Transmission Problems for Elliptic Second-Order Equations in Non-Smooth Domains (Frontiers in Mathematics)

Frontiers in Mathematics

Advisory Editorial Board Leonid Bunimovich (Georgia Institute of Technology, Atlanta) Benoît Perthame (Université Pierre et Marie Curie, Paris) Laurent Saloff-Coste (Cornell University, Ithaca) Igor Shparlinski (Macquarie University, New South Wales) Wolfgang Sprössig (TU Bergakademie Freiberg) Cédric Villani (Ecole Normale Supérieure, Lyon)

Mikhail Borsuk

Transmission

Problems

for Elliptic Second-Order

Equations in Non-Smooth Domains

Mikhail Borsuk Department of Mathematics and Informatics University of Warmia and Mazury M. Oczapowskiego str. 2 10-719 Olsztyn Poland e-mail: [email protected]

2010 Mathematics Subject Classification: 35J25, 35J60, 35J65, 35J70, 35J85, 35B05, 35B45, 35B65 ISBN 978-3-0346-0476-5 DOI 10.1007/978-3-0346-0477-2

e-ISBN 978-3-0346-0477-2

Library of Congress Control Number: 2010930426 © Springer Basel AG 2010 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained. Cover design: deblik, Berlin Printed on acid-free paper Springer Basel AG is part of Springer Science+Business Media www.birkhauser-science.com

To my wife Tatiana

Contents Preface

xi

Introduction

1

1 Preliminaries 1.1 List of symbols . . . . . . . . . . . . . . . . . . . . . . . 1.2 Operators and formulae related to spherical coordinates 1.3 The quasi-distance function rε and its properties . . . . 1.4 Function spaces . . . . . . . . . . . . . . . . . . . . . . . 1.5 Some inequalities . . . . . . . . . . . . . . . . . . . . . . 1.6 Sobolev embedding theorems . . . . . . . . . . . . . . . 1.7 The Cauchy problem for a differential inequality . . . . 1.8 Additional auxiliary results . . . . . . . . . . . . . . . . 1.8.1 Stampacchia’s Lemma . . . . . . . . . . . . . . . 1.8.2 Other assertions . . . . . . . . . . . . . . . . . .

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2 Eigenvalue problem and integro-differential inequalities 2.1 Eigenvalue problem for the m-Laplacian in a bounded domain on the unit sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 The Friedrichs-Wirtinger type inequality . . . . . . . . . . . . . . . 2.3 The Hardy and Hardy-Friedrichs-Wirtinger type inequalities . . . . 2.4 Auxiliary integro-differential inequalities . . . . . . . . . . . . . . . 3 Best possible estimates of solutions to the transmission problem for linear elliptic divergence second-order equations in a conical domain 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Local estimate at the boundary . . . . . . . . . . . . . . . . . . . . 3.3 Global integral estimates . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Local integral weighted estimates . . . . . . . . . . . . . . . . . . . 3.5 The power modulus of continuity at the conical point for weak solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 Proof of Theorem 3.3 . . . . . . . . . . . . . . . . . . . . .

5 5 6 8 9 10 11 13 13 13 14 17 17 21 25 28

33 33 37 43 50 56 56

viii

Contents

3.6 3.7

3.5.2 Remark to Theorem 3.7 . . . . . . . . . . . . . . . . . . . . Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

58 59 64

4 Transmission problem for the Laplace operator with N different media 67 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 4.2 Auxiliary statements and inequalities . . . . . . . . . . . . . . . . . 73 4.2.1 The eigenvalue problem . . . . . . . . . . . . . . . . . . . . 73 4.2.2 The comparison principle . . . . . . . . . . . . . . . . . . . 74 4.3 The barrier function. The preliminary estimate of the solution modulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 4.4 Local estimate at the boundary . . . . . . . . . . . . . . . . . . . . 82 4.5 Global integral estimates . . . . . . . . . . . . . . . . . . . . . . . . 82 4.6 Local integral weighted estimates . . . . . . . . . . . . . . . . . . 90 4.7 The power modulus of continuity at the conical point for weak solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 4.8 Appendix: Eigenvalue transmission problem in a composite plane domain with an angular point . . . . . . . . . . . . . . . . . . . . . 96 4.8.1 Four-media transmission problem . . . . . . . . . . . . . . . 98 4.8.2 Three-media transmission problem . . . . . . . . . . . . . . 101 4.8.3 Two-media transmission problem . . . . . . . . . . . . . . . 103 5 Transmission problem for weak quasi-linear elliptic equations in a conical domain 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Local estimate at the boundary . . . . . . . . . . . . . . . . . . 5.3 Global integral estimate . . . . . . . . . . . . . . . . . . . . . . 5.4 Local integral weighted estimates . . . . . . . . . . . . . . . . . 5.5 The power modulus of continuity at the conical point for weak solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.1 Proof of Theorem 5.3. . . . . . . . . . . . . . . . . . . . 5.5.2 Remark to Theorem 5.5 . . . . . . . . . . . . . . . . . . 5.6 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Transmission problem for strong quasi-linear elliptic equations in a conical domain 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Comparison principle . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Maximum principle . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Local estimate at the boundary . . . . . . . . . . . . . . . . . . 6.5 Integral estimates . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 The power modulus of continuity at the conical point for weak solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . .

. . . .

105 105 108 108 114

. . . .

. . . .

120 120 122 123

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. . . . .

135 135 139 141 148 155

. . 161 . . 162

Contents 6.7.1

ix The barrier function. The preliminary estimate of the solution modulus . . . . . . . . . . . . . . . . . . . . . . . . 162

7 Best possible estimates of solutions to the transmission problem for a quasi-linear elliptic divergence second-order equation in a domain with a boundary edge 171 7.1 Introduction. Assumptions . . . . . . . . . . . . . . . . . . . . . . . 171 7.2 The comparison principle . . . . . . . . . . . . . . . . . . . . . . . 176 7.3 Construction of the barrier function . . . . . . . . . . . . . . . . . 177 7.4 The case aγ+ = γa− . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 + − 7.4.1 The barrier function . . . . . . . . . . . . . . . . . . . . . . 180 7.4.2 Properties of the eigenvalue λ for (CPE) . . . . . . . . . . . 185 7.4.3 Perturbation of problem (M iP ) . . . . . . . . . . . . . . . . 187 7.4.4 Estimates of the (T DQL) solution modulus . . . . . . . . . 190 7.4.5 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 7.5 The case aγ+ = γa− . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 + − 7.5.1 Properties of solutions to the Sturm-Liouville boundary problem (StL) . . . . . . . . . . . . . . . . . . . . . . . . . 198 7.5.2 Estimates of the (T DQL) solution modulus . . . . . . . . . 203 Bibliography

209

Index

215

Notation Index

217

Preface The goal of this book is to investigate the behaviour of weak solutions to the elliptic transmisssion problem in a neighborhood of boundary singularities: angular and conic points or edges. We consider this problem both for linear and quasi-linear (very little studied) equations. In style and methods of research, this book is close to our monograph [14] together with Prof. V. Kondratiev. The book consists of an Introduction, seven chapters, a Bibliography and Indexes. Chapter 1 is of auxiliary character. We recall the basic definitions and properties of Sobolev spaces and weighted Sobolev-Kondratiev spaces. Here we recall also the well-known Stampacchia’s Lemma and derive a generalization for the solution of the Cauchy problem – the Gronwall-Chaplygin type inequality. Chapter 2 deals with the eigenvalue problem for m-Laplace-Beltrami operator. By the variational principle we prove a new integro-differential FriedrichsWirtinger type inequality. This inequality is the basis for obtaining of precise exponents of the decreasing rate of the solution near boundary singularities. Chapter 3 deals with the investigation of the transmission problem for linear elliptic second order equations in the domains with boundary conic point. Chapter 4 is devoted to the transmission problem in conic domains with N different media for an equation with the Laplace operator in the principal part. Chapters 5, 6 and 7 deal with the investigation of the transmission problem for quasi-linear elliptic second order equations in the domains with boundary conic point (Chapters 5–6) or with an edge at the boundary of a domain. All results are given in the book with complete proofs. The book is based on the author’s research he had made over the past years (see [8, 9, 10, 11, 12, 13]). I would like to express my gratitude to Dr. Mikhail Kolev from University of Warmia and Mazury in Olsztyn who improved my English and Dr. Mykhaylo Plesha from Lviv (Ukraine) who executed figures in TEX. It also should be mentioned that the work on the book was made possible by the support of the Polish Ministry of Science and Higher Education through Grant Nr N201 381834. This help is gratefully acknowledged. Olsztyn, Poland Spring 2010

Introduction Transmission problems appear frequently in various fields of physics and technics. For instance, one of the important problems of the electrodynamics of solid media is the research on electromagnetic processes in ferromagnetic media with different dielectric constants. This type of problem appears as well in solid mechanics if a body consists of composite materials. Let us mention also vibrating folded membranes, composite plates, folded plates, junctions in elastic multi-structures etc. In this work we obtain estimates of the weak solutions to the elliptic transmission problem near singularities on the boundary (conical boundary point or edge). Analogous results were established in [14] for the Dirichlet, mixed and Robin problems in domains with singularities on the boundary without interfaces. Many mathematicians have considered transmission problems for linear elliptic equations. For the first time, M. Schechter [65], O.A. Oleinik [61], V.A. Il’in [31], O.A. Ladyzhenskaya and N.N. Ural’tseva [43], Z.G. Sheftel [66], M.V. Borsuk [6] studied general interface problems for linear second-order elliptic operators in smooth domains. V.A. Il’in and I.A. Shishmarev [31, 32] investigated the classical solvability of the Dirichlet and Neumann problems for general (non-divergence) linear elliptic second-order equations with discontinuous coefficients. They used the potential method. Van Tun [69] extended their results and established Schauder estimates. M. Schechter [65], O.A. Oleinik [61] as well as O.A. Ladyzhenskaya and N.N. Ural’tseva [43] (see also [44]) investigated the weak solvability of the Dirichlet transmission problem for divergence linear elliptic second-order equations and established the smoothness of weak solutions near the interface. Z.G. Sheftel [66] extended the S. Agmon, A. Douglis and L. Nirenberg results [1] to general boundary value problems for elliptic equations of any order. M.V. Borsuk [6] derived exact Schauder estimates of solutions to the transmission problem for general linear elliptic second-order equations with the third boundary condition. The exactness of these estimates consists in their explicit dependence on the smoothness of coefficients in estimating constants. A. Lorenzi [50] proved an existence and uniqueness theorem for solutions in W 2,p (Rn ) of second-order linear elliptic equations, whose coefficients are constant-valued in the half-spaces Rn+ and Rn− . M. Costabel and E. Stephan [21] applied a direct boundary integral equation method for transmission problems.

M. Borsuk, Transmission Problems for Elliptic Second-Order Equations in Non-Smooth Domains, Frontiers in Mathematics, DOI 10.1007/978-3-0346-0477-2_1, © Springer Basel AG 2010

1

2

Introduction

Starting in the 1970s many mathematicians have studied linear transmission problems in non-smooth domains in some particular linear cases (see for example [15, 16, 17, 19, 51, 53, 54, 55, 56] and references cited in [58, 59]). T. Petersdorf [62] proved an existence and uniqueness theorem for solutions in W 1 (G), where G is the number of adjacent Lipschitz domains, of the transmission problem for the homogeneous Helmholz equation: he reduced the transmission problem to a system of boundary integral equations. R. Kellogg [35, 36, 37, 38], Ben M’Barek and M. Merigot [3], K. Lemrabet [45], M. Dobrowolski [24] investigated the behavior of solutions of the transmission Dirichlet and Neumann problems for the Laplace operator in a neighborhood of an angular boundary point of a plane-bounded domain. K. Lemrabet [46] studied also the case of bounded threedimensional domains. H. Blumenfeld [5] studied the regularity of solutions of the mixed transmission problem in a polygon and derived the approximate solution by the finite element method. M. Dauge and S. Nicaise [22, 23, 58] extended some of the V. Kondratiev–V. Maz’ya–P. Grisvard classical results, concerning the singular behavior of the weak solution of a boundary value problem in a non-smooth domain, to the transmission problems for the Laplace operator with oblique derivative boundary and interface condition on a two-dimensional polygonal topological network: they obtained index formulae, a calculation of the dimension of the kernel, an expansion of weak solutions into regular and singular parts and established higher regularity results. L. Escauriaza, E. Fabes, G. Verchota [27] investigated the regularity for weak solutions to transmission problems in plane domains with 1 internal Lipschitz boundaries. The H 1+ 4 -regularity result for the Laplace interface problem in two dimensions was obtained by M. Petzoldt [63]. General linear two- or three-dimensional interface problems in polygonal and polyhedral domains were considered by S. Nicaise, A.-M. Sändig in [59]: they studied the solvability, the regularity and the solution asymptotics in weighted Sobolev spaces. They investigated also [60] the regularity and boundary integral equations for two- or three-dimensional transmission problems with the Laplace and elasticity operators (see also the overview by D. Knees–A.-M. Sändig [40]). Regularity results in terms of weighted Sobolev-Kondratiev spaces were obtained as well by W. Chikouche, D. Mercier and S. Nicaise [19] for two- and three-dimensional transmission problems for the Laplace operator using a penalization technique. D. Kapanadze and B.-W. Schulze studied boundary-contact problems with conical [33] singularities and edge [34] singularities at the interfaces for general linear, any order, elliptic equations (as well as systems). They constructed parametrices and showed the regularity with asymptotics of solutions in weighted Sobolev-Kondratiev spaces. Concerning the transmission problem for quasi-linear elliptic equations we know a few works. At first the quasi-linear transmission problem was investigated only in smooth domains. M.V. Borsuk [6] and then V.Ja. Rivkind–N.N. Ural’tseva [64] proved the classical solvability of the transmission problem for quasi-linear elliptic second-order equations with co-normal derivative boundary and interface conditions. Later N. Kutev–P.L. Lions [41] investigated the transmission problem for general uniform elliptic second-order equations with Dirichlet boundary

Introduction

3

condition. They proved by the regularization method the existence and the uniqueness of C1 (G)∩C 0 (G)− solutions with appropriate conditions on the nonlinearities and the regularity on each side of the interface. We know only two papers, namely that by D. Knees [39] and by C. Ebmeyer, J. Frehse and M. Kassmann [26], which are concerned with the study of the regularity of weak solutions of special quasi-linear transmission problems on polyhedral domains. In particular, the regularity in the Nikolskii spaces up to the transmission surface and the boundary for the Dirichlet transmission problems are proved in [26]. A principal new feature of our work is the consideration of our estimates of weak solutions to the transmission problem for linear elliptic equations with minimal smooth coefficients in n-dimensional conic domains. Our examples demonstrate this fact. Moreover we consider the estimates of weak solutions for general divergence quasi-linear elliptic second-order equations in n-dimensional conic domains or in domains with edges. We shall study the following elliptic transmission problems: • for the linear transmission problem ⎧  ij  ∂ ⎪ L[u] ≡ ∂x a (x)uxj + ai (x)uxi + a(x)u = f (x), ⎪ i ⎪ ⎪ ⎪ ⎪ ⎪

 ⎨  1 x [u]Σ0 = 0, S[u] ≡ ∂u + σ ∂ν Σ0 |x| |x| u(x) = h(x), ⎪ ⎪ ⎪ ⎪ ⎪

 ⎪ ⎪ ⎩B[u] ≡ ∂u + 1 γ x u = g(x), ∂ν

|x|

|x|

x ∈ G \ Σ0 ; x ∈ Σ0 ;

(L)

x ∈ ∂G \ {Σ0 ∪ O};

• for the Laplace operator with N different media and mixed boundary condition ⎧ N −1 ⎪ ⎪ au − pu(x) = f (x), x ∈ G\ ∪ Σk ; ⎪ ⎪ k=1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨[u]Σk = 0, x ∈ Σk ,

(LN ) ∂u 1 + |x| βk (ω)u(x) = hk (x), k = 1, . . . , N − 1; Sk [u] ≡ a ∂nk ⎪ ⎪ Σk ⎪ ⎪ ⎪ ⎪ ⎪ ⎪   ⎪ ⎪ N −1 ⎪ ∂u 1 ⎪ x ∈ ∂G \ ∪ Σk ∪ O , ⎩B[u] ≡ α(x) · a ∂n + |x| γ(ω)u(x) = g(x), k=1



0, if x ∈ D and D ⊆ ∂G is the part of the 1, if x ∈ /D boundary ∂G where we consider the Dirichlet boundary condition; where a > 0, p ≥ 0; α(x) =

4

Introduction

• for weak nonlinear equations ⎧  d  q ij ⎪ ⎪ |u| a (x)uxj + b(x, u, ∇u) = 0, q ≥ 0, x ∈ G \ Σ0 ; ⎪− ⎪ ⎪ dxi ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ [u]Σ0 = 0,

 (WQL)  1 x q ⎪ u · |u| + σ = h(x, u), x ∈ Σ ; ⎪S[u] ≡ ∂u 0 ⎪ ∂ν |x| |x| Σ0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪

 ⎪ ⎪ ⎩B[u] ≡ ∂u + 1 γ x u · |u|q = g(x, u), x ∈ ∂G \ {Σ0 ∪ O}; ∂ν |x| |x| here q ≥ 0; if q = 0 and b(x, u, ∇u) = ai (x)uxi + a(x)u − f (x) then we have linear transmission problem (L); • for general elliptic divergence quasi-linear equations ⎧ d ⎪ ⎪ ai (x, u, ∇u) + b(x, u, ∇u) = 0, x ∈ G \ Σ0 ; ⎪− ⎪ dx ⎪ i ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ [u]Σ0 = 0,

 (QL)  1 x q+m−2 ⎪ u · |u| + = h(x, u), x ∈ Σ ; ⎪S[u] ≡ ∂u m−1 σ 0 ⎪ ∂ν |x| |x| Σ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪

 ⎪ ⎪ 1 x ⎩B[u] ≡ ∂u + m−1 u · |u|q+m−2 = g(x, u), γ |x| x ∈ ∂G \ {Σ0 ∪ O}; ∂ν |x| • and for elliptic divergence quasi-linear equations with triple degeneration ⎧ d i − dxi a (x, u, ∇u)+b(x, u, ∇u)+a0c(x, u) = f (x), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪[u]Σ0 = 0, ⎪ ⎪ ⎪ ⎪ ⎨  S[u] ≡ aai (x, u, ∇u)ni Σ0 ⎪ ⎪ ⎪ + β0 rτ −m+1 u|u|q+m−2 = h(x, u), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ B[u] ≡ aai (x, u, ∇u)ni ⎪ ⎪ ⎩ + γ(ω)rτ −m+1 u|u|q+m−2 = g(x, u),

x ∈ G \ Σ0 ; x ∈ Σ0 ; (TDQL) x ∈ Σ0 ;

x ∈ ∂G \ {Σ0 ∪ Γ0 },

where q ≥ 0, m > 1, a > 0, a0 ≥ 0, β0 > 0, τ ≥ m − 2 are given numbers and Σ0 contains an edge Γ0 . Summation over repeated indices from 1 to n is understood; here everywhere denotes the conormal derivative operator.

