacian with Dirichlet or with Neumann conditions on the cross-sections Vi are exactly those which make the results obtained in Proposition 8.7 available, i.e. there exist δi ≥ 0 such that π −Δp,w,i + δi ∈ RH∞ (Lp (Vi )) and φR,∞ −Δp,w,i +δi < 2 . Recall that the shift δi is inserted to assure injectivity in case of Neumann boundary conditions. Again these results remain true for the canonical extensions of the operators to Lp (Ω). Step 2: As above we first consider the case N = 2. Unlike in the proof of Theorem 10.5 we have −Δp,w,i + δi ∈ RH∞ (Lp (Vi , E)) a priori only for E := C instead of more general Banach spaces E. Moreover, D(Δp,w,i , E) is in general no longer a subset of W 2,p (Vi , E). By these facts we first have to show that (10.18)

λ − Δp,w,1 : D(Δp,w,1 , D(Δp,w,2 )) → Lp (V1 , D(Δp,w,2 ))

defines an isomorphism. This before was guaranteed by known results (see step 2 of the proof of Theorem 10.5). Let Δp,w,1 be either the Dirichlet or the Neumann Laplacian in Lp (V1 ) and let λ ∈ ρ(Δp,w,1 ). By Fubini’s theorem we see that (10.19)

−Δp,w,1 + δ ∈ RH∞ (Lp (V1 , W k,p (V2 )))

and

φR,∞ −Δp,w,1 +δ
2. We first assume ∂Vi = ∅ for all ' i = 1, . . . , N . W.l.o.g. ' we can rearrange the different sets Vi such that Γ0 = li=1 ∂Vi and Γ1 = N i=l+1 ∂Vi . Here l = N corresponds to the pure Dirichlet Laplacian, l = 0 to the pure Neumann

Laplacian, and 0 < Dirichlet-Neumann Laplacian. Set Ω0 := li=1 Vi and

lN< N to the mixed N p Ω1 := i=l+1 Vi and let ΔD p,w,0 and Δp,w,1 denote the extended L (Ω)-realizations of the Dirichlet problem on Ω0 and of the Neumann problem on Ω1 , respectively. Then according to the first step of the proof by iteration we see that −ΔD p,w,0 and ∞ −ΔN p,w,1 + δ for every δ > 0 admit each a bounded RH -calculus with angle less than π2 . Moreover, the argumentation on commutativity of resolvents performed in


167

N N D step 2 applies to ΔD p,w,0 and Δp,w,1 . As Δp,w = Δp,w,0 +Δp,w,1 , Proposition 4.19a) gives the desired result. In case ∂Vi = ∅ for some i = 1, . . . , N , we repeat the former step with the Laplacian on Ω0 × Ω1 subject to mixed Dirichlet-Neumann boundary conditions and the Laplacian defined on the whole space. The proof is now complete.