∂ ∂ν

Chapter 1

Preliminaries 1.1 List of symbols Let G ⊂ Rn , n ≥ 2 be a bounded domain with boundary ∂G that is a smooth surface everywhere except at the origin O ∈ ∂G and near the point O it is a conical surface with vertex at O. We assume that G = G+ ∪ G− ∪ Σ0 is divided into two subdomains G+ and G− by a Σ0 = G ∩ {xn = 0}, where O ∈ Σ0 . Let us fix some notation used throughout the work: • [l] – the integral part of l (if l is not integer); • R – the set of real numbers; • R+ – the set of positive numbers; • Rn – the n-dimensional Euclidean space, n ≥ 2; • N – the set of natural numbers; • N0 = N ∪ {0} – the set of non-negative integers; • S n−1 – a unit sphere in Rn centered at O; • (r, ω), ω = (ω1 , ω2 , . . . , ωn−1 ) – spherical coordinates of x ∈ Rn with pole O: x1 = r cos ω1 , x2 = r cos ω2 sin ω1 , . . . , xn−1 = r cos ωn−1 sin ωn−2 . . . sin ω1 , xn = r sin ωn−1 sin ωn−2 . . . sin ω1 ; • C – a rotational cone {x1 > r cos ω20 } with the vertex at O; • ∂C – the lateral surface of C : {x1 = r cos ω20 }; • Ω – a domain on the unit sphere S n−1 with smooth boundary ∂Ω obtained by the intersection of the cone C with the sphere S n−1 ; • ∂Ω = ∂C ∩ S n−1 ; • Gba = {(r, ω) | 0 ≤ a < r < b; ω ∈ Ω} ∩ G – a layer in Rn ;

M. Borsuk, Transmission Problems for Elliptic Second-Order Equations in Non-Smooth Domains, Frontiers in Mathematics, DOI 10.1007/978-3-0346-0477-2_2, © Springer Basel AG 2010

5

6

Chapter 1. Preliminaries • Γba = {(r, ω) | 0 ≤ a < r < b; ω ∈ ∂Ω} ∩ ∂G – the lateral surface of layer Gba ; • Gd = G \ Gd0 ; Γd = ∂G \ Γd0 , d > 0; • Σba = Gba ∩ {xn = 0} ⊂ Σ0 ; Σd = Σ0 \ Σd0 , d > 0; • Ωρ = Gd0 ∩ {|x| = ρ};

0 < ρ < d;

• Ω+ = Ω ∩ {xn > 0}, Ω− = Ω ∩ {xn < 0} =⇒ Ω = Ω+ ∪ Ω− ∪ σ0 ; • σ0 = Σ0 ∩ Ω; ∂± Ω = Ω± ∩ ∂C; ∂Ω± = ∂± Ω ∪ σ0 ; −k

d • G(k) := G22−(k+1) , k = 0, 1, 2, . . .; d   f+ (x), u+ (x), x ∈ G+ , f (x) = • u(x) = u− (x), x ∈ G− ; f− (x),

x ∈ G+ , x ∈ G−

etc.;

• [u]Σ0 denotes the saltus of  the function u(x)  on crossing Σ0 , i.e.,    lim u± (y); [u]Σ0 = u+ (x) −u− (x) , where u± (x) = Σ0

Σ0

Σ0

G± y→x∈Σ0

→ → • ni = cos(− n , xi ), i = 1, . . . , n, where − n denotes the unit outward vector with respect to G+ (or G) normal to Σ0 (respectively ∂G \ O).

1.2 Operators and formulae related to spherical coordinates Let us recall first some well-known formulae related to the spherical coordinates (r, ω1 , . . . , ωn−1 ) centered at the conical point O: dx = rn−1 drdΩ,

dΩρ = ρn−1 dΩ,

(1.2.1)

dΩ = J(ω)dω denotes the (n − 1)-dimensional area element of the unit sphere, where J(ω) = sinn−2 ω1 sinn−3 ω2 . . . sin ωn−2 ,

(1.2.2)

dω = dω1 . . . dωn−1 ,

(1.2.3)

ds = rn−2 drdσ denotes the (n−1)-dimensional area element on ∂C and dσ denotes the (n − 2)-dimensional area element on ∂Ω; 2

|∇u| =



∂u ∂r

2 +

1 2 |∇ω u| , r2

(1.2.4)

1.2. Operators and formulae related to spherical coordinates

x1

7

:

V0 60

*0

*0

O Figure 1

where |∇ω u| denotes the projection of the vector ∇u onto the tangent plane to the unit sphere at the point ω:  ∇ω u = 2

|∇ω u| =

n−1  i=1 2

1 qi



∂u ∂ωi

2

∂u 1 ∂u 1 ,..., √ √ q1 ∂ω1 qn−1 ∂ωn−1

 ,

(1.2.5)

, where q1 = 1, qi = (sin ω1 · · · sin ωi−1 )2 , i ≥ 2,

1 ∂ u n − 1 ∂u + 2 Δω u, denotes the Laplace operator, + ∂r2 r ∂r r   n−1 n−1 1 ∂ ∂u 1  ∂  J(ω) ∂u   n−i−1 sin = · ωi Δω u = J(ω) i=1 ∂ωi qi ∂ωi ∂ωi q sinn−i−1 ωi ∂ωi i=1 j Δu =

(1.2.6) denotes the Beltrami-Laplace operator,

divω u =

  n−1 J(ω) 1  ∂ u √ i , u = (u1 , . . . , un−1 ). J(ω) i=1 ∂ωi qi

(1.2.7)

8

Chapter 1. Preliminaries

Lemma 1.1. Assume for some d > 0 that Gd0 is the rotational cone with the vertex at O and the aperture ω0 , and let   n   ω0  2 d 2 ω0 2 Γ0 = (r, ω)x1 = cot , ω0 ∈ (0, 2π) . x ; |ω1 | = (1.2.8) 2 i=2 i 2 Then xi cos( n, xi )|Γd0 = 0, and cos( n, x1 )|Γd0 = − sin

ω0 . 2

(1.2.9)

Proof. By virtue of (1.2.8) we can rewrite the equation of Γd0 in the form F (x) ≡ ∂F n  ∂xi x21 − cot2 ω20 x2i = 0. We use the formula cos( n, xi ) = |∇F | , ∀i = 1, . . . , n. Because of

i=2 ∂F ∂x1 =

2x1 ,

∂F ∂xi

= −2 cot2

ω0 2 xi ,

∀i = 2, . . . , n, we obtain

∂F  1 2 xi  d= |∇F | ∂xi Γ0 |∇F |

 xi cos( n, xi )Γd = 0

 x21

n ω0  2 − cot x 2 i=2 i 2

   

Γd 0

= 0.

Furthermore from 2



|∇F | =   ⇒ |∇F |2 

  n  2 4 ω0 2 + = 4 x1 + cot x ∂xi 2 i=2 i i=2   cos2 ω20 4x21 2 = = 4x1 1 + , 2 ω0 sin 2 sin2 ω20

∂F ∂x1

Γd 0

2

2 n   ∂F

we get   cos( n, x1 )

Γd 0

= −2x1

sin ω20 ω0 = − sin , 2x1 2

since ∠ ( n, x1 ) >

π . 2



1.3 The quasi-distance function rε and its properties Let us assume that the cone K is contained in a rotational cone C with the opening angle ω0 . Furthermore, let us suppose that the axis of C coincides with {(x1 , 0, . . . , 0) : x1 > 0}. In this case we define the quasi-distance rε (x) as follows. ¯ and consider the unit radius-vector We fix the point Q = (−1, 0, . . . , 0) ∈ S n−1 \ Ω l = OQ = {−1, 0, . . . , 0}. We denote by r the radius-vector of the point x ∈ G ¯ d / G0 for all ε ∈]0, d[, and introduce the vector rε = r − ε l for each ε > 0. Since ε l ∈ ¯ it follows that rε (x) = | r − εl| = 0 for all x ∈ G. It is easy to verify that rε (x) has the following properties (see in detail §1.4 [14]): 1. thereexists h > 0 such that: rε (x) ≥ hr and rε (x) ≥ hε, ∀x ∈ G, where 1, if x1 ≥ 0, h= sin ω20 , if x1 < 0;

1.4. Function spaces

9

2. if x ∈ Gd , then rε (x) ≥

for all ε ∈]0, d2 [ ; ¯ 3. lim+ rε (x) = r, for all x ∈ G; d 2

ε→0

4. |∇rε |2 = 1, and rε =

n−1 rε .

1.4 Function spaces We use the standard function spaces: • C k (G± ) with the norm |u± |k,G± ; • the Lebesgue space Lp (G± ), p ≥ 1 with the norm u± p,G± ; • the Sobolev space W k,p (G± ) with the norm u± p,k;G± and introduce their direct sums • Ck (G) = C k (G+ )  C k (G− ) with the norm |u|k,G = |u+ |k,G+ + |u− |k,G− ; • Lp (G) = Lp (G+ )  Lp (G− ) with the norm   p1    p1 uLp (G) = |u+ |p dx + |u− |p dx ; G+

G−

• Wk,p (G) = W k,p (G+ )  W k,p (G− ) with the norm  k  p1   k  p1   β p β p up,k;G = |D u+ | dx + |D u− | dx . G+ |β|=0

G− |β|=0

k (G) For any integer k ≥ 0 and real α we define the weighted Sobolev space Vp,α  as the space of distributions u ∈ D (G) with the finite norm  k  p1  α+p(|β|−k) β p uVp,α k (G) = r |D u+ | dx G+ |β|=0



+

k 

 p1 rα+p(|β|−k) |Dβ u− |p dx

G− |β|=0 k− 1

and Vp,α p (∂G) as the space of functions ϕ, given on ∂G, with the norm ϕ k− p1 = inf ΦVp,α k (G) , where the infimum is taken over all functions Φ Vp,α (∂G)   such that Φ = ϕ in the sense of traces. We write ∂G

Wk (G) ≡ Wk,2 (G),

◦ k

k Wα (G) ≡ V2,α (G),

1 ◦ k− 2

Wα

k− 1

(∂G) ≡ V2,α 2 (∂G).

10

Chapter 1. Preliminaries

1.5 Some inequalities In this section we recall some elementary inequalities (see e.g. [2, 30]) which will be frequently used throughout this book. Lemma 1.2 (Cauchy’s Inequality). For any a, b ≥ 0 and ε > 0, we have ab ≤

ε 2 1 a + b2 . 2 2ε

(1.5.1)

Lemma 1.3 (Young’s Inequality). For any a, b ≥ 0, ε > 0 and p, q > 1 with 1 1 p + q = 1, we have  q 1 b 1 p . (1.5.2) ab ≤ (εa) + p q ε Lemma 1.4 (Hölder’s Inequality). For any non-negative real numbers ai , bi , i = 1, . . . , n, and p, q ∈ R with 1p + 1q = 1, we have n 

 ai b i ≤

i=1

n 

1/p  api

i=1

n 

1/q bqi

(1.5.3)

.

i=1

Lemma 1.5. (Theorem 41 [30]). For any non-negative real numbers a, b and m ≥ 1 we have mam−1 (a − b) ≥ am − bm ≥ mbm−1 (a − b). (1.5.4) Lemma 1.6 (Jensen’s Inequality (Theorem 65 [30])). Let ai , i = 1, . . . , n, be any non-negative real numbers and p > 0. Then  n p n n    p λ ai ≤ ai ≤Λ api , (1.5.5) i=1

where λ = min(1, n

p−1

i=1

) and Λ = max(1, n

i=1 p−1

).

Lemma 1.7. For any a, b ∈ R and m > 1 we have |b|m ≥ |a|m + m|a|m−2 a(b − a). Proof. By the Young inequality (1.5.2) with ε = 1, p = m, q =

(1.5.6) m m−1 ,

we obtain

m|a|m−2 ab ≤ m|b| · |a|m−1 ≤ |b|m + (m − 1)|a|m =⇒ (1.5.6). Theorem 1.8 (Hölder’s Inequality, see Theorem 189 [30]). Let p, q > 1 with 1p + 1q = 1 and u ∈ Lp (G), v ∈ Lq (G). Then  |uv|dx ≤ uLp(G) vLq (G) . G

If p = 1, then (1.5.7) is valid with q = ∞.



(1.5.7)

1.6. Sobolev embedding theorems

11

Corollary 1.9. Let 1 ≤ p ≤ q and u ∈ Lq (G). Then uLp(G) ≤ (meas G)1/p−1/q uLq (G) .

(1.5.8)

Corollary 1.10 (Interpolation inequality). Let 1 < p ≤ q ≤ r and 1/q = λ/p + (1 − λ)/r. Then the inequality uLq (G) ≤ uλLp(G) u1−λ Lr (G)

(1.5.9)

holds for all u ∈ Lr (G). Theorem 1.11 (Clarkson’s Inequality, see §3.2, Chapter I [68]). Let u, v ∈ Lp (G). Then       u + v p  u − v p 1

p p     u + ≤ + v Lp (G) Lp (G) , 2 ≤ p < ∞;  2  p  2  p 2 L (G) L (G)  p  p 1  u + v  p−1  p−1

1  u − v  p−1 1     upLp(G) + vpLp (G) + ≤ , 1 ≤ p ≤ 2.   p   2 2 2 2 L (G) p L (G)

Theorem 1.12 (Fatou’s Theorem, see Theorem 19 §6, Chapter III [25]). Let {fk } ∈ L1 (G), k ∈ N, be a sequence of non-negative functions which is convergent almost everywhere in G to the function f . Then   f dx ≤ sup fk dx. (1.5.10) G

G

We need also the following well-known inequalities: Theorem 1.13. (See e.g. (6.23), (6.24) Chapter I [42] or Lemma 6.36 [49]). Let ∂G be piecewise smooth and u ∈ W 1,1 (G). Then there is a constant c > 0 that depends only on G such that   |u|ds ≤ c (|u| + |∇u|) dx for each Γ ⊆ ∂Ω; (1.5.11) Γ

 ∂G

v 2 ds ≤

G



(δ|∇v|2 + cδ v 2 )dx, ∀v(x) ∈ W 1,2 (G), ∀δ > 0.

(1.5.12)

G

1.6 Sobolev embedding theorems We now recall the well-known Sobolev inequalities and Kondrashov compactness results which are frequently referred as the embedding theorems (see [68], §§1.4.5– 1.4.6 [52], §7.7[29]).

12

Chapter 1. Preliminaries

Theorem 1.14 (Sobolev inequalities (see e.g. [70, Theorem 2.4.1], [29, Theorem 7.10])). Let G be a bounded open domain in Rn and p > 1. Then  W01,p (G)

→

np (G) L n−p

for p < n,

C 0 (G)

for p > n.

(1.6.1)

Furthermore, there exists a constant c = c(n, p) such that for all u ∈ W01,p (G) we have uLnp/(n−p)(G) ≤ c∇uLp(G) (1.6.2) for p < n and sup |u| ≤ c(meas G)1/n−1/p ∇uLp(G)

(1.6.3)

G

for p > n. The following Embedding Theorems 1.15–1.20 were proved first by Sobolev [68] and can be found with complete proofs in [52, Section 1.4]. Let G be a C 0,1 bounded domain in Rn . Theorem 1.15. Let k ∈ N and p ∈ R with p ≥ 1, kp < n. Then the embedding W k,p (G) → Lq (G)

(1.6.4)

is continuous for 1 ≤ q ≤ np/(n − kp) and compact for 1 ≤ q < np/(n − kp). If kp = n, then the embedding (1.6.4) is continuous and compact for any q ≥ 1. Theorem 1.16. (For the proof see, for example, (2.19) §2, chapter II [43]). Let u ∈ W 1 (G). Then u2L

2p p−2

(G)

≤ δ∇u2L2 (G) + c(δ, p, n, meas G)u2L2(G) ,

p > n, ∀δ > 0. (1.6.5)

Theorem 1.17. Let k ∈ N0 , m ∈ N and let p, q ∈ R with p, q ≥ 1. If kp < n, then the embedding W m+k,p (G) → W m,q (G) (1.6.6) is continuous for any q ∈ R satisfying 1 ≤ q ≤ np/(n − kp). If k = np, then the embedding (1.6.6) is continuous for any q ≥ 1. Theorem 1.18. Let k, m ∈ N0 and p > 1. Then the embedding W k,p (G) → C m+β (G) is continuous if (k − m − 1)p < n < (k − m)p

and

0 < β ≤ k − m − n/p,

(1.6.7)

and compact if the inequality for β in (1.6.7) is sharp. If (k − m − 1)p = n, then the embedding is continuous for any β ∈ (0, 1).

1.7. The Cauchy problem for a differential inequality

13

Theorem 1.19. Let u ∈ W k,p (G) with k ∈ N, p ∈ R, kp > n and p > 1. Then u ∈ C m (G) for 0 ≤ m < k − n/p and there exists a constant c, independent of u, such that sup |Dα u(x)| ≤ cuW k,p (G)

x∈G

for all |α| < k − n/p. Theorem 1.20. Let G be a lipschitzian domain and Ts ⊂ G be a piecewise C k smooth s-dimensional manifold. Let k ≥ 1, p > 1, kp ≤ n, n − kp < s ≤ n, 1 ≤ q ≤ q∗ = sp/(n − kp). Then the embedding W k,p (G) → Lq (Ts ) is continuous and the inequality uLq (Ts ) ≤ cuW k,p (G) (1.6.8) holds. If q < q∗, then the above embedding is compact.

1.7 The Cauchy problem for a differential inequality Theorem 1.21. (See Theorem 1.57 [14]). Let V () be a monotonically increasing, non-negative differentiable function defined on [0, 2d] and satisfying the problem  V  (ρ) − P()V () + N (ρ)V (2ρ) + Q(ρ) ≥ 0, 0 < ρ < d, (CP) V (d) ≤ V0 , where P(), N (), Q() are non-negative continuous functions defined on [0, 2d] and V0 is a constant. Then we have d   d  d  τ   V () ≤ exp B(τ )dτ V0 exp − P(τ )dτ + Q(τ ) exp − P(σ)dσ dτ 







(1.7.1) with

2  B() = N () exp P(σ)dσ .

(1.7.2)



1.8 Additional auxiliary results 1.8.1 Stampacchia’s Lemma Lemma 1.22. (See Lemma 3.11 of [57]). Let ϕ : [k0 , ∞) → R be a non-negative and non-increasing function which satisfies ϕ(h) ≤

C [ϕ(k)]β (h − k)α

for

h > k > k0 ,

(1.8.1)

14

Chapter 1. Preliminaries

where C, α, β are positive constants with β > 1. Then ϕ(k0 + d) = 0, where dα = β−1 αβ/(β−1) C |ϕ(k0 )| 2 . Proof. For each s = 1, 2, . . . we let ks = k0 + d − {ϕ(ks )}. From (1.8.1) we obtain ϕ(ks+1 ) ≤

C2(s+1)α [ϕ(ks )]β , dα

d 2s

and consider the sequence

s = 1, 2, . . . .

(1.8.2)

We now prove by induction that ϕ(ks ) ≤

ϕ(k0 ) α < 0. , where μ = 2−sμ 1−β

(1.8.3)

For s = 0 the claim is evident. Let us suppose that (1.8.3) is valid up to s. By (1.8.2) and the definition of dα if follows that ϕ(ks+1 ) ≤ C

2(s+1)α [ϕ(k0 ]β ϕ(k0 ) ≤ −(s+1)μ . dα 2−sβμ 2

(1.8.4)

Since the right-hand side of (1.8.4) tends to zero as s → ∞, we obtain 0 ≤ ϕ(k0 + d) ≤ ϕ(ks ) → 0. 

1.8.2 Other assertions Lemma 1.23. (See Lemma 2.1 [20]). Let us consider the function  η(x) =

eκx − 1, x ≥ 0, −e−κx + 1, x ≤ 0,

where κ > 0. Let a, b be positive constants, m > 1. If κ > (2b/a) + m, then we have a aη  (x) − b|η(x)| ≥ eκx , ∀x ≥ 0; x  2m , ∀x ≥ 0. η(x) ≥ η m

(1.8.5) (1.8.6)

Moreover, there exist some d ≥ 0 and M > 0 such that x  m x  m , η  (x) ≤ M η , η(x) ≤ M η m m |η(x)| ≥ x, ∀x ∈ R. Proof. We refer to Lemma 1.60 [14] for the proof.

∀x ≥ d,

(1.8.7) (1.8.8) 

1.8. Additional auxiliary results

15

Lemma 1.24. (See Lemma 4.1 in Chapter 2 [18]). Let ψ(s) be a bounded nonnegative function defined on the interval [0, ]. Suppose that for any 0 ≤ σ < s ≤  the function ψ(s) satisfies ψ(σ) ≤ δψ(s) +

A + B, (s − σ)α

where δ ∈ (0, 1), A, B and α are non-negative constants. Then   A + B , 0 ≤ r < R ≤ , ψ(r) ≤ C (R − r)α

(1.8.9)

where C depends only on α, δ. Theorem 1.25 (S. Bernstein [4]). If y is a bounded analytic solution of the equation y  = f (x, y, y  ), a0 < x < b0 ,

where |f (x, y, y  )| < Ay 2 + B,

then y  is also bounded:  

|y |
0, σ(ω) > 0, γ(ω) > 0. We characterize the first eigenvalue ϑ(m) of the eigenvalue problem for the m-Laplacian by    a|∇ω ψ|m dΩ + σ(ω)|ψ|m dσ + γ(ω)|ψ|m dσ σ0 Ω ∂Ω  . (2.1.1) ϑ(m) = inf m 1,m a|ψ| dΩ ψ∈W (Ω) ψ=0

Ω

Let us introduce the following functionals on C0 (Ω) ∩ W1,m (Ω):    m m F [ψ] = a|∇ω ψ| dΩ + σ(ω)|ψ| dσ + γ(ω)|ψ|m dσ, Ω

σ0



∂Ω

m

a|ψ| dΩ,

G[ψ] = Ω



H[ψ] =

   m m m a |∇ω ψ| − ϑ|ψ| dΩ + σ(ω)|ψ| dσ + γ(ω)|ψ|m dσ

Ω

σ0

∂Ω

and the corresponding forms   m−2 1 ∂ψ ∂η dΩ + σ(ω)|ψ|m−2 ψηdσ F (ψ, η) = a|∇ω ψ| qi ∂ωi ∂ωi σ0 Ω  + γ(ω)|ψ|m−2 ψηdσ,  G(ψ, η) =

∂Ω

a|ψ|m−2 ψηdΩ. Ω

2.1. Eigenvalue problem for the m-Laplacian in a bounded domain Further we define the set K =

19

 ! "  ψ ∈ C0 (Ω) ∩ W1,m (Ω) G[ψ] = 1 . Since

K ⊂ C0 (Ω) ∩ W1,m (Ω), F [ψ] is bounded from below for ψ ∈ K. The greatest lower bound of F [ψ] for this family we denote by ϑ: inf F [ψ] = ϑ. Now our aim ψ∈K

is to establish Theorem 2.2. Let Ω ⊂ S n−1 be a domain with smooth boundary ∂Ω and γ(ω) be a positive bounded piecewise smooth function on ∂Ω, σ(ω) be a positive continuous function on σ0 . There exist ϑ > 0 and a function ψ ∈ K such that F (ψ, η) − ϑG(ψ, η) = 0 for arbitrary η ∈ C0 (Ω) ∩ W1,m (Ω). In particular F [ψ] = ϑ. Proof. Because F [v] is bounded from below for v ∈ K, there is ϑ = inf F [v]. Let v∈K

us consider a sequence {vk } ⊂ K such that lim F [vk ] = ϑ (such a sequence exists k→∞

by the definition of the infimum). By K ⊂ W1,m (Ω), sequence {vk } is bounded in W1,m (Ω) and therefore compact in Lm (Ω). Choosing a subsequence, if needed, we can assume that {vk } is converging in Lm (Ω). As a result we obtain the following property of the functional G: given any ε > 0 we can find N (ε) such that G[vk − vl ] ≤ a∗ · vk − vl m Lm (Ω) < ε,

∀ε > 0

(2.1.2)

for all k, l > N (ε). Now we use the inequality (1.5.6):    vk + vl m m m m−2   vk (vl − vk ), m > 1.  2  ≥ |vk | + 2 |vk | By integrating this inequality over Ω:       vk + vl m m m   a dΩ ≥ a|vk | dΩ + a|vk |m−2 vk (vl − vk )dΩ 2  2 Ω

Ω

Ω

m m−1 ,

q = m, we get and applying the Young inequality with p = m  m m m − 1 m−1 1   δ |vk |m + m |vl − vk |m ,  vk |vk |m−2 (vl − vk ) ≤ |vk |m−1 |vl − vk | ≤ 2 2 2 2δ for all δ > 0. This fact yields that        vk + vl m m m − 1 m−1 1 m   δ a dΩ ≥ 1 − a|vk | dΩ − m a|vl − vk |m dΩ, 2  2 2δ Ω

k +vl

m m−1 m−1 − 2δε1m , ∀δ, ε1 2 δ 1 m setting ε = mϑ 2 ε1 , we get

1−



Ω

Ω

 m  m−1 ≥ 1 − m−1 G[vk ]− 2δ1m G[v for all δ > 0. This implies that G 2 2 δ  l − v k ], l > for all δ > 0. By using G[vk ] = G[vl ] = 1 and G[vl −vk ] < ε1 , we obtain G vk +v 2 v

m−1

> 0 for large k, l. Now, by choosing δ m = ε1 m and  G

 ε vk + vl >1− 2 ϑ

(2.1.3)

20

Chapter 2. Eigenvalue problem and integro-differential inequalities

for large k, l. The functionals F [v] and G[v] are homogeneous functionals and F [v] does not change under the passage from v to cv (c = therefore their ratio G[v] const = 0). Hence F [v] = inf F [v] = ϑ. inf v∈K v∈W1,m (Ω) G[v] l Therefore F [v] ≥ ϑG[v] for all v ∈ W1,m (Ω). Since vk +v ∈ W1,m (Ω) together 2 with vk , vl ∈ K, then    

vk + vl ε vk + vl ≥ ϑG >ϑ 1− = ϑ − ε, k, l > N (ε). F 2 2 ϑ

Let us take k and l large enough so that F [vk ] < ϑ + ε and F [vl ] < ϑ + ε. We apply the Clarkson inequalities (see Theorem 1.11): 1) m ≥ 2: 

   vl − vk 1 1 vl + vk F ≤ F [vl ] + F [vk ] − F 2 2 2 2 < ϑ + ε − (ϑ − ε) = 2ε 2) 1 < m ≤ 2: F

1 m−1



1      m−1 1 1 vl − vk vk + vl 1 m−1 ≤ F [vk ] + F [vl ] −F 2 2 2 2 2−m 1 1 2ε < (ϑ + ε) m−1 − (ϑ − ε) m−1 < (θ + ε) m−1 , m−1

by inequality (1.5.4). Consequently, F [vk − vl ] → 0,

as k, l → ∞.

(2.1.4)

From (2.1.2), (2.1.4) it follows that vk − vl W1,m (Ω) → 0, as k, l → ∞. Thus, {vk } is a Cauchy sequence in W1,m (Ω) and hence by the completeness of W1,m (Ω) there exists a function ψ ∈ W1,m (Ω) such that vk − ψW1,m (Ω) → 0, as k → ∞. Moreover,  F [vk ] − F [ψ] =

 a (|∇ω vk |m − |∇ω ψ|m ) dΩ +

Ω

σ0

 γ(ω) (|vk |m − |ψ|m ) dσ.

+ ∂Ω

σ(ω) (|vk |m − |ψ|m ) dσ

2.2. The Friedrichs-Wirtinger type inequality

21

Now, by (1.5.4) and the Hölder inequality, we have         a (|∇ω vk |m − |∇ω ψ|m ) dΩ ≤ m a|∇vk |m−1 |∇ω (vk − ψ)|dΩ     Ω Ω ⎛ ⎞1/m ⎛ ⎞(m−1)/m   ≤ m ⎝ a|∇ω (vk − ψ)|m dΩ⎠ · ⎝ a|∇ω vk |m dΩ⎠ → 0, as k → ∞, Ω

Ω

since vk ∈ W1,m . Furthermore, by using (1.5.6), the Hölder inequality for integrals and (1.5.11), we obtain   m m σ(ω) (|vk | − |ψ| ) dσ ≤ m σ(ω)|vk |m−1 · |vk − ψ|dσ σ0

σ0

≤ m max |σ(ω)| · vk − ψLm (σ0 ) · vk Lm (σ0 ) σ0

≤ cvk − ψW1,m (Ω) · vk W1,m (Ω) → 0, as k → ∞, and also 

 γ(ω) (|vk |m − |ψ|m ) dσ ≤ m

∂Ω

γ(ω)|vk |m−1 · |vk − ψ|dσ

∂Ω

≤ m max |γ(ω)| · vk − ψLm (∂Ω) · vk Lm (∂Ω) ∂Ω

≤ cvk − ψW1,m (Ω) · vk W1,m (Ω) → 0, as k → ∞. Therefore we get F [ψ] = lim F [vk ] = ϑ. Analogously it can be seen that G[ψ] = 1. k→∞

Suppose now that η is some function from C0 (Ω) ∩ W1,m (Ω). Consider the [ψ+εη] ratio F G[ψ+εη] . It is a continuously differentiable function of ε on some interval around the point ε = 0. This ratio has a minimum at ε = 0 equal to ϑ and therefore, by the Fermat Theorem, we have   F (ψ, η)G[ψ] − F [ψ]G(ψ, η) F [ψ + εη] = 0, =m G[ψ + εη] ε=0 G2 [ψ] which because of F [ψ] = ϑ, G[ψ] = 1, gives F (ψ, η) − ϑG(ψ, η) = 0, for all η ∈ C0 (Ω) ∩ W1,m (Ω). 

2.2 The Friedrichs-Wirtinger type inequality Now from the variational principle we obtain The Friedrichs-Wirtinger type inequality. Let ϑ(m) be the least positive eigenvalue of the problem (N EV P ) (it exists according to Theorem 2.2). Let Ω ⊂ S n−1 and

22

Chapter 2. Eigenvalue problem and integro-differential inequalities

ψ ∈ W1,m (Ω), γ(ω) be a positive bounded piecewise smooth function on ∂Ω, σ(ω) be a positive continuous function on σ0 . Then ⎫ ⎧     ⎬ ⎨ 1 a|ψ|m dΩ ≤ a|∇ω ψ|m dΩ + σ(ω)|ψ|m dσ + γ(ω)|ψ|m dσ (W )m ⎭ ϑ(m) ⎩ Ω

Ω

with the sharp constant

σ0

∂Ω

1 ϑ(m) .

Proof. By approximation arguments, it is clearly that we only need to consider the above described functionals F [ψ], G[ψ], H[ψ] on C1 (Ω) ∩ W2,m (Ω). We will find the minimum of the functional F [ψ] on the set K. For this we investigate the minimization of the functional H[ψ] on all functions ψ(ω) ∈ C1 (Ω) ∩ W2,m (Ω) which satisfy the boundary conditions from (N EV P ). We use formally the Lagrange multipliers and get the Euler equation from the condition δH[ψ] = 0. Calculating the first variation δH, we have ⎫  ⎧*N −1  +m  ⎨  1 ∂ψ 2 2 ⎬ m m 2 2 δH[ψ] = δ a − ϑ(ψ ) dΩ + σ(ω)(ψ 2 ) 2 dσ ⎩ ⎭ q ∂ωi i=1 i σ0 Ω   γ(ω)(ψ 2 ) 2 dσ m

+ ∂Ω

 J(ω) m−2 ∂ψ · δψdω · |∇ω ψ| = −m a qi ∂ωi i=1 Ω   ∂ψ · δψdσ − mϑ a|ψ|m−2 ψ · δψdΩ + m a|ψ|m−2 − ∂→ ν Ω ∂Ω     ∂ψ · δψdσ + m +m a|ψ|m−2 − σ(ω)|ψ|m−2 ψ · δψdσ ∂→ τ σ0 σ0  m−2 ψ · δψdσ + m γ(ω)|ψ| 

∂Ω



= −m

N −1 

∂ ∂ωi



! " a divω (|∇ω ψ|m−2 ∇ω ψ) + ϑ|ψ|m−2 ψ · δψdΩ

Ω

  ∂ψ m−2 + σ(ω)|ψ| a|ψ| ψ · δψdσ +m → ∂− τ σ0    ∂ψ m−2 + γ(ω)|ψ| a|ψ|m−2 → +m ψ · δψdσ. ∂− ν  

m−2

∂Ω

2.2. The Friedrichs-Wirtinger type inequality

23

Hence, because of δH[ψ] = 0, ∀δψ ∈ C0 (Ω) ∩ W1,m (Ω), we get the eigenvalue problem (N EV P ) for the m-Laplacian . Conversely, let ϑ, ψ(ω) be a weak solution of the eigenvalue problem for the m-Laplacian. From the definition of the weak eigenfunction, by setting η = ψ(ω), we obtain 0 = F [ψ] − ϑG[ψ]

=

(by K)

F [ψ] − ϑ ⇒ ϑ = F [u],

consequently, the required minimum is the least eigenvalue of the eigenvalue problem for the m-Laplacian. The existence of a function ψ ∈ K such that F [ψ] ≤ F [v] for all v ∈ K has been proved above.  Remark 2. For m = 2 the eigenvalue problem (N EV P ) takes the form ⎧ ⎪ ⎪ a± (ω ψ± + ϑψ± ) = 0, a± are  positive constants; ω ∈ Ω± , ⎪ ⎨  ∂ψ a ∂→ + σ(ω)ψ  = 0, [ψ]σ0 = 0, − τ σ σ0  0 ⎪ ⎪  ± ⎪ ⎩ a± ∂ψ + γ (ω)ψ = 0,  → − ± ± ∂ν

(EV P )

∂± Ω

and we set λ=

2−n+

, (n − 2)2 + 4ϑ . 2

(2.2.1)

Then (W )m takes the form  aψ 2 (ω)dΩ Ω

⎫ ⎧   ⎬ ⎨ 1 ≤ a|∇ω ψ|2 dΩ + σ(ω)ψ 2 (ω)dσ + γ(ω)ψ 2 (ω)dσ , (W )2 ⎭ λ(λ + n − 2) ⎩ Ω

σ0

∂Ω

for all ψ(ω) ∈ W1 (Ω). Remark 3. The constants in (W )m and (W )2 are the best possible ones. Corollary 2.3. Let ϑ(m) be the least positive eigenvalue of the problem (N EV P ). Let v(x) ∈ W1,m (Gd0 ), γ(ω) be a positive bounded piecewise smooth function on ∂Ω, σ(ω) be a positive continuous function on σ0 . Then for any  ∈ (0, d) and for all α,  arα |v|m dx G 0

≤

1 ϑ(m)



G 0

 arα+m |∇v|m dx +

rα+m Σ 0

σ(ω) m |v| ds + rm−1

 rα+m Γ 0

γ(ω) m . |v| ds , rm−1 (2.2.2)

24

Chapter 2. Eigenvalue problem and integro-differential inequalities

provided that integrals on the right are finite. In particular,  a|v|m dx ≤ G 0

m

⎧ ⎪ ⎨

 ϑ(m) ⎪ ⎩

 a|∇v|m dx +

G 0

Σ 0

σ(ω) m |v| ds + rm−1

 Γ 0

⎫ ⎪ ⎬

γ(ω) m |v| ds . ⎪ rm−1 ⎭ (H − W )m

Proof. Consider the inequality (W )m for the function v(r, ω). By multiplying it by rα+n−1 and integrating over r ∈ (0, ), we obtain the desired inequality (2.2.2). Setting in it α = 0 we get (H − W )m .  Similarly we have ◦ 1

Corollary 2.4. Let u ∈ C0 (G) ∩ Wα−2 (G), and λ be as above in (2.2.1). Let σ(ω), ω ∈ σ0 ; γ(ω), ω ∈ ∂Ω be non-negative bounded piecewise smooth functions. Then 

1 λ(λ + n − 2)

arα−4 u2 dx ≤

Gd 0

 +



arα−2 |∇u|2 dx +

Gd 0

.

rα−3 γ(ω)u2 (x)ds ,



rα−3 σ(ω)u2 (x)ds

Σd 0

(2.2.3)

∀α.

Γd 0 ◦ 1

Lemma 2.5. Let v ∈ C0 (G) ∩ W0 (G) and σ(ω) ≥ 0, γ(ω) ≥ 0. Then for any ε > 0 and for all α, 

arεα−2 r−2 v 2 dx

Gd 0

≤/ c·

⎧ ⎪ ⎨ ⎪ ⎩

arεα−2 |∇v|2 dx +

Gd 0

c= /

1 λ(λ + n − 2)



r−1 rεα−2 σ(ω)v 2 (x)ds +

Σd 0

 (α−2)sgn(α−2) 2 , h

 Γd 0

r−1 rεα−2 γ(ω)v 2 ds

⎫ ⎪ ⎬ ⎪ ⎭

,

(2.2.4)

where h is defined in Subsection 1.3. Proof. Multiplying both sides of the Friedrichs-Wirtinger inequality (W )2 by (r + ε)α−2 rn−3 and integrating over r ∈ ( 2 , ), we obtain

2.3. The Hardy and Hardy-Friedrichs-Wirtinger type inequalities 

25

a(r + ε)α−2 r−2 v 2 dx

G /2

≤

1 λ(λ + n − 2)





a(r + ε)α−2 |∇v|2 dx +

G /2

r−1 (r + ε)α−2 γ(ω)v 2 ds

Γ /2



+

 r−1 (r + ε)α−2 σ(ω)v 2 ds ,

Σ /2

for all ε > 0 or, because of the property 1) of rε §1.3, rε ≤ r + ε ≤ h2 rε we find  arεα−2 r−2 v 2 dx G /2

≤/ c·

⎧ ⎪ ⎨ ⎪ ⎩

arεα−2 |∇v|2 dx +

G /2





r−1 rεα−2 σ(ω)v 2 ds +

Σ /2

r−1 rεα−2 γ(ω)v 2 ds

Γ /2

⎫ ⎪ ⎬ ⎪ ⎭

,

for all ε > 0. Letting ρ = 2−k d, (k = 0, 1, 2, . . .) and summing the obtained inequalities over all k we get the desired inequality (2.2.4). 