Remark 10.10. There are some situations in which the assertion remains true for δ = 0. In fact, the shift δ > 0 is only required to overcome the lack of injectivity in ’Neumann cross-sections’. Thus, if we assume e.g. pure Dirichlet boundary conditions the assertion remains true for δ = 0 since then every −ΔD p,w,i as defined in (10.16) is injective. Furthermore, if at least one Vi , say Vi0 , is bounded and the operator on Vi0 is the Dirichlet Laplacian −ΔD p,w,i0 , we obtain 0 ∈ ρ(−Δp,w ) (i.e. in particular δ = 0) for the full Laplacian with mixed Dirichlet-Neumann boundary conditions −Δp,w on Ω. This follows thanks to the spectral property D ∞ p σ(−ΔD p,w,i0 ) ⊂ (c, ∞) for some c > 0 which gives −Δp,w,i0 −δi0 ∈ H (L (Vi0 )) for each δi0 < c by Cauchy’s theorem and step 1 of the proof. Due to the fact that we may choose the shifts δi > 0 for the Neumann-Laplacians on other cross-sections arbitrarily small, the sum over all δi , i = i0 , can be absorbed by −δi0 . Remark 10.11. We emphasize again that classes of unbounded Lipschitz domains and of mixed boundary conditions are treated simultaneously. Moreover, Theorem 10.9 yields more regularity for u ∈ D(Δp,w ) than indicated by (10.17) if at least one cross-section Vi is smooth. Also observe that the assertions of Remarks 8.25, 8.26, and 8.27 carry over to the present situation. Remark 10.12. Finally we note that Remark 10.7 also applies to the situation in Theorem 10.9 if only one non-smooth domain is considered. Recall that a bounded H∞ -calculus for an E-valued realization of the Laplacian in non-smooth Lipschitz domains was deduced from Fubini’s theorem especially for E := Lp (G). Since Fubini’s theorem fails to be true for integrability with respect to different values of p, the present proof does not apply in full generality. However, if only VN is non-smooth, we can apply Theorem 10.5, i.e. it is possible to consider Lp (V , Lp+1 (V+1 , . . . , LpN (VN )) . . . )-valued realizations of −Δp−1 ,−1 on standard domains V−1 , = 2, . . . , N (cf. Remark 8.24). Given a bounded Lipschitz domain V ⊂ R2 , the V -dependent range for p such ∞ p that −ΔD p ∈ RH (L (V )) extends to (4 + ε) 0. By our technique this range is preserved for higher dimensional Lipschitz cylinders provided the roughness of the boundary is of two dimensional character. Theorem 10.13. Let Vi be a C 2 standard domain in Rni , ni ∈ N, or a bounded Lipschitz domain in R2 and assume that Δp,w,i is the Dirichlet Laplacian on Lipschitz cross-sections Vi . Then there exists ε > 0 depending only on the Lipschitz character of the different Vi such that for all (4 + ε) 0 we have −Δp,w + δ ∈ RH∞ (Lp (Ω)) and φR,∞ −Δp,w +δ < π/2, where Δp,w is defined as in (10.17).

168


Proof. We make use of the better range (4+ε) < p < 4+ε of p in Proposition 8.7. Now we can go on as in the proof of Theorem 10.9. Remark 10.14. Observe that in addition to the better range of p, all observations given in Remarks 10.10, 10.11, and 10.12 apply also here. The result on non-commuting operators from the previous section can be established particularly in the context of the Dirichlet Laplacian in cylindrical Lipschitz domains. Theorem 10.15. Let 1 n1 and r ∈ W 2,∞ (V1 ) else, (ii) r(x1 ) ∈ Σϑ for all x1 ∈ V1 and some 0 ≤ ϑ < π − φR∞ −ΔD , p,w

(iii)

∇r Δr , r r

2

− 2 |∇r| ∈ L∞ (V1 ), r2

(iv) r−1 ∈ L∞ (V1 ) or 0 ∈ ρ(Δp,2 ), and let φ > ϑ. Then there exists ε > 0 depending only on the Lipschitz character of Vi such that for all (3 + ε) < p < 3 + ε (respectively (4 + ε) < p < 4 + ε) there is a δ ≥ 0 such that for −Δr,p + δ := −Δp,w,1 − Mr Δp,w,2 + δ defined on D(Δr,p ) := D(Δp,w,1 ) ∩ D(Mr Δp,w,2 ) we have that −Δr,p + δ ∈ RH∞ (Lp (Ω)) R∞ and φR,∞ −Δr,p +δ ≤ φ + φ−ΔD . p,w

Proof. We try to mimic the proof of Theorem 10.8. By the fact that Δru = uΔr + ∇r · ∇u + rΔu we see that also here we have Mr (D(Δp,w,1 )) ⊂ D(Δp,w,1 ). Completely analogous to the preceding we therefore arrive at (10.13) with A1 = −Δp,w,1 , A2 = −Δp,w,2 , and |∇r|2 ∇r Δr Ru = u. · ∇u + −2 2 r r r We have to show that R is relatively bounded. Since D(Δp,w,1 ) ⊂ W 2,p (V1 ) fails to be true in general, this is not so obvious as above. Recall that by the results obtained in [JK95] we know that D((−Δp,w,1 )1/2 ) = W01,p (V1 ). Since Δp,w,1 ∈ H∞ (Lp (V1 )) this yields W01,p (V1 ) = D((−Δp,w,1 )1/2 ) = [Lp (V1 ), D(Δp,w,1 )]1/2 . Hence, the interpolation inequality 1/2

≤ C(ε u D(Δp,w,1 ) + C(ε) u p ) u W 1,p ≤ C u D(Δp,w,1 ) u 1/2 p


169

holds for all u ∈ D(Δp,w,1 ) and ε > 0. This implies Ru p ≤ C u W 1,p ≤ C(ε Δp,w,1 u p + C(ε) u p )

(u ∈ D(Δp,w,1 ), ε > 0)

from which we deduce that μ − Δp,w,1 − R is sectorial for some μ ≥ 0. The remaining proof now copies verbatim from Theorem 10.8. Example 10.16. Theorem 10.15 in particular covers the case of heat conduction in a bounded Lipschitz cylinder with either in longitudinal direction or in crosssections non-constant heat conductivity coefficient (see also Example 8.28).

A Notation and vector-valued function spaces Throughout this thesis the symbols X, Y and Z stand for Banach spaces. Given a closed operator A, we denote by D(A), N (A), and R(A) the domain of definition, the kernel, and the range of A, respectively. Furthermore, by ρ(A) and σ(A) we denote the resolvent set and the A. The symbol L(X, Y ) stands for the Banach space of all bounded linear operators from X to Y equipped with operator norm · L(X,Y ) . As an abbreviation we set L(X) := L(X, X). In the following we collect some definitions and facts on function spaces. This will be done in a vector-valued context, i.e. the functions under consideration are allowed to take values in rather arbitrary Banach spaces. For a more general introduction and further results we refer to the monographs [Ama95] and [Ama09]. Since many vector-valued function spaces will define Banach spaces again, we will find it convenient to employ E and F as further symbols for Banach spaces which merely occur as image spaces. Let E be an arbitrary Banach space and let G ⊂ Rn denote an arbitrary domain. For 1 ≤ p ≤ ∞ we denote by Lp (G, E) the E-valued Lebesgue or Bochner spaces. Endowed with the norm

1 f (x) pE dx p , 1 ≤ p < ∞, G f p := f p,G := f p,G,E := ess supx∈G f (x) E , p = ∞ Lp (G, E) is a Banach space. For m ∈ N0 ∪{∞} we denote by C m (G, E) the space of all m-times continuously differentiable functions. In particular C(G, E) := C 0 (G, E) denotes the space of continuous functions. Its subspace of bounded and uniformly continuous functions is denoted by BUC(G, E). Finally for α ∈ (0, 1) we denote by C m,α (G, E) the space of m-times continuously differentiable functions whose m-th derivative is H¨ older continuous with H¨ older exponent α. The subindex 0, e.g. C0∞ (G, E), will be used to indicate the subspace of in G compactly supported functions of the respective function space. Finally, C∞ (Rn , E) denotes the closed subspace of BUC(Rn , E) with respect to · ∞ , consisting of all continuous functions vanishing at infinity. We agree to drop the indication E in the notation of all function spaces if E = C. Let m ∈ N0 and K ⊂ G be bounded. For f ∈ C ∞ (G, E) define the seminorms pm,K (f ) := max ∂ α f L∞ (K,E) . |α|≤m

Then the space of smooth E-valued functions endowed with these seminorms E(G, E) := C ∞ (G, E), pm,K ; m ∈ N0 , K ⊂ G bounded T. Nau, Lp-Theory of Cylindrical Boundary Value Problems, DOI 10.1007/978-3-8348-2505-6, © Vieweg+Teubner Verlag | Springer Fachmedien Wiesbaden 2012