2.3 The Hardy and Hardy-Friedrichs-Wirtinger type inequalities We recall first the classical Hardy inequality (see Theorem 330 [30]): Theorem 2.6. Let d > 0 and v ∈ C 0 [0, d] ∩ W 1,2 (0, d) with v(0) = 0. Then d r 0

4 v (r)dr ≤ (4 − n − α)2

n−5+α 2



d r

n−3+α

0

∂v ∂r

2 dr

(2.3.1)

for α < 4 − n provided that the integral on the right-hand side is finite. Corollary 2.7. Let 0 < ε < d and v ∈ C 0 [ε, d] ∩ W 1,2 (ε, d), with v(ε) = 0. Then d r ε

4 v (r)dr ≤ (4 − n − α)2

n−5+α 2



d r

n−3+α

∂v ∂r

2 dr,

α < 4 − n.

(2.3.2)

ε

Proof. We apply the inequality (2.3.1) to the function v(r) extended by zero into [0, ε). 

26

Chapter 2. Eigenvalue problem and integro-differential inequalities Now using the Hardy inequality we establish:

Proposition 2.8 (The Hardy-Friedrichs-Wirtinger type inequality). ◦ 1

Let u ∈ C0 (G) ∩ Wα−2 (G) and σ(ω), γ(ω) be positive bounded piecewise smooth functions. Then    α−4 2 ar u dx ≤ H(λ, n, α) arα−2 |∇u|2 dx + rα−3 σ(ω)u2 (x)ds. Gd 0

Σd 0

Gd 0



 +

r

α−3

2

γ(ω)u (x)ds ,

Γd 0

H(λ, n, α) =

1 , α ≤ 4 − n. λ(λ + n − 2) + 14 (4 − n − α)2

(2.3.3)

Proof. For α = 4 − n the required inequality (2.3.3) coincides with (2.2.3). Now, let α < 4 − n.  We shall show that u(0) = 0. In fact, from the representation u(0) = u(x) − u(x) − u(0) by the Cauchy inequality we have 12 |u(0)|2 ≤ |u(x)|2 + |u(x) − u(0)|2 . Putting v(x) = u(x) − u(0) we obtain    1 2 α−4 α−4 2 |u(0)| r dx ≤ r u (x)dx + rα−4 |v|2 dx < ∞. (2.3.4) 2 Gd 0

Gd 0

Gd 0

◦ 1

(The first integral from the right is finite due to u ∈ Wα−2 (G), and the second inted  gral is also finite, by virtue of Theorem 2.6.) Since rα−4 dx = meas Ω rα+n−5 dr 0

Gd 0

= ∞, by α+n−4 < 0, the assumption u(0) = 0 contradicts (2.3.4). Thus u(0) = 0. Therefore we can use Theorem 2.6:   4 arα−4 u2 dx ≤ arα−2 u2r dx. (2.3.5) |4 − n − α|2 Gd 0

Gd 0

Multiplying (W )2 by rn−5+α and integrating over r ∈ (0, d) we obtain    1 1 α−4 2 ar u dx ≤ arα−2 2 |∇ω u|2 dx + rα−3 σ(ω)u2 (x)ds λ(λ + n − 2) r Gd 0

Gd 0

Σd 0

 +

r

α−3

 2

γ(ω)u (x)ds . (2.3.6)

Γd 0 2

Adding the inequalities (2.3.6), (2.3.5) and using the formula |∇u| = 2 1 r 2 |∇ω u| , we get the desired inequality (2.3.3).

 ∂u 2 ∂r

+ 

2.3. The Hardy and Hardy-Friedrichs-Wirtinger type inequalities

27

Lemma 2.9. Let u ∈ C 0 (Gdε ) ∩ W 1 (Gdε ), u(ε) = 0 and σ(ω), γ(ω) be positive bounded piecewise smooth functions. Then for any ε > 0, it holds that  ar



α−4 2

u dx ≤ H(λ, n, α)

Gd ε

ar

α−2



2

|∇u| dx +

rα−3 σ(ω)u2 (x)ds

Σd ε

Gd ε





+

r

α−3

2

γ(ω)u (x)ds , (2.3.7)

Γd ε

where H(λ, n, α) is determined by (2.3.3). Proof. We have inequality (W )2 . Multiplying it by rn−5+α and integrating over r ∈ (ε, d) we obtain (2.3.7) for α = 4 − n. If α < 4 − n, we consider the inequality (2.3.2) and integrate it over Ω; then we have 1 (4 − n − α)2 4



Gd ε

rα−4 u2 dx ≤



rα−2 u2r dx.

Gd ε

Adding this inequality to the above one for α = 4 − n and using the formula  2 |∇u|2 = ∂u + r12 |∇ω u|2 , the desired result follows.  ∂r

Lemma 2.10. Let u ∈ C0 (G)∩W1 (G), u(0) = 0 and σ(ω), γ(ω) be positive bounded piecewise smooth functions. Then, for any ε > 0, it holds that 

arεα−4 u2 dx

 ≤ H(λ, n, α)

Gd 0

arεα−2 |∇u|2 dx +



rεα−3 σ(ω)u2 (x)ds

Σd 0

Gd 0



+

 rεα−3 γ(ω)u2 (x)ds , (2.3.8)

Γd 0

H(λ, n, α) =

1 , α ≤ 4 − n. λ(λ + n − 2) + 14 (4 − n − α)2

Proof. We perform the change of variables yi = xi − εli , i = 1, . . . , n and use the

28

Chapter 2. Eigenvalue problem and integro-differential inequalities

inequality (2.3.7); thus we obtain   arεα−4 u2 (x)dx = a|y|α−4 u2 (y + εl)dy Gd 0

ε Gd ε



 +



a|y|α−2 |∇y u(y + εl)|2 dy +

≤ H(λ, n, α) ε Gd ε

|y|α−3 σ(y + εl)u2 (y + εl)ds

ε Σd ε



|y|α−3 γ(y + εl)u2 (y + εl)ds

ε Γd ε



arεα−2 |∇u|2 dx

= H(λ, n, α)

 +

rεα−3 σ(x)u2 (x)ds

 +

Σd 0

Gd 0

 rεα−3 γ(x)u2 (x)ds

Γd 0

. 

2.4 Auxiliary integro-differential inequalities Lemma 2.11. Let Gd0 be the conical domain, v(, ·) ∈ W1,m (Ω) for almost all  ∈ (0, d) and    σ(ω) m γ(ω) m m V () = a|∇v| dx + |v| ds + |v| ds < ∞. (2.4.1) rm−1 rm−1 G 0

Σ 0

Γ 0

Let ϑ(m) be the smallest positive eigenvalue of the problem (N EV P ) and γ(ω) be a positive bounded piecewise smooth function on ∂Ω, σ(ω) be a positive continuous function on σ0 . Then for almost all  ∈ (0, d),  ∂v   av |∇v|m−2 dΩ ≤ Ξ(m) · (2.4.2) 1 V (), ∂r mϑ m Ω

  3 m 2

where Ξ(m) =

, 2 m−1 2−m (m − 1) m · 2 2 ,

m ≥ 2, 1 < m ≤ 2.

(2.4.3)

Proof. Writing the function V () in spherical coordinates 

 r

V () = 0

n−1







m

a|∇v(r, ω)| dΩ dr +

r

n−m−1

0

Ω

∂Ω

r 0

γ(ω)|v(r, ω)| dσ dr 

 +

 m

n−m−1

 m

σ(ω)|v(r, ω)| dσ dr σ0

2.4. Auxiliary integro-differential inequalities

29

and differentiating it with respect to  we obtain  V  () = n−1 a|∇v(, ω)|m dΩ Ω

+



n−m−1



 m

m

γ(ω)|v(, ω)| dσ +

(2.4.4)

σ(ω)|v(, ω)| dσ . σ0

∂Ω

m ≥ 2. Using the Cauchy inequality and next the Young inequality with p = we have     ∂v ∂v m−2 n−1 m−2  av |∇v| dΩ =  av |∇v|  dΩ  ∂r ∂r Ω

Ω

      ∂v v n m−2  · · |∇v| a =  dΩ   ∂r r= Ω     2  2       v ∂v ε 1   +   · |∇v|m−2  ≤ n a  2  2ε  ∂r  

Ω





≤

n

a Ω

m 2,

p =

m m−2

r=

dΩ

r=

εδ m

  m  2  v  2 1  ∂v  m m−2    + m − 2 · ε δ − m−2 |∇v| + · |∇v|    m 2 2ε  ∂r 

dΩ, r=

for all ε, δ > 0. Applying now the Friedrichs-Wirtinger type inequality (W )m we obtain          v(, ω) m  v(, ω) m ∂v εδn m−2  dσ + γ(ω)   av |∇v| dΩ ≤ σ(ω)     dσ ∂r mϑ(m)   Ω

 + n Ω



σ0

∂Ω

 m 2 ε − m−2 εδ  ∇ω v  δ a + (m − 2) |∇v|m mϑ(m)    2m   2  1  ∂v  m−2  + · |∇v|  dΩ, ∀ε, δ > 0. (2.4.5)  2ε  ∂r  r=

    Because |∇v|2 = vr2 + r12 |∇ω v|2 and  ∇ω v  ≤ |∇v|, we get:  m  2 εδ  ∇ω v  1  ∂v  + · |∇v|m−2 mϑ(m)    2ε  ∂r    2  2  εδ  ∇ω v  1  ∂v  εδ |∇v|m , · |∇v|m−2 = ≤ +     mϑ(m)  2ε ∂r mϑ(m)

30

Chapter 2. Eigenvalue problem and integro-differential inequalities

if we choose ε > 0 from the equality εδ 1 = . 2ε mϑ(m)

(2.4.6)

Therefore from (2.4.5) it follows that  Ω

∂v ε av |∇v|m−2 dΩ ≤ ∂r m

εδ n−m + mϑ(m)



2 m − 2 − m−2 δ + δ ϑ(m) 2





 n Ω





γ(ω)|v(, ω)| dσ , ∀δ > 0. (2.4.7)

m

m

σ(ω)|v(, ω)| dσ + σ0

a|∇v(, ω)|m dΩ

∂Ω

m−2

Let us put δ = ϑ m (m). Then, by (2.4.6), ε = and (2.4.4) we derive the required (2.4.2).

,m 2

1

ϑ m (m). Thus from (2.4.7)

1 < m ≤ 2. Because of |∇v| ≥ |vr | and 1 < m < 2, we have |∇v|m−2 ≤ |vr |m−2 . Therefore m using the Young inequality with p = m, p = m−1 :  Ω

∂v av |∇v|m−2 dΩ = n−1 ∂r  ≤ n Ω

 Ω

       v ∂v  m−2  n m−1 av |∇v| a   · |vr |  dΩ ≤   dΩ   ∂r  r=

- ε  v m m − 1 1   + ε− m−1 a m  m

r=

Ω

 m .  ∂v      dΩ, ∀ε > 0.  ∂r   r=

Next, applying the Friedrichs-Wirtinger type inequality (W )m we obtain  Ω

∂v εn av |∇v|m−2 dΩ ≤ ∂r mϑ(m) +

1 n  m



 a Ω



       v(, ω) m  v(, ω) m  dσ + γ(ω)   σ(ω)     dσ  

σ0

 m 1 ε  ∇ω v  + (m − 1)ε− m−1 ϑ(m)   

∂Ω

 m    ∂v       ∂r  

dΩ,

∀ε > 0.

r=

Now we choose ε = (m − 1)ϑ(m)

m−1 m

1

=⇒ (m − 1)ε− m−1 =

ε ϑ(m)

(2.4.8)

2.4. Auxiliary integro-differential inequalities

31

and therefore the above inequality gives  Ω

∂v εn av |∇v|m−2 dΩ ≤ ∂r mϑ(m)  + Ω

But, because of |∇v|2 = vr2 +



      v(, ω) m  v(, ω) m   dσ   σ(ω)  dσ + γ(ω)     

σ0

      ∇ω v m  ∂v m   +  a    ∂r    

1 2 r 2 |∇ω v|

∂Ω



dΩ .

(2.4.9)

r=

and the Jensen inequality (1.5.5), we get

  2  m2      ∇ω v m  ∂v m  2−m m  +   =  ∇ω v   + (vr2 ) 2 ≤ 2 2 |∇v|m ,     ∂r    

1 < m ≤ 2.

Hence and from (2.4.9) it follows that  Ω

2−m

∂v 2 2 ε n  av |∇v|m−2 dΩ ≤ ∂r mϑ(m)

 σ0

 m      v(, ω) m   dσ + γ(ω)  v(, ω)  dσ σ(ω)       

+ Ω

  m a|∇v|  



∂Ω

dΩ .

r=

Substituting here ε from (2.4.8) and recalling the definition (2.4.3)-(2.4.4) we get the desired inequality (2.4.2). 

For the case m = 2 we establish the best possible result. Lemma 2.12. Let Gd0 be the conical domain and ∇u(, ·) ∈ L2 (Ω) a.e.  ∈ (0, d) and λ be defined by (2.2.1). Assume that for a.e.  ∈ (0, d),  U (ρ) =

ar

2−n

2



|∇u| dx +

G o

r

1−n

2



σ(ω)u (x)ds +

Σ 0

r1−n γ(ω)u2 (x)ds < ∞.

Γ 0

(2.4.10) Then  Ω

  ∂u n − 2 2  + u  a u  ∂r 2

r=

dΩ ≤

  U (). 2λ

(2.4.11)

32

Chapter 2. Eigenvalue problem and integro-differential inequalities

Proof. Writing U () in spherical coordinates,  r

U () = 0

2−n



+

a|∇u| dΩ rn−1 dr Ω

 r

 2

1−n

0



2

2

γ(ω)|u| dσ rn−2 dr

σ(ω)|u| dσ + σ0



 ∂Ω

         1 1 2 2 2 2 = r a ur + 2 |∇ω u| dΩdr + σ(ω)|u| dσ + γ(ω)|u| dσ dr r r 0

0

Ω

σ0

∂Ω

and differentiating with respect to , we obtain     2  ∂u 1  2  U () = a  + |∇ω u|  dΩ  ∂r  r= Ω ⎞ ⎛   1⎝ + σ(ω)u2 (, ω)dσ + γ(ω)u2 (, ω)dσ ⎠ .  σ0

(2.4.12)

∂Ω

ε 2 Moreover, by the Cauchy inequality, we have for all ε > 0: ρu ∂u ∂r ≤ 2 u +   1 2 ∂u 2 . Then 2ε ρ ∂r     2   ∂u n − 2 2  ε+n−2 2 ∂u + u  dΩ ≤ a u au2 dΩ + a dΩ.  ∂r 2 2 2ε ∂r r=

Ω

Ω

Ω

Thus choosing ε = λ we obtain, by the Friedrichs-Wirtinger inequality (W )2 ,    ∂u n − 2 2  + u  dΩ a u  ∂r 2 Ω

r=

  2 2 ∂u a|∇ω u| dΩ + a dΩ 2ε ∂r Ω Ω    ε+n−2   2 2 U ().  + σ(ω)u (, ω)dσ + γ(ω)u (, ω)dσ = 2λ(λ + n − 2) 2λ

ε+n−2 ≤ 2λ(λ + n − 2)



2

σ0

∂Ω

Chapter 3

Best possible estimates of solutions to the transmission problem for linear elliptic divergence secondorder equations in a conical domain 3.1 Introduction Let G ⊂ Rn , n ≥ 2 be a bounded domain with the boundary ∂G that is a smooth surface everywhere except at the origin O ∈ ∂G and near the point O it is a conical surface with the vertex at O. We assume that G = G+ ∪ G− ∪ Σ0 is divided into two subdomains G+ and G− by a Σ0 = G ∩ {xn = 0}, where O ∈ Σ0 (see Figure 1). We consider the elliptic transmission problem ⎧  ij  ∂ ⎪ L[u] ≡ ∂x a (x)uxj + ai (x)uxi + a(x)u = f (x), ⎪ i ⎪ ⎪ ⎪ ⎪ ⎪

 ⎨  1 x [u]Σ0 = 0, S[u] ≡ ∂u ∂ν Σ0 + |x| σ |x| u(x) = h(x), ⎪ ⎪ ⎪ ⎪ ⎪

 ⎪ ⎪ ⎩B[u] ≡ ∂u + 1 γ x u = g(x), ∂ν |x| |x|

x ∈ G \ Σ0 ; x ∈ Σ0 ;

(L)

x ∈ ∂G \ {Σ0 ∪ O}

(summation over repeated indices from 1 to n is understood); here: • •

∂ ∂ν

∂ = aij (x)ni ∂x , j

 ∂u

∂ν Σ0

denotes the saltus of the co-normal derivative of the function u(x) on

M. Borsuk, Transmission Problems for Elliptic Second-Order Equations in Non-Smooth Domains, Frontiers in Mathematics, DOI 10.1007/978-3-0346-0477-2_4, © Springer Basel AG 2010

33

34

Chapter 3. Elliptic transmission problem in a conical domain crossing Σ0 , i.e., 

∂u ∂ν

 Σ0

= aij + (x)

∂u+  ∂u−  ni  −aij ni  . − (x) ∂xj ∂xj Σ0 Σ0

In this chapter we obtain the best possible estimates of the weak solutions of the problem (L) near the conical boundary point. Analogous results were established earlier in [14] for the Dirichlet and Robin problems in the case of a conical domain without interfaces. Definition 3.1. A function u(x) is called a weak solution of the problem (L) pro◦ 1

vided that u(x) ∈ C0 (G) ∩ W0 (G) and satisfies the integral identity  G

! ij " a (x)uxj ηxi − ai (x)uxi η(x) − a(x)uη(x) dx + 

+

1 γ(ω)u(x)η(x)ds = r

∂G





Σ0

1 σ(ω)u(x)η(x)ds r 

h(x)η(x)ds −

g(x)η(x)ds + Σ0

∂G



f (x)η(x)dx (II) G

◦ 1

for all functions η(x) ∈ C0 (G) ∩ W0 (G). Without loss of generality, we assume that there exists d > 0 such that Gd0 is a rotational cone with the vertex at O and the aperture ω0 , thus   n   ω0  2 d 2 ω0 2 , ω0 ∈ (0, 2π) . (3.1.1) x ; r ∈ (0, d), ω1 = Γ0 = (r, ω)x1 = cot 2 i=2 i 2 Lemma 3.2. Let u(x) be a weak solution of (L). For any function η(x) ∈ C0 (G) ∩ ◦ 1

W0 (G) the equality  .   aij (x)uxj ηxi + f (x) − ai (x)uxi − a(x)u η(x) dx G 0

 aij (x)uxj η(x) cos(r, xi )dΩ +

= Ω

+

   1 g(x) − γ(ω)u(x) η(x)ds r

Γ 0

   1 h(x) − σ(ω)u(x) η(x)ds r

Σ 0

holds for a.e.  ∈ (0, d).

(II)loc

3.1. Introduction

35

Proof. Let χ (x) be the characteristic function of the set G0 . We consider the integral identity (II) replacing in it η(x) by η(x)χ (x). As a result we obtain  .   aij (x)uxj ηxi + f (x) − ai (x)uxi − a(x)u η(x) dx G 0

 



=−

aij (x)uxj η(x)χxi dx +

G 0

Γ 0

 1 g(x) − γ(ω)u(x) η(x)ds r

   1 h(x) − σ(ω)u(x) η(x)ds. + r Σ 0

Because of formula (7 ) Subsection 3 §1 Chapt. 3 [28], xi χxi = − δ( − r), r where δ( − r) is the Dirac distribution lumped on the sphere r = , we get (see Example 4 Subsection 3 §1 Chapt. 3 [28])   xi ij − a (x)uxj η(x)χxi dx = aij (x)uxj η(x) δ( − r)dx r G 0

G 0



aij (x)uxj η(x) cos(r, xi )dΩ .

= Ω



Hence it follows the required statement. Assumptions. (a) the condition of the uniform ellipticity: 2 ν± ξ 2 ≤ aij ± (x)ξi ξj ≤ μ± ξ ,

∀x ∈ G± ,

∀ξ ∈ Rn ;

ν± , μ± = const > 0, and aij (0) = aδij ,  a+ , x ∈ G+ , j where δi is the Kronecker symbol and a = a− , x ∈ G− , constants; we write: a∗ = min{a+ , a− } > 0, ν∗ = min{ν− , ν+ };

a± are positive

a∗ = max{a+ , a− } > 0; μ∗ = max(μ− , μ+ );

(b) aij (x) ∈ C0 (G), ai (x) ∈ Lp (G), a(x), f (x) ∈ Lp/2 (G) ∩ L2 (G); n < p ≤ 2n, for which the inequalities   12 n ij ij 2 |a± (x) − a± (y)| ≤ a± A(|x − y|); i,j=1

36

Chapter 3. Elliptic transmission problem in a conical domain  12  n i 2 |a± (x)| + |x|2 |a± (x)| ≤ a± A(|x|) |x| i=1

hold for x, y ∈ G, where A(r) is a monotonically increasing, non-negative function, continuous at 0, A(0) = 0; (c) a(x) ≤ 0 in G; σ(ω) ≥ ν0 > 0 on σ0 ; γ(ω) ≥ ν0 > 0 on ∂G; (d) there exist numbers f1 ≥ 0, g1 ≥ 0, h1 ≥ 0, s > 1 such that |f (x)| ≤ f1 |x|s−2 ,

|g(x)| ≤ g1 |x|s−1 ,

|h(x)| ≤ h1 |x|s−1 ;

(e) γ(ω) is a positive bounded piecewise smooth function on ∂Ω, σ(ω) is a positive continuous function on σ0 . Our main result is the following theorem. Theorem 3.3. Let u be a weak solution of the problem (L) and λ be defined by (2.2.1). Let assumptions (a)–(e) be satisfied with A(r) Dini-continuous at zero. Then there existd ∈ (0, 1) and positive constants C0 , C1 , C2 depending only n   i 2 , ω0 , s, meas G, diam G and on the quantity on n, ν∗ , μ∗ , p, λ,   |a (x)|  i=1

1 0

A(r) r dr

Lp/2 (G)

such that the inequality

⎧ ⎪|x|λ , ⎨

 ⎪

 1 1 1 |u(x)| ≤ C0 u2,G + f1 + √ g1 + √ h1 · |x|λ ln |x| , ⎪ ν0 ν0 ⎪ ⎩|x|s ,

if s > λ, if s = λ, (3.1.2) if s < λ

holds for all x ∈ Gd0 . If, in addition, 0 (G), aij (x) ∈ C1 (G), σ(ω) ∈ C 1 (σ0 ), γ(ω) ∈ C1 (∂G), f (x) ∈ Vp,2p−n 1−1/p

1−1/p

h(x) ∈ Vp,2p−n (σ0 ), g(x) ∈ Vp,2p−n (∂G); p > n and there is the number

τs =: sup −s h >0

1− 1 p Vp,2p−n (Σ ) /2

+ g

 1− 1 p Vp,2p−n (Γ ) /2

,

(3.1.3)

then for all x ∈ Gd0 the inequality

⎧ ⎪|x|λ−1 , ⎨

 ⎪

 1 1 1 |∇u(x)| ≤ C1 u2,G + f1 + √ g1 + √ h1 + τs · |x|λ−1 ln |x| , ⎪ ν0 ν0 ⎪ ⎩|x|s−1 ,

if s > λ, if s = λ, if s < λ (3.1.4)

3.2. Local estimate at the boundary

37

2 (G), then holds. Furthermore, if u ∈ Vp,2p−n 2 uVp,2p−n (G 0)

⎧ ⎪ λ , ⎨

 ⎪

 1 1 ≤ C2 u2,G + f1 + √ g1 + √ h1 + τs · λ ln 1 , ⎪ ν0 ν0 ⎪ ⎩ s ,

if s > λ, if s = λ,

(3.1.5)

if s < λ.

3.2 Local estimate at the boundary We derive here a result asserting the local boundedness (near the conical point) of the weak solution of problem (L). Theorem 3.4. Let u(x) be a weak solution of the problem (L) and assumptions (a)–(c) be satisfied. If, in addition, h(x) ∈ L∞ (Σ0 ), g(x) ∈ L∞ (∂G), then the inequality .

sup |u(x)| ≤ C −n/t ut,G0 + 2(1−n/p) f p/2,G0 +  g∞,Γ0 + h∞,Σ0 Gκ 0

(3.2.1) holds for any t > 0, κ ∈ (0, 1) and  ∈ (0, d), where C > 0 is a constant depending n   i 2 only on n, ν∗ , μ∗ , t, p, κ,  .  |a (x)|  Lp/2 (G)

i=1

Proof. We apply the Moser iteration method. First we assume that t ≥ 2. We consider the integral identity (II) and make the coordinate transformation x = x . Let G be the image of G, ∂G be the image of ∂G, Σ0 be the image of Σ0 , and we have dx = n dx , ds = n−1 ds . In addition, we denote v(x ) = u(x ), F (x ) = 2 f (x ), G(x ) = g(x ), H(x ) = h(x ).

(3.2.2)

Then (II) means  . aij (x )vxj ηxi − ai (x )vxi η(x ) − 2 a(x )v(x )η(x ) dx G



+ Σ0



1 σ(ω)v(x )η(x )ds + |x |

∂G

1 γ(ω)v(x )η(x )ds |x | (II)

for all η(x ) ∈ C0 (G ) ∩ W0 (G ). Now we define the quantity k by

 k = k() = ν∗−1 Fp/2,G10 + G∞,Γ10 + H∞,Σ10

(3.2.3)

Σ0

∂G

H(x )η(x )ds −



F (x )η(x )dx

=

G(x )η(x )ds +





G

◦ 1

38

Chapter 3. Elliptic transmission problem in a conical domain

and set

v(x ) = |v(x )| + k.

(3.2.4)

Now we observe that 1 1 1 1 1 |F | · kv = |F |(v − |v|) · v = |F | · v 2 − |F | · |v|v ≤ |F | · v 2 ; k k k k k (3.2.5) 1 1 |H|v ≤ |H| · v 2 ; |G|v ≤ |G| · v 2 k k

|F |v =

in the same way. As the test function in the integral identity (II) we choose η(x ) = ζ 2 (|x |)vv t−2 (x ), where ζ(|x |) ∈ C∞ 0 ([0, 1]) is a non-negative function to be further specified. By the chain and the product rules, η is a valid test function in (II) and also ηxi = v t−2 vxi ζ 2 (|x |) + (t − 2)v t−3 |v|vxi ζ 2 (|x |) + 2ζζxi vv t−2 (x ), so that, by substitution into (II) taking into account that a(x ) ≤ 0 in G , v ≤ |v| ≤ v and t ≥ 2, we obtain   1 aij (x )vxi vxj v t−2 ζ 2 (|x |)dx + σ(ω)v 2 v t−2 (x )ζ 2 (|x |)ds |x | G10

Σ10



1 γ(ω)v 2 v t−2 (x )ζ 2 (|x |)ds ≤  |x |

+ Γ10



|a (x )ζxi vxj |v ij

t−1





+



ζ(|x |)dx +

G10



|ai (x )vxi |v t−1 (x )ζ 2 (|x |)dx

G10



+2



|H(x )|v t−1 (x )ζ 2 (|x |)ds +

Σ10



|G(x )|v t−1 (x )ζ 2 (|x |)ds

Γ10

|F(x )|v t−1 (x )ζ 2 (|x |)dx .

G10

By the ellipticity condition, assumption (c) and taking into account (3.2.5), it follows that    1  2 t−2 2   2μ|∇ v| · |∇ ζ| · v t−1 ζ(|x |)+ |F(x )| · v t ζ 2 (|x |) ν|∇ v| · v ζ (|x |)dx ≤ k G10



+

n 

 |a (x)| i

2

i=1

+

1 H∞,Σ10 · k

 Σ10

G10

1 2



|∇ v| · v



1 ζ (|x |) dx + G∞,Γ10 · k

t−1 2

v t (x )ζ 2 (|x |)ds .







v t (x )ζ 2 (|x |)ds

Γ10

(3.2.6)

3.2. Local estimate at the boundary

39

We estimate every term by the Cauchy inequality for any ε > 0: 

√   μ∗ t t   t−1  −1  2 2 ν|∇ v| · v ζ · √ v |∇ ζ| 2μ|∇ v| · |∇ ζ| · v ζ ≤ 2 ν∗ ≤ εν|∇ v|2 · v t−2 ζ 2 + 

 n 

 12 |a (x)| i

2

(μ∗ )2 t  2 v |∇ ζ| ; εν∗

· |∇ v| · v t−1 ζ 2

i=1

⎛  12 ⎞  n 

√  t t 1 ≤ ν|∇ v| · v 2 −1 · ⎝ √ v 2 ·  |ai (x)|2 ⎠ ζ 2 ν∗ i=1 ≤

n ε 1 2 i ν|∇ v|2 · v t−2 ζ 2 +  |a (x)|2 · v t ζ 2 . 2 2εν∗ i=1

For the estimating integrals over the boundaries we apply the inequality (1.5.12). Then from (3.2.6) it follows that   3 ν|∇ v|2 · v t−2 ζ 2 (|x |)dx ≤ ε ν|∇ v|2 · v t−2 ζ 2 (|x |)dx (3.2.7) 2 G10

 

+ G10

+

G10

 n

1   (μ∗ )2 t 1 v · |∇ ζ|2 + 2 |ai (x)|2 + |F(x )| · v t ζ 2 (|x |) dx εν∗ 2εν∗ i=1 k

    1

1 G∞,Γ10 + H∞,Σ10 · δ|∇ (ζv t/2 )|2 + c0 v t ζ 2 dx , k δ

∀ε, δ > 0.

G10

From relations 

|∇ (ζv t/2 )|2 ≤ 2 ζ 2 |∇ (v t/2 )|2 + v t |∇ ζ|2 ,

|∇ (v t/2 )|2 =

t2 t−2  2 v |∇ v| (3.2.8) 4

follows the inequality |∇ (ζv t/2 )|2 ≤

t2 t−2  2 2 v |∇ v| ζ + 2v t |∇ ζ|2 . 2

Now, from (3.2.7) and (3.2.9), by choosing ε =  1 ν|∇ v|2 · v t−2 ζ 2 (|x |)dx 2 G10

δt2 G∞,Γ10 + H∞,Σ10 · ≤ · 2ν∗ k



G10

1 3

(3.2.9)

we find that

ν|∇ v|2 · v t−2 ζ 2 (|x |)dx

(3.2.10)

40

Chapter 3. Elliptic transmission problem in a conical domain 

+ G10

⎞  −1 n 1 1 G |F(x )| + c δ + H  0 ∞,Γ0 ∞,Σ0 ⎝ 3 ⎠ · v t ζ 2 (|x |)dx |ai (x)|2 + 2ν∗ i=1 k ⎛

 

+ G10

2

 3(μ∗ )2 2δ

G∞,Γ10 + H∞,Σ10 + v t · |∇ ζ|2 dx , ν∗ k

∀δ ∈ (0, 1].

We choose now δ = 2t12 ; by the definition of the number k in (3.2.3), the last inequality (3.2.10) can be rewritten in the following way:  ν|∇ v|2 · v t−2 ζ 2 (|x |)dx G10



2

≤ 8c0 ν∗ t ·   +4 G10

2









ζ (|x |)v (x )dx + t

G10

12(μ∗ )2 4ν∗ + 2 ν∗ t

n 32  i |F(x )| |a (x)|2 + 2ν∗ i=1 k

  · |∇ ζ|2 v t (x )dx G10



· v t ζ 2 (|x |)dx .

But, by (3.2.8), the last inequality means  ν|∇ (v t/2 )|2 ζ 2 (|x |)dx G10



4

≤ 2c0 ν∗ t ·

2









ζ (|x |)v (x )dx + t

G10

+ t2

 

G10

3(μ∗ )2 t2 + ν∗ ν∗

n 32  i |F(x )| |a (x)|2 + 2ν∗ i=1 k

  · |∇ ζ|2 v t (x )dx G10



· v t ζ 2 (|x |)dx .

Since t ≥ 2, we can rewrite the above inequality as 

ν|∇ (v t/2 )|2 ζ 2 (|x |)dx ≤ C1 t4

G10

+ C2 t

2

G10

ν

  ν |∇ ζ|2 + ζ 2 (|x |) v t dx

G10







n 

|F(x )|  |a (x)| + k i=1 2

i

2

 · v t ζ 2 (|x |)dx , (3.2.11)

where constants C1 , C2 depend only on c0 , ν∗ , μ∗ and are independent of t. Setting √ (3.2.12) w(x ) = ν · v t/2 (x )

3.2. Local estimate at the boundary

41

from (3.2.11) we obtain 

|∇ w|2 ζ 2 (|x |)dx ≤ C1 t4

G10

+ C2 t

2

 

G10



  2  |∇ ζ| + ζ 2 (|x |) w2 (x )dx

G10 n 

|F(x )|  |a (x)| + k i=1 2

i



2

· w2 (x )ζ 2 (|x |)dx . (3.2.13)

The desired iteration process can now be developed from (3.2.13). By the Sobolev Imbedding Theorem 1.15, we have ζw22n/

,G10 n−2 /

≤ C∗ ·



  2   |∇ ζ| + ζ 2 w2 (x ) + ζ 2 (|x |)|∇ w|2 dx ,

(3.2.14)

G10

where n / = n for n > 2, / 2 ∈ (2, p) and C ∗ depends only on n. Using the Hölder inequality for integrals 

|F (x )| · w2 (x )ζ 2 (x )dx ≤ F p/2,G10 · wζ22p

1 p−2 ,G0

,

p > 2,

(3.2.15)

G10

we get from (3.2.13)–(3.2.15): ζw

2

2n / ,G10 n−2 /

≤ C3 t

4





 |∇ ζ|2 + ζ 2 (|x |) w2 (x )dx

G1

0   n     )| |F(x   + C4 t2 2 |ai (x)|2 +   k  i=1

· wζ22p

p/2,G10

1 p−2 ,G0

,

p > n. (3.2.16)

By the interpolation inequality for Lp -norms ζw

2p 1 p−2 ,G0

≤ εζw

n /

2n / ,G10 n−2 /

/ + ε n−p ζw2,G10 ,

p > n, ∀ε > 0,

(3.2.17)

from (3.2.16)–(3.2.17) it follows that ζw

2n / ,G10 n−2 /

 1/2 n 

,  n / |F(x )|   2 i  2 / ≤ t C4  |a (x)| + × εwζ 2n/ ,G1 + ε n−p ζw2,G10  0 n−2 /  k  i=1 p/2,G10 , + t2 C3 · (ζ + |∇ ζ|)w2,G10 , p > n, ∀ε > 0.

42

Chapter 3. Elliptic transmission problem in a conical domain

Choosing ε =

1 √ 2t C4

 n  2  |ai (x)|2 +  i=1

− 12  |F (x )|  , k  p/2,G10

we obtain

p

≤ Ct p−n/ (ζ + |∇ ζ|)w2,G10 , n < p ≤ 2n, (3.2.18)   n   ∗ i 2  and is independent of where C depends only on c0 , n, ν∗ , μ , p,  |a (x)|  ζw

2n / ,G10 n−2 /

i=1

p/2,G

t. Recalling the definition of w by (3.2.12), we finally establish from (3.2.18) the inequality ζ · v t/2 

p

≤ Ct p−n/ (ζ + |∇ ζ|) · v t/2 2,G10 ,

2n / ,G10 n−2 /

n < p ≤ 2n.

(3.2.19)

This inequality can now be iterated to yield the desired estimate. κ+(1−κ)2−j For all κ ∈ (0, 1) we define sets G(j) ≡ G0 , j = 0, 1, 2, . . .. It is easy κ    to verify that G0 ≡ G(∞) ⊂ · · · ⊂ G(j+1) ⊂ G(j) ⊂ · · · ⊂ G(0) ≡ G10 . Now we consider the sequence of cut-off functions ζj (x ) ∈ C∞ (G(j) ) such that 0 ≤ ζj (x ) ≤ 1 in G(j) and ζj (x ) ≡ 1 in G(j+1) , ζj (x ) ≡ 0 |∇ ζj | ≤

for |x | > κ + 2−j (1 − κ);

2j+1 1−κ

κ + 2−j−1 (1 − κ) < |x | < κ + 2−j (1 − κ).

for

 n/ j , j = 0, 1, 2, . . .. Now we rewrite We define also the number sequence tj = t n/ −2 the inequality (3.2.19) replacing ζ(|x |) by ζj (x ) and t by tj ; then taking tj -th root, we obtain 2/tj  j 2p C · 1 vtj+1 ,G(j+1) ≤ · 4 tj · (tj ) p−n/ tj vtj ,G(j) . 1−κ After iteration, we find that  vtj+1 ,G(j+1) ≤

p p−n /

Ct · 1−κ

Notice that the series but the series

∞  j=0



∞  j=0

1 tj

=

n / 2t

∞ 

n / n /−2 j tj

 p−n/ n/ 2 j=0 t1j

∞ 

·4

j=0

j tj

· vt,G10 .