172


is a Fréchet space. The space of E-valued test functions, that is, the space C0∞ (G, E) equipped with the inductive limit topology as presented e.g. in [Ama03, Section 1.1] is denoted by D(G, E). Also recall the Schwartz space of smooth, rapidly decreasing E-valued functions on Rn denoted by S(Rn , E). Setting qk,m (f ) :=

sup x∈Rn , |α|≤m

(1 + |x|2 )k ∂ α f (x) E ,

it consists of all f ∈ C ∞ (Rn , E) such that qk,m (f ) < ∞ for each k, m ∈ N0 . Endowed with the topology induced by {qk,m ; k, m ∈ N0 }, S(Rn , E) becomes a Fréchet space. As an abbreviation let F ∈ {E, D, S}. In what follows set S(G, E) := S(Rn , E), i.e. in the context of rapidly decreasing functions G is always assumed to be the whole space. Again we abbreviate F(G) := F(G, C) and set F (G, E) := L(F(G), E) endowed with the topology of uniform convergence on bounded subsets of F(G, E). In particular, T ∈ D (G, E) if and only if for each compact set K ⊂ G there exist m ∈ N0 and C > 0 such that T (ϕ) E ≤ Cpm,K (ϕ)

(A.1)

holds for all ϕ ∈ C0∞ (G, E) with supp ϕ ⊂ K. A sequence (Tj )j∈N ⊂ D (G, E) is said to converge to T ∈ D (G, E), i.e. Tj → T if and only if Tj (ϕ) → T (ϕ) for all ϕ ∈ C0∞ (G). As usual, we call D (G, E) the space of E-valued distributions on G, E (G, E) the subspace of E-valued distributions with compact support and S (Rn , E) the space of E-valued tempered distributions. It is well-known that D(G, E) → E(G, E)

and

D(Rn , E) → S(Rn , E) → E(Rn , E)

are dense embeddings. Since C0∞ (Rn , E) is dense in Lp (Rn , E) for 1 ≤ p < ∞ the same is true for S(Rn , E). Furthermore, the relations between D, S and E immediately yield E (G, E) → D (G, E)

and

E (Rn , E) → S (Rn , E) → D (Rn , E).

We write f ∈ Lploc (G, E) if ϕf ∈ Lp (G, E) for all ϕ ∈ C0∞ (Rn , E). Given any function f ∈ L1loc (G, E) we set Tf (ϕ) := [f ](ϕ) := (f, ϕ) := f ϕdx, (ϕ ∈ C0∞ (G)). G

Then Tf : D(G) → E is linear and for each compact set K ⊂ G it holds that Tf (ϕ) E ≤ f 1,K,E p0,K (ϕ)

(ϕ ∈ C0∞ (G), supp ϕ ⊂ K).


173

Taking C = f 1,K,E and m = 0 in (A.1) shows Tf ∈ D (G, E). Moreover, the mapping L1loc (G, E) → D (G, E); f → Tf is linear and injective so that L1loc (G, E) → D (G, E). For f ∈ L1loc (G, E) the distributions Tf are called regular distributions. Restricted to smaller function spaces sometimes more can be said about the according regular distributions. For example, it is easily seen that C0 (G, E) → E (Rn , E). In view of Fourier transform we will make use of the important fact that Lp (Rn , E) → S (Rn , E) for 1 ≤ p ≤ ∞. For the sake of simplicity we will frequently write f ∈ F (G, E) instead of Tf ∈ F (G, E) if no confusion seems likely. α For α ∈ Nn 0 the distributional derivative ∂ T of T ∈ D (G, E) is defined by α ∂ T (ϕ) := (−1)|α| T (∂ α ϕ) (ϕ ∈ D(G)). In view of (A.1) it is easily verified that Tj → T implies ∂ α Tj → ∂ α T . Along the same lines derivation of tempered distributions defines a continuous mapping in S (Rn , E). For each m ∈ N0 and 1 ≤ p ≤ ∞ the E-valued Sobolev space W m,p (G, E) is defined as the space of all f ∈ Lp (G, E) such that ∂ α f ∈ Lp (G, E) for all |α| ≤ m. Endowed with the norm ⎧ 1 p ⎪ ⎪ ⎪ ⎨ ∂ α f pp , 1 ≤ p < ∞, f m,p := f m,p,G := f m,p,G,E := |α|≤m ⎪ ⎪ ⎪ max ∂ α f ∞ , p=∞ ⎩ |α|≤m