(3.2.20)

is convergent by the d’Alembert ratio test,

as a geometric series. Therefore from (3.2.20) we get

vtj+1 ,G(j+1) ≤ Cvt,G10 . Consequently, letting j → ∞, we have sup v(x ) ≤ x ∈Gκ 0

Cvt,G10 . Hence, because of the definitions of the function v(x ) by (3.2.4) and of the number k by (3.2.3), we obtain 

sup |v(x )| ≤ C vt,G10 + Fp/2,G10 + G∞,Γ10 + H∞,Σ10 . x ∈Gκ 0

3.3. Global integral estimates

43

Returning to the variables x, u, by (3.2.2), we obtain the required estimate (3.2.1) in the case t ≥ 2. Let now 0 < t < 2. We consider (3.2.1) with t = 2: sup |u(x)| ≤ C− 2 u2,G0 + K(), n

(3.2.21)

x∈Gκ 0

.

. K() = C 2(1−n/p) f p/2,G0 +  g∞,Γ0 + h∞,Σ0

where

2 t

Now, using the Young inequality with q = ⎛ n n ⎜ C− 2 u2,G0 = C− 2 ⎝

 ≤



we can write

⎞1/2 ⎟ ut · u2−t ⎠

· C− 2 ut,G ≤ n

t/2

0

G 0

2 2−t

G 0

1−t/2 sup |u(x)|

and q  =

n 1 sup |u(x)| + C1 − t ut,G0 . (3.2.22) 2 G0

Let us define the function ψ(s) = sup |u(x)|. Then from (3.2.21)–(3.2.22) it follows x∈Gs0

that

1 n ψ() + C1 − t ut,G0 + K(), κ ∈ (0, 1). (3.2.23) 2 Further we apply Lemma 1.24. Letting r = κ, R = , δ = 12 , α = nt , A = C1 ut,G0 , B = K() from (3.2.23) we obtain the validity of required estimate (3.2.1) in the case 0 < t < 2. Thus, the proof of Theorem 3.4 is complete.  ψ(κ) ≤

3.3 Global integral estimates In this section we derive a global estimate for the Dirichlet integral. Theorem 3.5. Let u(x) be a weak solution of problem (L) and assumptions (a)–(c) be satisfied. If, in addition, h(x) ∈ L2 (Σ0 ), g(x) ∈ L2 (∂G), then the inequality 



2

ν|∇u| dx + G

≤C

⎧ ⎨ ⎩

Σ0

σ(ω) 2 u (x)ds + r

u2 (x)dx +

G



γ(ω) 2 u (x)ds r

∂G

 G

1 f 2 (x)dx + ν0

 Σ0

1 h2 (x)ds + ν0



g 2 (x)ds

⎭

(3.3.1)

∂G

  n   i 2  holds, where constant C > 0 depends only on p, n, ν∗ ,  |a (x)|  i=1

meas G, diam G.

⎫ ⎬

Lp/2 (G)

and

44

Chapter 3. Elliptic transmission problem in a conical domain

Proof. We put in (II) η(x) = u(x) and apply the classical Hölder inequality. By assumptions (a), (c), we obtain    σ(ω) 2 γ(ω) 2 2 u (x)ds + u (x)ds ν|∇u| dx + (3.3.2) r r Σ0

G

∂G

2     3 n 3 |ai (x)|2 |u||∇u|dx + |u||h(x)|ds + |u||g(x)|ds + |u||f (x)|dx. ≤ 4 i=1

G

Σ0

G

∂G

Further, by assumptions (b), (c), the Cauchy inequality and the integral Hölder inequality, we have: 2   3   n n 3 ε 1 4 |ai (x)|2 |u||∇u|dx ≤ ν|∇u|2 dx + |ai (x)|2 |u|2 dx 2 2εν ∗ i=1 i=1 G

G

ε ≤ 2



ν|∇u|2 dx +

G

1 2εν∗

G

⎞ p−2 ⎛   p2 ⎞ p2 ⎛ p   n 2p i 2 ⎠ ⎝ ⎠ ⎝ p−2 |a (x)| dx · |u| dx , G

i=1

G

p > 2, for all ε > 0. Now we apply the inequality (1.6.5), Theorem 1.16; hence it follows that 2  n    3 n   3 ε 1  i 2 4 |ai (x)|2 |u||∇u|dx ≤ ν|∇u|2 dx + |a (x)|    2 2εν∗2  i=1 i=1 Lp/2 (G) G G    δν|∇u|2 + c(δ, p, n, meas G)u2 dx, ∀ε > 0, ∀δ > 0, p > n. (3.3.3) × G

If we choose δ =

ε2 ν∗2  n   i  |a (x)|2    i=1

 (1 − ε) ⎛

2



ν|∇u| dx + G

Σ0

, then from (3.3.2)–(3.3.3) we get Lp/2 (G)

σ(ω) 2 u (x)ds + r

  n     i 2 ≤ c ⎝ε, p, n, ν∗ ,  |a (x)|    i=1 Lp/2 (G)   + |u||g(x)|ds + |u||f (x)|dx. ∂G



γ(ω) 2 u (x)ds r ∂G ⎞   , meas G⎠ · |u|2 dx + |u||h(x)|ds G

Σ0

G

By the Cauchy inequality, by virtue of the assumption (c), we have

(3.3.4)

3.3. Global integral estimates

45

   σ(ω) r |u| |h(x)| ds |u||h(x)|ds = r σ(ω) Σ0 Σ0   1 σ(ω) 2 diamG ≤ u (x)ds + h2 (x)ds; 2 r 2ν0 Σ0 Σ0       γ(ω) r |u| |g(x)| ds |u||g(x)|ds = r γ(ω) ∂G ∂G   γ(ω) 2 1 diamG u (x)ds + ≤ g 2 (x)ds; 2 r 2ν0 ∂G ∂G    1 1 |u||f (x)|dx ≤ |u|2 dx + |f |2 dx. 2 2  



G

G

Hence and from (3.3.4) with ε =

G

1 2

we derive the required inequality (3.3.1)



Now we will derive a global estimate for the weighted Dirichlet integral. Theorem 3.6. 1 Let u(x) be a weak solution of problem (L) and λ be as above in (2.2.1). Let assumptions (a)–(c) be satisfied with function A(r) that is continuous at zero. If, in addition, 

◦ 0

f (x) ∈ Wα (G),

rα−1 h2 (x)ds < ∞,

Σ0



rα−1 g 2 (x)ds < ∞,

4 − n ≤ α ≤ 2,

∂G

◦ 1

then u(x) ∈ Wα−2 (G) and 

  a rα−2 |∇u|2 + rα−4 u2 dx +

G

≤C

- 

 Σ0

 u2 + (1 + rα )f 2 (x) dx +

 Σ0

G



rα−3 σ(ω)u2 (x)ds + rα−1 h2 (x)ds +



rα−3 γ(ω)u2 (x)ds

∂G

. rα−1 g 2 (x)ds , (3.3.5)

∂G

where the constant C > 0 depends only on p, n, ν∗ , μ∗ , a∗ , ν0 , α, λ,   n    |ai (x)|2  and meas G.   i=1

Lp/2 (G)

i Proof. We put in (II) η(x) = rεα−2 u(x), ηxi = rεα−2 uxi + (α − 2)rεα−3 xi −εl rε u(x). As a result we have

1 See

also below Subsection 3.5.2

46 

Chapter 3. Elliptic transmission problem in a conical domain arεα−2 |∇u|2 dx +

G

=

2−α 2

−

arεα−4 (xi − εli )(u2 )xi dx

G





 aij (x) − aij (0) rεα−4 (xi − εli )uxj u(x)dx

 aij (x) − aij (0) rεα−2 uxi uxj dx +

G

rεα−2 u(x)h(x)ds

 i  a (x)uxi + a(x)u − f (x) rεα−2 u(x)dx

rεα−2 u(x)g(x)ds.

+

Σ0

 G





+

r−1 rεα−2 γ(ω)u2 (x)ds

∂G

G





r−1 rεα−2 σ(ω)u2 (x)ds +

Σ0



+ (2 − α) 



(3.3.6)

∂G

We transform the first integral on the right-hand side by integrating by parts:  arεα−4 (xi − εli ) G

 a+ rεα−4 (xi

= G+



=−

∂u2 dx ∂xi

∂u2 − εli ) + dx + ∂xi

 a− rεα−4 (xi − εli ) G−

   ∂ rεα−4 (xi − εli ) dx + au2 a+ u2+ rεα−4 (xi − εli )ni ds ∂xi

G

∂G+



a− u2− rεα−4 (xi − εli )ni ds

+ ∂G−



=−

∂u2− dx ∂xi

au2

G

   ∂ rεα−4 (xi − εli ) dx + au2 rεα−4 (xi − εli )ni ds ∂xi



+ [a]Σ0

∂G

u2 rεα−4 (xi − εli )ni ds,

(3.3.7)

Σ0

because of [u]Σ0 = 0. Now, we make elementary calculations:  α−4  ∂ i rε (xi − εli ) = nrεα−4 + (α − 4)(xi − εli )rεα−5 xi −εl = (n + α − 4)rεα−4 ; 1) ∂x rε i       2) because of ni  = cos(xn , xi ) = δin , (xi − εli )ni  = δin (xi − εli ) Σ Σ Σ0 0 0     = (xn − εln ) = xn  = 0, since Σ0 = {xn = 0} ∩ G and ln = 0; Σ0

Σ0

3.3. Global integral estimates

47

  3) from the representation ∂G = Γd0 ∪ Γd and by (1.2.9), (xi − εli )ni  d = Γ0

−ε sin ω20 , therefore 

au2 rεα−4 (xi −εli )ni ds

∂G



ω0 = −ε sin 2

au2 rεα−4 ds+



au2 rεα−4 (xi −εli )ni ds.

Γd

Γd 0

Hence and from (3.3.7) it follows that 2−α 2

 arεα−4 (xi G

−ε

∂u2 (2 − α)(4 − n − α) − εli ) dx = ∂xi 2

ω0 2−α sin 2 2



2−α 2

au2 rεα−4 ds +

Γd 0





arεα−4 u2 dx

G

au2 rεα−4 (xi − εli )ni ds.

(3.3.8)

Γd

From (3.3.6), (3.3.8) with regard to 4−n ≤ α ≤ 2 we obtain the following equality:    2−α 1 α−2 ω0 α−2 2 2 α−4 arε |∇u| dx + ε au rε ds + σ(ω)u2 (x)ds sin r 2 2 r ε  + ∂G

1 α−2 2−α r γ(ω)u2 (x)ds ≤ r ε 2 

+ (2 − α)  −





au2 rεα−4 (xi − εli )ni ds

Γd

 aij (x) − aij (0) uxj rεα−4 (xi − εli )u(x)dx

G

 α−2

rε



 aij (x) − aij (0) uxi uxj dx +

G

rεα−2 u(x)h(x)ds + Σ0

(3.3.9)

 i  a (x)uxi + a(x)u − f (x) rεα−2 u(x)dx

G





+

Σ0

Γd 0

G

rεα−2 u(x)g(x)ds. ∂G

Next we estimate the integral over Γd . Because on Γd : rε ≥ hr ≥ hd

⇒

(α − 3) ln rε ≤ (α − 3) ln(hd), α−3

since α ≤ 2, we have rεα−3 |Γd ≤ (hd) 2−α 2

 Γd

au2 rεα−4 (xi

and therefore:

2−α − εli )ni ds ≤ 2



arεα−3 u2 ds

Γd



≤ cδ Gd

u2 dx + δ



Gd

2−α α−3 (hd) ≤ 2



au2 ds

Γd

|∇u|2 dx, ∀δ > 0 (3.3.10)

48

Chapter 3. Elliptic transmission problem in a conical domain

by (1.5.12). By the Cauchy inequality and γ(ω) ≥ ν0 > 0,  δ

1  1 , 1 1 rg 2 (x), ∀δ > 0; ug ≤ r 2 , |g| r− 2 γ(ω)|u| ≤ r−1 γ(ω)u2 + 2 2δν0 γ(ω) taking into account property 1) of rε we obtain    δ 1 1 rεα−2 |u||g|ds ≤ rεα−2 γ(ω)u2 ds + rα−1 g 2 (x)ds, ∀δ > 0. (3.3.11) 2 r 2δν0 ∂G

∂G

∂G

Similarly, because of σ(ω) ≥ ν0 > 0,    δ 1 α−2 α−2 1 2 σ(ω)u ds + rε |u||h(x)|ds ≤ rε rα−1 h2 (x)ds, ∀δ > 0 2 r 2δν0 Σ0

Σ0

Σ0

(3.3.12) and  rεα−2 uf (x)dx ≤ G

δ 2

 G

ar−2 rεα−2 u2 dx +

1 2a∗ δ



rα f 2 (x)dx, ∀δ > 0.

(3.3.13)

G

Further, we use the representation G = Gd0 ∪ Gd . We estimate integrals over Gd0 . By assumption (b) and the Cauchy inequality, we obtain  

  aij (x) − aij (0) rεα−2 uxi uxj + rεα−4 (xi − εli )u(x)uxj

Gd 0

. + rεα−2 ai (x)uxi u(x) + rεα−2 a(x)u2 (x) dx 

≤ A(d) a rεα−2 |∇u|2 + rεα−3 |∇u| · |u(x)| + r−1 rεα−2 |∇u| · |u(x)| Gd 0

 + r−2 rεα−2 u2 (x) dx    ≤ 2A(d) a rεα−2 |∇u|2 + r−2 rεα−2 u2 + rεα−4 u2 dx.

(3.3.14)

Gd 0

Next, we estimate integrals over Gd . By assumptions (a), (c) and the Cauchy inequality and taking into account the inequality (3.3.3), we get   Gd

  aij (x) − aij (0) rεα−2 uxi uxj + rεα−4 (xi − εli )u(x)uxj

. + rεα−2 ai (x)uxi u(x) + rεα−2 a(x)u2 (x) dx

3.3. Global integral estimates

≤ μ∗





49

 3rεα−2 |∇u|2 + rεα−4 |u|2 dx +

Gd

⎛

  n    ∗ i 2 ⎝ ≤ C p, n, ν∗ , μ , α, d,  |a (x)|   



2 3 n 3 α−2 rε |u||∇u|4 |ai (x)|2 dx

Gd

⎠·

Lp/2 (G)

i=1

i=1

⎞





 ν|∇u|2 + u2 dx.

(3.3.15)

Gd

Thus, from (3.3.9)–(3.3.15) we derive:    1 α−2 ω0 2−α sin r arεα−2 |∇u|2 dx + ε au2 rεα−4 ds + σ(ω)u2 (x)ds 2 2 r ε Σ0

Γd 0

G



1 α−2 r γ(ω)u2 (x)ds r ε ∂G    ≤ 2A(d) a rεα−2 |∇u|2 + r−2 rεα−2 u2 + rεα−4 u2 dx

+

(3.3.16)

Gd 0

⎛

⎞   n        ∗ i 2 ⎝ ⎠ + C p, n, ν∗ , μ , α, d,  ν|∇u|2 + u2 dx |a (x)|  ·   i=1 Lp/2 (G) Gd    1 1 δ + rα−1 g 2 (x)ds + rα−1 h2 (x)ds + ar−2 rεα−2 u2 dx 2δν0 2δν0 2 Σ0 G ∂G    1 1 1 δ δ + rα f 2 (x)dx + rεα−2 σ(ω)u2 ds + rεα−2 γ(ω)u2 ds, ∀δ > 0. 2a∗ δ 2 r 2 r Σ0

G

∂G

By the inequality rε ≥ hr (see §1.3), we have rεα−4 ≤ h−2 r−2 rεα−2 . Hence, by Lemma 2.5, from (3.3.16) it follows that    1 α−2 1 α−2 α−2 2 2 r r arε |∇u| dx + σ(ω)u (x)ds + γ(ω)u2 (x)ds r ε r ε Σ0

G

∂G

≤ c(λ, ω0 ) (δ + A(d))     α−2 2 −1 α−2 2 −1 α−2 2 × arε |∇u| dx + r rε σ(ω)u (x)ds + r rε γ(ω)u ds Σ0

G

⎛

∂G

⎞   n        ⎠· ν|∇u|2 + u2 dx (3.3.17) |ai (x)|2  + C ⎝p, n, ν∗ , μ∗ , α, d,    i=1 Lp/2 (G) G    1 1 1 α−1 2 α−1 2 r g (x)ds + r h (x)ds + rα f 2 (x)dx, ∀δ > 0. + 2δν0 2δν0 2a∗ δ ∂G

Σ0

G

50

Chapter 3. Elliptic transmission problem in a conical domain

1 and next d > 0 such that, by the continuity of A(r) at zero, Choosing δ = 4c(λ,ω 0) 1 c(λ, ω0 )A(d) ≤ 4 we derive



arεα−2 |∇u|2 dx

G

 + Σ0

⎛

1 α−2 r σ(ω)u2 (x)ds + r ε



1 α−2 r γ(ω)u2 (x)ds r ε ∂G ⎞    ⎠ ν|∇u|2 + u2 + rα f 2 (x) dx

  n     i 2 ≤ C ⎝p, n, ν∗ , μ , a∗ , α, λ,  |a (x)|    i=1 Lp/2 (G) G    1 1 rα−1 g 2 (x)ds + rα−1 h2 (x)ds , ∀ε > 0. + ν0 ν0 ∗

(3.3.18)

Σ0

∂G

Now, we observe that the right-hand side of (3.3.18) does not depend on ε. Therefore we can perform the passage to the limit as ε → +0 by the Fatou Theorem. Then we get 

arα−2 |∇u|2 dx +

G



rα−3 σ(ω)u2 (x)ds +

Σ0

⎛



rα−3 γ(ω)u2 (x)ds

∂G

⎞    n       ∗ i 2 ⎝ ⎠ ≤ C p, n, ν∗ , μ , a∗ , α, λ,  ν|∇u|2 + u2 + rα f 2 (x) dx |a (x)|    i=1 Lp/2 (G) G    1 1 rα−1 g 2 (x)ds + rα−1 h2 (x)ds . (3.3.19) + ν0 ν0 Σ0

∂G

Applying Theorem 3.5 and Corollary 2.4 (see inequality (2.2.3)), from (3.3.19) follows the required estimate (3.3.5). 