W m,p (G, E) becomes a Banach space. By this definition W 0,p (G, E) = Lp (G, E). For 1 ≤ p < ∞ we denote by W0m,p (G, E) the closure of C0∞ (G, E) in W m,p (G, E). m,p (G, E) for ≤ p ≤ ∞ and m ∈ N which We further define the Fréchet spaces Wloc p consist of all u ∈ Lloc (G, E) such that uϕ ∈ W m,p (G, E) for all ϕ ∈ C0∞ (G). In the subsequent lines, we briefly focus on the task to find a primitive with respect to xj of a distribution T ∈ D (Rn , E), that is, a distribution F ∈ D (Rn , E) such that ∂j F = T . For that purpose, given n ∈ N, n ≥ 2, and x ∈ Rn for each j ∈ {1, . . . , n} we define (xj , x ) := (x , xj ) := (x1 , . . . , xj−1 , xj , xj+1 , . . . , xn ) = x. Then for ϕ ∈ C0∞ (Rn )there exists ψ ∈ C0∞ (Rn ) such that ϕ = ∂j ψ if and only if the function Φ(x ) := R ϕ(x , s)ds fulfills Φ ≡ 0. Let Dj (Rn ) denote the subspace of functions ϕ ∈ D(Rn ) for which one of these equivalent conditions holds true. Let κ ∈ C0∞ (R) be such that R κ(s)ds = 1. Then every ϕ ∈ D(Rn ) admits a unique decomposition of the form ϕ(x , xj ) = ϕ0 (x , xj ) + κ(xj )Φ(x )

174


such that ϕ0 ∈ Dj (Rn ). This allows to define a projection onto Dj (Rn ) given by Pj (ϕ) := ϕ0 = ϕ − κ ⊗ Φ, where κ ⊗ Φ(x) := κ(xj )Φ(x ) denotes the tensor xj Pj ϕ(x , s)ds defines a test function product of functions. Hence, ψϕ (x) := −∞ ∞ n ψ ∈ C0 (R ) and F (ϕ) := −T (ψϕ )

(A.2)

fulfills ∂j F = T . Moreover, ∂j F = 0 if and only if F is independent of xj which means (A.3)

F (ϕ) = F (ϕ(· + cej ))

(ϕ ∈ C0∞ (Rn ), c ∈ R).

For T ∈ D (Rn , E) this implies (A.4)

∇T = 0

⇐⇒

T = Tη = η

with

η∈E

(see [Wal94, Chapter 6]). It is easily seen that a regular distribution Tf is independent of xj if and only if f (x , xj ) = f (x , xj + c) for all c ∈ R, i.e. if f (x) = f (x ) x-almost everywhere. Given T ∈ F (Rn , E) and ϕ ∈ F(Rn ), for F ∈ {D, E} the convolution T ∗ ϕ is a well-defined element of E(Rn , E) (see [Ama95, Section III 4.2]). To be more precise, let ϕ(x) ˇ := ϕ(−x) and τa ϕ(x) := ϕ(x − a) with fixed a ∈ Rn for x ∈ Rn . Then (A.5)

ˇ T ∗ ϕ(x) := T (τx ϕ)

(x ∈ Rn )

simultaneously defines the bilinear and separately continuous mappings F (Rn , E) × F(Rn ) → E(Rn , E)

and

E (Rn , E) × D(Rn ) → D(Rn , E).

This allows to define the convolution of an E-valued and a scalar-valued distribution T and F , if at least one of them is compactly supported, as for ϕ ∈ D(Rn ) (A.6)

(T ∗ F ) ∗ ϕ = T ∗ (F ∗ ϕ)

is a well-defined element in E(Rn , E) → D (Rn , E). Again this defines the bilinear and separately continuous mappings D (Rn , E) × E (Rn ) → D (Rn , E)

and

E (Rn , E) × D (Rn ) → D (Rn , E).

A proof of this result and of hypocontinuity of the above mappings can be found in [Ama03, Proposition 1.2.3]. We close this chapter with a well-known and useful example which immediately follows from (A.5) and (A.6). Lemma A.1. Let δ ∈ E (Rn ) denote the Dirac distribution, i.e. δ(ϕ) := ϕ(0) for ϕ ∈ E(Rn ). Then T ∗ δ = T for T ∈ D (Rn , E).