3.4 Local integral weighted estimates Theorem 3.7. Let u(x) be a weak solution of problem (L) and assumptions (a)– (d) be satisfied with A(r) which is Dini-continuous at zero. Let λ be as above in ◦ 1

(2.2.1). Then u(x) ∈ W2−n (G) and there exist d ∈ (0, 1e ) and a constant C > 0 1 depending only on n, s, λ, a∗ , ω0 and on A(r) r dr such that the inequality 0





a r G 0

2−n

2

|∇u| + r

 u (x) dx +

−n 2

 r Σ 0

1−n

2



σ(ω)u (x)ds + Γ 0

r1−n γ(ω)u2 (x)ds

3.4. Local integral weighted estimates

51

⎧ ⎪2λ , ⎨  ⎪



1 1 ≤ C u22,G + f12 + g12 + h21 · 2λ ln2 1 , ⎪ ν0 ν0 ⎪ ⎩2s ,

if s > λ, if s = λ,

(3.4.1)

if s < λ

holds for almost all  ∈ (0, d). ◦ 1

Proof. By Theorem 3.6 u(x) ∈ W2−n (G), so it is enough to prove the estimate (3.4.1). Putting η(x) = r2−n u(x) in (II)loc , according to the definition (2.4.10) we have      ∂u  U () =  au(x)  dΩ + r2−n u(x) aij (x) − aij (0) uxj cos(r, xi )dΩ ∂r +

r=

Ω

 r

2−n

Ω



u(x)g(x)ds +

Γ 0

r

2−n

    −r2−n aij (x) − aij (0) uxi uxj u(x)h(x)ds +

Σ 0

G 0

+ (n − 2)r−n u(x)aij (x)xi uxj + r2−n u(x)ai (x)uxi + r2−n a(x)u2 (x)  − r2−n u(x)f (x) dx.

(3.4.2)

Now, we transform some integrals from the right-hand side:  (n − 2) r−n u(x)aij (x)xi uxj dx G 0



= (n − 2)

r

−n

5 xi

G 0

  a ∂u2 · + u(x) aij (x) − aij (0) uxj 2 ∂xi

6 (3.4.3)

dx;

by the Gauss-Ostrogradskiy divergence theorem,      ∂u2 ar−n xi dx = − au2 (x) nr−n − nr−n dx + −n au2 (x)xi ni dΩ ∂xi G 0

G 0



+ [a]Σ0

r−n u2 (x)xi ni ds +

Σ 0

Hence, since by Lemma 2.1      ni  = 0 and ni  = ; Γ 0

we have

Ω

n−2 2

 ar G 0

−n



Ω

ar−n u2 (x)xi ni ds.

(3.4.4)

Γ 0

  xi ni 

Σ0

  = xi cos(xn , xi )

∂u2 n−2 xi dx = ∂xi 2

 Ω

Σ0

  = xn 

au2 (x)dΩ.

Σ0

= 0,

(3.4.5)

52

Chapter 3. Elliptic transmission problem in a conical domain

By a(x) ≤ 0 and Lemma 2.12, from (3.4.2)–(3.4.5) we derive      2−n U () +  u(x) aij (x) − aij (0) uxj cos(r, xi )dΩ U () ≤ 2λ Ω

 +

r

2−n



u(x)g(x)ds +

Γ 0

r

2−n

    −r2−n aij (x) − aij (0) uxi uxj u(x)h(x)ds +

Σ 0

G 0

   + (n−2)r−n u(x) aij (x)−aij (0) xi uxj +r2−n u(x)ai (x)uxi −r2−n u(x)f (x) dx. Hence, by virtue of assumption (b), it follows that     U () ≤ U () + A() a|u||∇u|dΩ + r2−n |u(x)||g(x)|ds 2λ Ω



r2−n |u(x)||h(x)|ds + c1 (n)A()

+ Σ 0



Γ 0

  a r2−n |∇u|2 + r1−n |u||∇u| dx

G 0



r2−n |u(x)||f (x)|dx.

+

(3.4.6)

G 0

Further, we shall derive an upper bound for each integral from the right-hand side. Applying the Cauchy and Friedrichs-Wirtinger inequalities (see (W )2 ) according to (2.4.12), we obtain     1 a|u||∇u|dΩ ≤ a 2 |∇u|2 + |u|2 dΩ ≤ c2 (λ)U  (); (3.4.7) 2 Ω Ω     1−n ar |u||∇u|dx ≤ a r2−n |∇u|2 + r−n |u|2 dx ≤ c3 (λ)U () (3.4.8) G 0

G 0

by virtue of inequality (2.2.3); and for all δ > 0,  Γ 0

   , 3−n 1 r r2−n |u||g|ds = γ(ω)|u| r 2 , |g| ds γ(ω)  Γ0   1 δ 1−n 2 r γ(ω)|u| ds + r3−n |g|2 ds; ≤ 2 2δν0 

Γ 0

 r Σ 0

1−n 2

2−n

|u||h|ds =

Γ 0



r Σ 0

1−n 2

   , 3−n 1 2 , σ(ω)|u| r |g| ds σ(ω)

(3.4.9)

3.4. Local integral weighted estimates δ ≤ 2  r

2−n

G 0

 r Σ 0

δ |u(x)||f (x)|dx ≤ 2a∗ ≤

1−n

 ar G 0

53

1 σ(ω)|u| ds + 2δν0 2

−n

1 |u| dx + 2δ 2

1 δ c4 (λ)U () + 2a∗ 2δ







r3−n |h|2 ds;

(3.4.10)

Σ 0

r4−n |f |2 dx

G 0

r4−n |f |2 dx

(3.4.11)

G 0

by virtue of inequality (2.2.3). Thus, from (3.4.6)–(3.4.11) we get  (1 + c6 (λ)A()) U  ()+ [1 − c5 (n, λ, a∗ )(δ + A())] U () ≤ 2λ ⎫ ⎧ ⎪ ⎪    ⎬ ⎨ 1 1 1 4−n 2 3−n 2 3−n 2 r |f | dx + r |g| ds + r |h| ds , ∀δ > 0. (3.4.12) + ⎪ 2δ ⎪ ν0 ν0 ⎭ ⎩   Γ0

G0

Σ0

However, by the condition (d),    1 1 4−n 2 3−n 2 r |f | dx + r |g| ds + r3−n |h|2 ds ν0 ν0 G 0

Γ 0

  1 1 1 ≤ c0 f12 + g12 + h21 · 2s , 2s ν0 ν0

Σ 0

where c0 depends only on meas Ω, meas ∂Ω, meas σ0 . Hence we derive the differential inequality (CP ) from §2.4 with 2λ · [1 − c5 (n, λ, a∗ )(δ + A())], ∀δ > 0; N () ≡ 0;    λ 1 1 Q() = c0 f12 + g12 + h21 · δ −1 2s−1 , ∀δ > 0; (3.4.13) s ν0 ν0   -  .  U0 = C u2 + (1 + r4−n )f 2 (x) dx + r3−n h2 (x)ds + r3−n g 2 (x)ds , P() =

Σ0

G

∂G

due to (3.3.5) with α = 4 − n. 1) s > λ. In this case we put δ = ε , for all ε > 0. Then 2λ · [1 − c5 (n, λ, a∗ )(ε + A())];    λ 1 2 1 2 2 Q() = c0 f1 + g1 + h1 · 2s−1−ε . s ν0 ν0 P() =

54

Chapter 3. Elliptic transmission problem in a conical domain 2λ 

Since P() = τ − 

−

K()  ,

where K() satisfies the Dini condition at zero we obtain

  τ

 2λ d K(r) K(s) τ + ds ≤ ln dr =⇒ P(s)ds = −2λ ln +  s τ r ⎛

exp ⎝−

d 

⎛ exp ⎝−

τ

0



⎞ ⎛ d 

 2λ

 2λ K(τ ) dτ ⎠ = K0 P(τ )dτ ⎠ ≤ exp ⎝ ; d τ d ⎞

0

⎞ ⎛ d 

 2λ

 2λ K(τ ) ⎠ dτ = K0 P(τ )dτ ⎠ ≤ exp ⎝ . τ τ τ ⎞

0



We have also: d 

   τ  d λc0 K0 1 2 1 2 2λ 2 f 1 + g 1 + h1  Q(τ ) exp − P(σ)dσ dτ ≤ τ 2s−2λ−ε−1 dτ s ν0 ν0 



λc0 K0 ≤ s

  s−λ d 1 1 f12 + g12 + h21 · 2λ , ν0 ν0 s−λ

since s > λ and we can choose ε = s − λ. Now we apply Theorem 1.21: then from (1.7.1), by virtue of the inequalities deduced above and according to (2.2.3) with α = 4 − n, we obtain the statement (3.4.1) for s > λ. 2) s = λ. In this case we can take in (3.4.13) any function δ() > 0 instead of δ > 0. Then we obtain the problem (CP ) with 2λ(1 − δ()) A() − c5 ; N () = 0;     1 1 Q() = c0 f12 + g12 + h21 · δ −1 ()2λ−1 . ν0 ν0 P() =

Now, we choose δ() =

2λ ln

1

 , 0 <  < d, where e is the Euler number. Then ed 

we derive τ

d

 2λ τ dσ A(τ ) 

dτ + c5 + − P(σ)dσ ≤ ln ed τ τ σ ln σ   0 ⎛ ⎞ d

 2λ ln ed  A(τ ) = ln dτ =⇒ + ln ⎝  ⎠ + c5 ed τ τ ln τ 0

3.4. Local integral weighted estimates

55

 ed   d    τ ln 2λ  A(τ )  dτ , ·  exp c5 exp − P(σ)dσ ≤ τ τ ln ed τ  0  d   d   A(τ )  2λ ed  exp c5 dτ exp − P(τ )dτ ≤ ln d  τ 0



and therefore d

 τ  Q(τ ) exp − P(σ)dσ dτ





  d dτ 1 2 1 2 2λ ed  2

 · ≤ c6 f1 + g1 + h1  ln ed ν0 ν0  τ δ(τ ) ln τ   

ed  1 1 . ≤ 2λc6 f12 + g12 + h21 · 2λ ln2 ν0 ν0  Now we apply Theorem 1.21, and from (1.7.1), by virtue of the inequalities deduced above, we establish U () ≤ / c6 (U0 + f12 +

1 1 2 1 g1 + h21 )2λ ln2 , γ0 σ0 

0 0. We perform the change of variables x = x and u(x ) = ψ()v(x ). Then the function v(x ) satisfies the problem

 ⎧ 2 ∂ ij  i  2   2 ⎪  a (x )v  xj + a (x )vxi +  a(x )v = ψ() f (x ), x ∈ G1/4 , ⎪ ∂xi ⎪ ⎪ ⎨  ∂v  + |x1 | σ(ω)v(x ) = ψ() h(x ), x ∈ Σ21/4 , [v(x )]Σ2 = 0, ∂ν (L )  Σ21/4 1/4 ⎪ ⎪ ⎪ ⎪ ⎩ ∂v  1   2 ∂ν  + |x | γ(ω)v(x ) = ψ() g(ρx ), x ∈ Γ1/4 . By the Sobolev Embedding Theorem 1.19, sup |∇ v(x )| ≤ cvW2,p (G11/2 ) ,

x ∈G11/2

(3.5.5)

p > n.

[66, 67] for the solution of the On the strength of the local Lp a priori estimate   2 and near smooth portions of the equation of the (L ) inside the domains G1/4 ±

boundaries Σ21/4 ∪ Γ21/4 we have

(3.5.6) vW2,p (G11/2 ) .  f Lp (G21/4 ) + hW 1−1/p,p (Σ2 ) + gW1−1/p,p(Γ2 ) + cvLp (G21/4 ) . ≤c 1/4 1/4 ψ() Returning back to the variables x, from (3.5.5) and (3.5.6) it follows that sup |∇u|

G /2

≤ c1−n/p −2 uLp(G2 ) + f p,G2 + gV1−1/p (Γ2 /4

/4

p,0

) /4

.

+ hV 1−1/p (Σ2 p,0

) /4

and 2 (G ) 2−n/p uVp,0 /2 2−n/p −2  uLp(G2 ) + f p,G2 + gV1−1/p (Γ2 ≤ c /4

or

/4

p,0

/4

+ hV 1−1/p (Σ2 ) p,0

/4

. )

58

Chapter 3. Elliptic transmission problem in a conical domain sup |∇u|

G /2

≤ c−1 |u|0,G2 + f V0

+ gV1−1/p

2 p,2p−n (G/4 )

/4

2 p,2p−n (Γ/4 )

+ hV 1−1/p (Σ2 p,2p−n

.

/4

)

and 2 uVp,2p−n (G ) /2 ≤ c |u|0,G2 + f V0 /4

2 p,2p−n (G/4 )

+ gV1−1/p

2 p,2p−n (Γ/4 )

+ hV 1−1/p (Σ2 p,2p−n

/4

. . )

Hence, by virtue of (3.1.2), (3.1.3) and assumption (d), desired results (3.1.4) and (3.1.5) follow. 

3.5.2 Remark to Theorem 3.7 Now we can state that Theorem 3.6 is true for α ∈ (4−n−2λ, 2], if a neighborhood of the conic point is convex. In fact, from estimate (3.1.2) we obtain u(0) = 0 and therefore we can apply Lemma 2.10 to the proof of Theorem 3.6 for α ∈ (4 − n − 2λ, 4 − n). In this case the equality (3.3.8) can be rewritten, by virtue of (2.3.8), in the form of the inequality  ∂u2 2−α → dx ≤ au2 rεα−4 (xi − εli ) cos(− n , xi )ds ∂xi 2 G Γd   (2 − α)(4 − n − α) H(λ, n, α) + rεα−2 |∇u|2 dx + rεα−3 σ(ω)u2 (x)ds 2

2−α 2



arεα−4 (xi − εli )

 +



Σd 0

Gd 0

rεα−3 γ(ω)u2 (x)ds ,

Γd 0

where H(λ, n, α) is determined by (2.3.3). By virtue of the convexity of Gd0 and the first property of rε (see §1.3), we have rε ≥ r. Therefore (3.3.17) takes the form     1 α−2 (2 − α)(4 − n − α) H(λ, n, α) r arεα−2 |∇u|2 dx + σ(ω)u2 (x)ds 1− 2 r ε  +

1 α−2 r γ(ω)u2 (x)ds r ε

G



≤ c(λ, ω0 ) (δ + A(d))

∂G



+ Σ0



r−1 rεα−2 σ(ω)u2 (x)ds +

 ∂G

Σ0

arεα−2 |∇u|2 dx

G

r−1 rεα−2 γ(ω)u2 ds



(3.5.7)

3.6. Appendix

59

⎛

⎞   n        ⎠ + C ⎝p, n, ν∗ , μ∗ , α, d,  ν|∇u|2 + u2 dx |ai (x)|2    i=1 Lp/2 (G) G    1 1 1 α−1 2 α−1 2 r g (x)ds + r h (x)ds + rα f 2 (x)dx, ∀δ > 0. + 2δν0 2δν0 2a∗ δ Σ0

∂G

G

But, by (2.3.3) and 4−n−2λ < α < 4−n, we verify that 1− (2−α)(4−n−α) H(λ, n, α) 2 > 0. Therefore from (3.5.7) we conclude the validity of (3.3.18) and hence we obtain the required statement.

3.6 Appendix We consider in detail the two-dimensional transmission problem for the Laplace operator in an angular domain and investigate the corresponding eigenvalue problem. Let domain G ⊂ R2 lie inside the corner G0 = {(r, ω) |r > 0; − ω20 < ω < ω20 }, ω0 ∈]0, 2π[; O ∈ ∂G and in some neighborhood of O boundary ∂G coincide with sides of the corner: ω = − ω20 and ω = ω20 . Let us write ω0 Γ± = {(r, ω) | r > 0; ω = ± }, Σ0 = {(r, ω) | r > 0; ω = 0} 2     = γ± = const > 0. We and put σ(ω) = σ(0) = σ = const ≥ 0, γ(ω) ω0 Σ0

ω=±

consider the following problem: ⎧ ⎪ ⎨a± u± = f±(x), [u]Σ0 = 0, a ∂∂u + σr u(x) = h(x), → − n Σ0 ⎪ ⎩ ± + 1r γ± u± (x) = g± (x), α± a± ∂u − ∂→ n

2

x ∈ G± ; x ∈ Σ0 ;

(3.6.1)

x ∈ Γ± \ O,

where α± ∈ {0; 1}. It is well known that the homogeneous problem (f (x) = h(x) = g(x) = 0) has a solution of the form u(r, ω) = rλ ψ(ω), where λ2 is an eigenvalue and ψ(ω) is an associated regular eigenfunction of the problem 

⎧  ω0 2 ⎪ ψ + λ ψ (ω) = 0, ω ∈ 0, + ⎪ 2 ; ⎪ +

⎪ ⎨  ω ∈ − ω20 , 0 ; ψ− + λ2 ψ− (ω) = 0, (3.6.2) ⎪   ⎪ ψ (0) = ψ (0); a ψ (0) − a ψ (0) = σψ(0); ⎪ + − + − + − ⎪ ⎩ ±α± a± ψ  (± ω20 ) + γ± ψ(± ω20 ) = 0. 1) case λ = 0. In this case we have ψ± (ω) = A± · ω + B± . From boundary conditions B+ = B− = B and for the finding of A+ , A− , B we have the system ⎧ ⎪ a+ A+ − a− A− − σB = 0, ⎨   α+ a+ + ω20 γ+ A+ + γ+ B = 0, ⎪   ⎩ − α− a− + ω20 γ− A− + γ− B = 0.

60

Chapter 3. Elliptic transmission problem in a conical domain

Since A2+ + A2− + B 2 = 0, the system determinant must be equal to zero; this means the equality



ω0 

ω0  ω0  γ+ α− a− + γ− + a+ γ+ α− a− + γ− σ α+ a+ + 2 2 2

ω0  γ+ = 0 + a− γ− α+ a+ + 2

(3.6.3)

holds. Thus, if equality (3.6.3) is satisfied, then λ = 0 and we have the corresponding eigenfunction ⎧

!  " ⎨a− γ− ω − ω0 γ+ − α+ a+ , ω ∈ 0, ω0 , 2 2

ψ(ω) = !  " ⎩a+ γ+ ω + ω0 γ− − α− a− , ω ∈ − ω0 , 0 , if σ = 0; 2 2 ⎧

    ⎨−γ+ α− a− + ω0 γ− (ω + a+ ) − a− γ− α+ a+ + ω0 γ+ , ω ∈ 0, ω0 , 2 σ σ 2 2 ψ(ω) =     ⎩γ− α+ a+ + ω0 γ+ (ω − a− ) − a+ γ+ α− a− + ω0 γ− , ω ∈ − ω20 , 0 , 2 σ σ 2 if σ = 0. 2) case λ = 0. Solving the problem equations we have ψ± (ω) = A± cos(λω)+ B± sin(λω). From boundary conditions we obtain A+ = A− = A and for the finding of A, B+ , B− we have the system ⎧ ⎪ σA − λa+ B+ + λa− B− = 0, ⎨    γ+ cos λω2 0 − λα+ a+ sin λω2 0 A + γ+ sin λω2 0 + λα+ a+ cos λω2 0 B+ = 0, ⎪    ⎩ γ− cos λω2 0 − λα− a− sin λω2 0 A − γ− sin λω2 0 + λα− a− cos λω2 0 B− = 0. 2 2 Since A2 + B+ + B− = 0, the system determinant must be equal to zero; this means that λ is determined from the transcendence equation

σ(λ2 α+ α− a+ a− + γ+ γ− ) + λ2 (a+ − a− )(α− a− γ+ − α+ a+ γ− ) + λ[σ(α− a− γ+ + α+ a+ γ− ) + (a+ + a− )(γ+ γ− − λ2 α+ α− a+ a− )] sin(λω0 ) + [σ(λ2 α+ α− a+ a− − γ+ γ− ) + λ2 (a+ + a− )(α− a− γ+ + α+ a+ γ− )] cos(λω0 ) = 0. (3.6.4) Now we investigate some special cases of boundary conditions. The Dirichlet problem: α± = 0, γ± = 1. Equation (3.6.4)takes the form σ(1 − cos(λω0 )) + λ(a+ + a− ) sin(λω0 ) = 0. Hence π , if σ = 0, where λ∗ is the least positive root of the tranwe derive λ = ω∗0 λ , if σ > 0, − λ, and the corresponding eigenfunction scendence equation tan λω2 0 = − a+ +a σ

61

y

y = tg

λω0 2

3.6. Appendix

= βλ −

π ω0

λ∗ 2π ω0

3π ω0

λ

Figure 2 ⎧   ⎨sin λ ω0 − ω , 2 ψ(ω) =   ⎩sin λ ω0 + ω , 2 2), we observe that

π ω0

ω ∈ 0, ω20 ,

By the graphical method (see Figure ω ∈ − ω20 , 0 .