B Related topics in the literature In the subsequent lines we comment on topics in the literature that are related to some parts of this thesis. As frequently mentioned throughout this thesis, E-valued parabolic boundary value problems in standard domains were extensively studied in [DHP03] and [DDH+ 04]. For classical papers on scalar-valued boundary value problems we refer to [DN55], [ADN59], [ADN64], and [Sol66] in the elliptic case and to [AV64], [ADF97], and [LSU68] in the parameter-elliptic and parabolic case. For a more comprehensive list see also [DHP03]. An approach to a class of elliptic differential operators with Dirichlet boundary conditions in uniform C 2 -domains can be found in [Kun03]. All cited results above are based on standard localization procedures for the domain contrary to the approach presented in this thesis. In [AB02] solution isomorphisms for the first order equation (B.1)

ut + Au = f

subject to pure periodic boundary conditions in Lebesgue and H¨ older spaces are established. A generalization of these results on equation (B.1) to pure periodic first order integro-differential equations in Lebesgue, Besov, and H¨ older spaces is given in [KL04]. Here the concept of 1-regularity in the context of sequences is introduced (cf. Remark 7.10). In [KL06] one finds a comprehensive treatment of periodic second order equations of type (B.2)

utt + aAut + αAu = f

in Lebesgue and H¨ older spaces. The special case a = 0 and α = 1 subject to pure periodic boundary conditions, pure Dirichlet conditions, and pure Neumann conditions in Lebesgue spaces is also treated in [AB02]. In [KLP09] more general equations are investigated in the mentioned spaces as well as in Triebel-Lizorkin spaces. Moreover, applications to nonlinear equations are presented. Note that all articles mentioned so far consider equations depending on a single variable related to time. In contrast to that, in this thesis we apply multiplier techniques with respect to multiple space variables. In particular, the solution isomorphisms for the one-dimensional equation (B.1) subject to pure periodic boundary conditions from [AB02] are extended to solution isomorphisms for the multi-dimensional equation (1.6) subject to ν-periodic boundary conditions in Theorem 7.15. In [AR09] various properties as e.g. Fredholmness of the operator ∂t − A(·) associated with the non-autonomous periodic first order Cauchy problem in the Lq -context are investigated. Results on this operator based on Floquet theory are T. Nau, Lp-Theory of Cylindrical Boundary Value Problems, DOI 10.1007/978-3-8348-2505-6, © Vieweg+Teubner Verlag | Springer Fachmedien Wiesbaden 2012

176


obtained in the PhD-thesis [Gau01]. We remark that in Floquet theory ν-periodic instead of pure periodic boundary conditions appear in a natural way. Maximal regularity of second order initial value problems of the type utt (t) + But (t) + Au(t) = f (t)

(t ∈ [0, T )),

u(0) = ut (0) = 0 is treated in [CS05] and [CS08]. In particular, q-independence of maximal regularity for second order problems of this type is shown. The same equation involving dynamic boundary conditions is studied in [XL04]. The non-autonomous second order problem involving t-dependent operators B(t) and A(t) appears in [BCS08]. We also refer to [XL98] for the investigation of higher order Cauchy problems. For results in boundary value problems in (0, 1) with operator-valued coefficients subject to numerous types of homogeneous and inhomogeneous boundary conditions we refer to [FLM+ 08], [FSY09], [FY10], and the references therein. Their approaches mainly rely on semigroup theory and do not allow for an easy generalization to (0, 1)n . In [FSY09], however, applications to boundary value problems in the cylindrical space domain (0, 1) × V can be found. As far as the author of this thesis knows, the usage of operator-valued multipliers to treat cylindrical boundary value problems was first carried out in [Gui04] and [Gui05] in a Besov space setting. In these papers the author constructs semiclassical fundamental solutions for a class of elliptic operators on infinite cylindrical domains Rn × V . This proves to be a strong tool for the treatment of related elliptic and parabolic (see [Gui04] and [Gui05]) as well as of hyperbolic (see [Gui05]) problems. In particular, this approach leads to semiclassical representation formulas for solutions of related elliptic and parabolic boundary value problems. Based on these formulas and on a multiplier result of Amann [Ama97], the author derives a couple of interesting results for these problems in a Besov space setting. In particular, the given applications include asymptotic behavior in the large, singular perturbations, exact boundary conditions on artificial boundaries, and the validity of maximum principles. With a remark on possible generalizations, in [Gui04] and [Gui05] the author assumes A1 = −Δ which avoids the investigation of Dore-Venni-type angle conditions. Very recently in [DCGL10] the wellposedness of a class of parabolic boundary value problems in a vector-valued H¨ older space setting is proved. In this article Ω = [0, L] × V , the first part is given by A1 = a(xn )∂n2m , xn ∈ [0, L], and A2 is uniformly elliptic. In contrast to [Gui04], [Gui05], and [DCGL10] here we present the Lp -approach to cylindrical boundary value problems. Therefore, the notion of R-boundedness comes into play which is not required in the framework of Besov or H¨ older spaces. Results on sums of closed operators were already applied to the second order problem (B.2) with a = 0 and α = 1 subject to Dirichlet boundary conditions in [HP98]. In the same article the method is applied to the Dirichlet problem (B.3)

Δu = f

in Ω,

u|∂Ω = 0,


177

where Ω defines a cone in R2 . In virtue of a change of variables, Ω is transferred to a strip-type domain. This allows for an application of the Dore-Venni-Theorem to deduce unique solvability by means of suitable weights. Note that in contrast to strip-type domains in this thesis cylindrical domains are treated where more than one cross-section has a non-vanishing boundary. The parabolic problem associated with (B.3) where the space domain is given by a cone in Rn or more general a wedge in Rm+n is investigated recently in [PS07]. A change of variables leads to the space domain Rm × M where M defines an open subset of the n-sphere with smooth boundary. For literature on the Laplacian in general bounded Lipschitz domains we refer to [JK95] ans [Woo07] for the Dirichlet problem and to [FMM98] and [Woo07] for the Neumann problem and to the references therein. For a treatment of the Laplacian with mixed boundary conditions in more general Lipschitz domains we refer to [HDKR08], [HDR09], and to the literature cited therein. For more classical results on the Helmholtz projection as well as on Stokes and Navier-Stokes equations we refer to the textbook [Gal94]. Out of the wide range of results on the Stokes operator on bounded domains we only name [NS03] where a bounded H∞ -calculus in Lp -spaces on bounded and exterior C 3 -domains subject to non-slip boundary conditions is proved and refer to the references therein. It is based on the related half space result from [DHP01], a localization procedure, and perturbation arguments for the H∞ -calculus. For a recent result on Stokes and Navier-Stokes equations with partial-slip boundary conditions in a half space see [Saa06]. The Helmholtz projection in layers and infinite cylinders Ω with a standard cross-section is constructed by Farwig in [Far03] by means of continuous Fourier multiplier results. Here the R-boundedness assumptions on the symbol are inferred from an equivalent condition involving arbitrary Muckenhoupt weights. Together with Ri, this author serializes results on resolvent estimates, maximal regularity and boundedness of the H∞ -calculus for the Stokes operator in Lpσ (Ω) in [FR07c], [FR07a], and [FR08]. In [FR07b] these results and results on bounded domains are merged to according results on domains with finite exits to infinity. For general unbounded domains of class C 1 the existence of the Helmholtz projection in L2 ∩ Lp for p > 2 and in L2 + Lp for 1 < p < 2 instead of in Lp is proved in [FKS07]. Resolvent estimates for the Stokes operator in infinite layers by means of continuous Fourier multipliers were also proved in [AW05a]. In the series [Abe05a]/[Abe05b], these estimates are extended to various boundary conditions and to layer-like domains. Moreover, a representation of the Helmholtz projection in Lp -spaces on this type of domains by means of singular Green operators is deduced. Furthermore, a bounded H∞ -calculus of the Stokes operator subject to Dirichlet boundary conditions is proved.

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Index α∗ , 12 α ≤ β, 10 [α, β], 10 BIP(X), 45 C∞ (Rn , E), 9 ∞ (Qn , E), 11 Cper ∞ (Rn+m , E), 20 Cper,n Cauchy problem, 51 class HT , 30 coarse decomposition, 33 contraction principle of Kahane, 26 convergence lemma, 45 operator-valued, 154 cylindrical boundary value problem, 103, 155 Dα , 10 (Rn , E), 20 Dν,per (Rn+m , E), 20 Dν,per,n n Dper (R , E), 11 (Rn+m , E), 20 Dper,n δ2π , 12 discrete derivative, 32 Δγ , 32 distribution ν-periodic, 20 partially ν-periodic, 20 partially periodic, 20 periodic, 11 Dunford calculus, 43, 44 operator-valued, 153 E(Σ), 43 F-calculus

boundedness, 44 R-boundedness, 46 fine decomposition, 33 Fourier coefficient, 10, 12 of derivatives, 16 partial, 20 Fourier multiplier continuous, 27 discrete, 28 of resolvent type, 56, 80 Fourier series representation result, 14, 22 uniqueness result, 15 Fourier transform, 9 of derivatives, 10 Gp (Ω), 145 Gpν,per (Ω), 140 γper , 143 H∞ -calculus boundedness, 44 R-boundedness, 46 H∞ (Σ), 42 H0∞ (Σ), 42 H∞ (Σσ , A), 153 H0∞ (Σσ , A), 153 H∞ (X), 44 Helmholtz decomposition, 135, 142, 145 ν-periodic, 140 intermediate condition, 36 kν , 85, 134 Kalton-Weis-Theorem, 154

T. Nau, Lp-Theory of Cylindrical Boundary Value Problems, DOI 10.1007/978-3-8348-2505-6, © Vieweg+Teubner Verlag | Springer Fachmedien Wiesbaden 2012

188

Lp(0) (Qn , E), 88 Lpσ (Ω), 145 Lpσ,ν,per (Ω), 141 Labbas-Terreni condition, 49 Lipschitz domain, 101 maximal regularity, 51, 53 multi-index, 10 multiplier result discrete, 35, 36 discrete, intermediate condition, 36 multiplier theorem continuous, 31 Michlin, 31 PΩ , 145 PΩ,ν , 140 PΩ,per,p , 144 PRm , 135 PΩ˜ , 142 parameter-elliptic boundary value problem, 100 cylindrical boundary value problem, 105, 156 differential operator, 100 polynomial, 56, 80 φA , 41 φ∞ A , 44 φR A , 46 φRF A , 46 φR∞ A , 46 polynomial angle of parameter-ellipticity, 56 degree, 56 elliptic, 56, 80 parameter-elliptic, 56, 80

Index

principal part, 56 trigonometric, 10 property (α), 31 pseudo-sectorial, 41 pseudo-R-sectorial, 46 ΨH∞ (X), 44 ΨRH∞ (X), 46 ΨRS(X), 46 ΨS(X), 41 R-boundedness, 25 resolvent commuting operators, 48 RH∞ -calculus, 46 RH∞ (X), 46 RS(X), 46 S(X), 41 sectorial, 41 Σφ , 41 Sobolev spaces ν-periodic, 18 homogeneous, periodic, 18 periodic, 16 standard domain, 99 θA , 45 T(Qn , E), 10 Volterra equation, 52 1,p (Qn , E), 138 W(0),per

1,p Wper, ˆ (Ω), 143 + m,p Wper (Qn , E), 16 m,p Wν,per (Qn , E), 18

Zα , 55

Lp-theory of cylindrical boundary value problems : an operator-valued fourier multiplier and functional calculus approach

Fourier Series and Boundary Value Problems

Fourier Analysis and Boundary Value Problems

Fourier analysis and boundary value problems

Fourier Series and Boundary Value Problems

Fourier Series, Transforms, and Boundary Value Problems

Boundary Value Problems for Functional Differential Equations

Polyharmonic boundary value problems

A unified approach to boundary value problems

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