< λ∗
0, σ equation tan λω2 0 = a+ +a · λ1 . For λ = ωπ0 we find the corresponding eigenfunction −

ψ(ω) =

⎧ ⎨a− sin πω , ω0 ⎩a+ sin πω , ω0

ω ∈ 0, ω20 ,

ω ∈ − ω20 , 0 .

62

Chapter 3. Elliptic transmission problem in a conical domain

For λ = λ∗ we find the corresponding eigenfunction ⎧  ⎨cos λ∗ ω − ψ(ω) =  ⎩cos λ∗ ω +



ω0 2 ,  ω0 2 ,

ω ∈ 0, ω20 ,

ω ∈ − ω20 , 0 . π ω0 .

y = tg

λω0 2

By the graphical method (see Figure 3), we observe that 0 < λ∗
0, λ = · sin(λω) · . u(r, ω) = r ln r ω0 a+ , ω ∈ − 2 , 0 , By direct calculations (see also the investigation of the Neumann problem in Appendix, §3.6), we verify that it satisfies the transmission problem ⎧  ij  ∂ ⎪ ai (x)uxi = 0, x ∈ G \ Σ0 ; ⎨ ∂xi a (x)uxj + ∂u [u]Σ0 = 0, x ∈ Σ0 ; ∂ν Σ0 = 0, ⎪ ⎩ ∂u = 0, x ∈ ∂G \ {Σ0 ∪ O}, ∂ν

3.7. Examples

65 y

≈ 1,04 2π x 3 4x y= 2 x −4 y = tg

0

3 4

1

2

3

x

≈ −1,43

Figure 5

where x2 x1 x2 2a 2a · 2 2 , a12 (x) = a21 (x) = · 2 , λ + 1 r ln(1/r) λ + 1 r ln(1/r) 2a x2 a22 (x) = a − · 2 1 , aij (0) = aδij , i, j = 1, 2; λ + 1 r ln(1/r) 1 1 a1 (x) = − A(r) cos ω, a2 (x) = − A(r) sin ω, r r d 2a A(r) A(r) = , =⇒ dr = +∞. (λ + 1) ln(1/r) r

a11 (x) = a −

0

Clearly, the equation is uniformly elliptic in Gd0 for 0 < d < e−2 with the ellipticity 2a constants ν = a − ln(1/d) and μ = a. Thus, we observe that leading coefficients of the equation are continuous but not Dini-continuous at zero. From the explicit form of solution u we have |u(x)| ≤ c|x|λ−ε ,

2 uVp,2p−n ≤ cλ−ε (G 0)

(3.7.1)

for all ε > 0. This example shows that it is not possible to replace λ − ε in (3.7.1)

66

Chapter 3. Elliptic transmission problem in a conical domain

by λ without additional assumptions regarding the continuity modulus of leading coefficients of the equation at zero. Example 3.2. Let (λ, ψ(ω)) be a solution of eigenvalue problem (3.6.2) (see Appendix, §3.6). Then function u(x) = rλ ln( 1r )ψ(ω) is a solution of the transmission problem ⎧ λ−2 ⎪ x ∈ G \ Σ0 ; ⎨u = −2λr  ψ(ω), ∂u σ [u]Σ0 = 0, a ∂→ + r u(x) = 0, x ∈ Σ0 ; (3.7.2) − n Σ0 ⎪ ⎩ ∂u 1 + γu (x) = 0, x ∈ ∂G \ {Σ ∪ O}, αa ∂ → − ± 0 r n where a > 0, σ > 0, γ > 0, α ∈ {0, 1}. All assumptions of Theorem 3.3 are fulfilled with s = λ. This example shows the precision of assumption (d) and estimate (3.1.2) for s = λ.

Chapter 4

Transmission problem for the Laplace operator with N different media 4.1 Introduction In this chapter we investigate the behavior of weak solutions to the transmission problem (LN ) for the Laplace operator with N different media in a neighborhood of the boundary conical point: ⎧ ⎪ ⎪ au − pu(x) = f (x), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

∂u 1 = 0, S [u] ≡ a [u] k Σk ∂nk Σ + |x| βk (ω)u(x) = hk (x), ⎪ ⎪ k ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩B[u] ≡ α(x) · a ∂u + 1 γ(ω)u(x) = g(x), ∂ n |x| 

N −1

x ∈ G\ ∪ Σk ; k=1

x ∈ Σk , k = 1, . . . , N − 1; x ∈ ∂G \ {O} , (LN )

0, if x ∈ D, 1, if x ∈ / D. A principal new feature of this chapter is the derivation of sharp estimates for the (LN ) solutions in n-dimensional (n ≥ 2) conic domains with N different media (N ≥ 2). We demonstrate some examples. where a > 0, p ≥ 0; α(x) =

Let G ⊂ Rn , n ≥ 2 be a bounded domain with boundary ∂G that is a smooth surface everywhere except at the origin O ∈ ∂G and near the point O it is a convex conical surface with vertex at O and the opening ω0 . We assume that

M. Borsuk, Transmission Problems for Elliptic Second-Order Equations in Non-Smooth Domains, Frontiers in Mathematics, DOI 10.1007/978-3-0346-0477-2_5, © Springer Basel AG 2010

67

68

Chapter 4. Laplace operator with N different media

G =

N 8

Gi is divided into N sub-domains Gi , i = 1, . . . , N by (N − 1) hyper-

i=1

planes Σk , k = 1, . . . , N − 1, where O belongs to every Σk and Gi ∩ Gj = ∅, i = j. Let φi be openings at the vertex O in domains Gi . Let us define the value θk = φ1 + φ2 + · · · + φk , thus ω0 = θN . In addition to Chapter 1 we introduce the following notation: • Ωi : a domain on the unit sphere S n−1 with boundary ∂Ωi obtained by the intersection of the domain Gi with the sphere S n−1 (i = 1, . . . , N ); thus N 8 Ω= Ωi ; i=1

• Σ=

N −1

Σk , where Σk = G ∩ {ω1 =

k=1

• σ=

N −1

ω0 2

− θk }, k = 1, . . . , N − 1;

σk , where σk = Σk ∩ Ω;

k=1

• Gba = {(r, ω) | 0 ≤ a < r < b; ω ∈ Ω} ∩ G: a layer in Rn ; • Γba = {(r, φ) | 0 ≤ a < r < b; φ ∈ ∂Ω} ∩ ∂G: the lateral surface of layer Gba ; • Gd = G \ Gd0 , Γd = ∂G \ Γd0 , d > 0; • (Gi )ba = {(r, ω) | 0 ≤ a < r < b; ω ∈ Ω} ∩ Gi i = 1, . . . , N ; • Σba = Gba ∩ Σ; Σd = Σ \ Σd0 , d > 0; • (Σk )ba = Gba ∩ Σk , k = 1, . . . , N − 1; • D ⊆ ∂G: the part of the boundary ∂G where we consider the Dirichlet boundary condition.   • u(x) = ui (x), x ∈ Gi ; f (x) = fi (x), x ∈ Gi ; a = ai , etc. Gi

• [u]Σk denotes the saltus of the function u(x) on crossing Σk , i.e.,       [u]Σk = uk (x) −uk+1 (x) , uk (x) = lim u(x), Gk x→x∈Σk Σk Σk Σk   uk+1 (x) = lim u(x); Σk

∂u • a ∂ nk

Σk

Gk+1 x→x∈Σk

denotes the saltus of the co-normal derivative of the function u(x)

on crossing Σk , i.e.,   ∂u ∂uk  ∂uk+1  a = ak  −ak+1  , ∂ nk Σk ∂ nk Σk ∂ nk Σk where nk denotes the outward unit vector with respect to Gk normal to Σk .

4.1. Introduction

69

x1

: V2 62

6N

VN

61 V 1

*0

*0

O Figure 6

Without loss of generality we assume that there exists d > 0 such that Gd0 is a convex rotational cone with the vertex at O and the aperture ω0 , thus   n   ω0  2 d 2 ω0 2 , ω0 ∈ (0, π) . (4.1.1) x ; r ∈ (0, d), ω1 = Γ0 = (r, ω)x1 = cot 2 i=2 i 2 We use the standard function spaces: • C k (Gi ) with the norm |ui |k,Gi , • Lebesgue space Lp (Gi ), p ≥ 1 with the norm ui p,Gi , • the Sobolev space W k,p (Gi ) with the norm ui k,p;Gi , and introduce their direct sums • Ck (G) = C k (G1 )  · · ·  C k (GN ) with the norm |u|k,G =

N  i=1

• Lp (G) = Lp (G1 )· · ·Lp (GN ) with the norm uLp(G) = • W

k,p

(G) = W

k,p

(G1 )  · · ·  W uk,p;G =

k,p

(GN ) with the norm

N   k  i=1

Gi |β|=0

 p1 |D ui | dx . β

p

N 

|ui |k,Gi ;



i=1

 Gi

 p1 |ui | dx ; q

70

Chapter 4. Laplace operator with N different media

We define the weighted Sobolev spaces: k k k (G) = Vp,α (G1 )  · · ·  Vp,α (GN ) for integer k ≥ 0 and real α, where • Vp,α k Vp,α (Gi ) denotes the space of all distributions u ∈ D (Gi ) satisfying for α k i = 1, . . . , N : r p +|β|−k |Dβ ui | ∈ Lp (Gi ); Vp,α (G) is a Banach space for the norm  p1 N   k  α+p(|β|−k) β p uVp,α k (G) = r |D ui | dx . i=1

Gi |β|=0

k− 1

• Vp,α p (∂G) is the space of functions ϕ, given on ∂G, with the norm ϕ

k− 1

Vp,αp (∂G)

= inf ΦVp,α k (G) ,

  where the infimum is taken over all functions Φ such that Φ = ϕ in the ∂G sense of traces. We write Wk (G) ≡ Wk,2 (G),

◦ k

k Wα (G) ≡ V2,α (G).

Definition 4.1. A function u(x) is called a weak solution of problem (LN ) provided ◦ 1

that u(x) ∈ C0 (G) ∩ W0 (G) and satisfies the integral identity 

 auxj ηxj dx + G

Σ



=

1 β(ω)u(x)η(x)ds + r

1 α(x) γ(ω)u(x)η(x)ds r

∂G





h(x)η(x)ds −

α(x)g(x)η(x)ds + Σ

∂G



(pu(x) + f (x)) η(x)dx

(II)

G ◦ 1

for all functions η(x) ∈ C0 (G) ∩ W0 (G). The summation over repeated indices from 1 to n is understood; here:  f (x)dx = G

N   i=1 G

 fi (x)dx,

h(x)ds =

N −1  

Σ

i

hk (x)ds, etc.

k=1 Σ

k

Remark 4. In the Dirichlet boundary condition  case (α(x) ≡ 0) we assume, with   = 0 =⇒ u = 0. out loss of generality, that g  ∂G∩D

∂G∩D

We assume that M0 = max |u(x)| is known. x∈G

4.1. Introduction

71

Lemma 4.2. Let u(x) be a weak solution of (LN ). For any function η(x) ∈ C0 (G)∩ ◦ 1

W0 (G) the equality  auxi ηxi G 0



+ Γ 0

 . ∂u + (f (x) + pu(x)) η(x) dx = a η(x)dΩ ∂r Ω

    1 1 h(x) − β(ω)u(x) η(x)ds α(x) g(x) − γ(ω)u(x) η(x)ds + r r 

(II)loc

Σ 0

holds for a.e.  ∈ (0, d). 

Proof. The proof is analogous to the proof of Lemma 3.2 Chapter 3. Let us define the numbers ⎧ ⎪ a∗ = min{a1 , . . . , aN } > 0, a∗ = max{a1 , . . . , aN } > 0; ⎪ ⎪ ⎪ ⎨p∗ = max{p1 , . . . , pN } ≥ 0; [a]Σk = ak − ak+1 , k = 1, . . . , N − 1; ⎪ ⎪   ⎪ ⎪ ⎩a0 = max [a]Σ  ; / a = max(a∗ , a0 ). k

(4.1.2)

1≤k≤N −1

Assumptions. (a) f (x) ∈ Lp/2 (G) ∩ L2 (G); p > n; a tan ω20 on ∂G; βk (ω) ≥ β0 > / a tan ω20 on Σk , k = 1, . . . , N − 1; (b) γ(ω) ≥ γ0 > / (c) there exist numbers f0 ≥ 0, g0 ≥ 0, h0 ≥ 0, s > 1, such that |f (x)| ≤ f0 |x|s−2 , |g(x)| ≤ g0 |x|s−1 , |hk (x)| ≤ h0 |x|s−1 , k = 1, . . . , N − 1. We formulate the main result as the following theorem. Let , 2 − n + (n − 2)2 + 4ϑ , λ= 2

(4.1.3)

where ϑ is the smallest positive eigenvalue of the problem (EV P N ) (see Subsection 4.2.1). Theorem 4.3. Let u be a weak solution of problem (LN ) and assumptions (a)–(c) be satisfied. Then there are d ∈ (0, 1) and C0 > 0 depending only on n, a∗ , a∗ , λ, ω0 , f0 , h0 , g0 , β0 , γ0 , s, M0 , meas G, diam G such that the inequality ⎧ λ ⎪ ⎪ ⎨|x| ,  if s > λ,

 1 1 1 , if s = λ, (4.1.4) |u(x)| ≤ C0 u2,G + f0 + √ g0 + √ h0 · |x|λ ln |x| ⎪ γ0 β0 ⎪ ⎩|x|s , if s < λ

72

Chapter 4. Laplace operator with N different media

holds for all x ∈ Gd0 . If, in addition, β(ω) ∈ C1 (Σ), γ(ω) ∈ C1 (∂G), 0 f (x) ∈ Vp,2p−n (G), 1−1/p

1−1/p

h(x) ∈ Vp,2p−n (Σ), g(x) ∈ Vp,2p−n (∂G); p > n and there is a number

τs =: sup −s h >0

1− 1 p Vp,2p−n (Σ ) /2

+ g

 1− 1 p Vp,2p−n (Γ ) /2

(4.1.5)

,

then

⎧ ⎪|x|λ−1 , ⎨

 ⎪

 1 1 1 , |∇u(x)| ≤ C1 u2,G + f0 + √ g0 + √ h0 + τs · |x|λ−1 ln |x| ⎪ γ0 β0 ⎪ ⎩|x|s−1 ,

if s > λ, if s = λ, if s < λ (4.1.6)

for all x ∈ Gd0 . Furthermore, the following is true: 2 • u ∈ Vp,2p−n (G), p > n and 2 uVp,2p−n (G 0)

⎧ ⎪ λ ,  ⎪ ⎨

 1 1 ≤ C2 u2,G + f0 + √ g0 + √ h0 + τs · λ ln 1 , ⎪ γ0 β0 ⎪ ⎩ s ,

• if

◦ 0



f (x) ∈ Wα (G),

rα−1 h2 (x)ds < ∞,

Σ



if s > λ, if s = λ,

(4.1.7)

if s < λ;

rα−1 g 2 (x)ds < ∞,

∂G

where 4 − n − 2λ < α ≤ 2,

(4.1.8)

◦ 1

then u(x) ∈ Wα−2 (G) and     a rα−2 |∇u|2 + rα−4 u2 dx + rα−3 β(ω)u2 (x)ds G



+ ∂G

≤C

Σ

α(x)rα−3 γ(ω)u2 (x)ds -  G

 u2 + (1 + rα )f 2 (x) dx +

(4.1.9) 

rα−1 h2 (x)ds +

Σ



. α(x)rα−1 g 2 (x)ds ,

∂G ∗

where the constant C > 0 depends only on n, a∗ , a , α, λ and the domain G.

4.2. Auxiliary statements and inequalities

73

4.2 Auxiliary statements and inequalities In addition to Lemma 1.1 by means of direct calculation we obtain Lemma 4.4. k = 1, . . . , N − 1

xi cos( nk , xi )|Σk = 0,

(4.2.1)

(it is the equation of Σk .) We will need some statements and inequalities.

4.2.1 The eigenvalue problem Let Ω ⊂ S n−1 with smooth boundary ∂Ω be the intersection of the cone C with → the unit sphere S n−1 . Let − ν be the exterior normal to ∂C at points of ∂Ω and → − τ k be the exterior with respect to Ωk normal to Σk (lying in the plane tangent to Ωk ), k = 1, . . . , N − 1. Let γ(ω), ω ∈ ∂Ω be a positive bounded piecewise smooth function, βk (ω) be a positive continuous function on Σk , k = 1, . . . , N − 1. We consider the eigenvalue problem for the Laplace-Beltrami operator ω on the unit sphere ⎧ ai (ω ψi + ϑψ ⎪ ⎪ i ) = 0, ω ∈ Ωi , ai are positive constants; i = 1, . . . , N, ⎨  a ∂∂ψ + βk (ω)ψ  = 0, k = 1, . . . , N − 1, [ψ]σk = 0, → − τk σ σk k  ⎪ ⎪ ⎩α(ω)a ∂ψ + γ(ω)ψ  = 0, → − ∂ν ∂Ω

(EV P N ) which consists in the determination of all values ϑ (eigenvalues) for which (EV P N ) has a non-zero weak solution (eigenfunction). From the variational principle we obtain the Friedrichs-Wirtinger type inequality(see the proof for N = 2 in §2.2): Theorem 4.5. Let ϑ be the smallest positive eigenvalue of the problem (EV P N ) and ψ ∈ W1 (Ω). Let γ(ω) be a positive bounded piecewise smooth function on ∂Ω, βk (ω) be a positive continuous function on σk , k = 1, . . . , N − 1. Then 



2

aψ (ω)dΩ ≤

ϑ Ω

2

a|∇ω ψ| dΩ +

N −1  

2

βk (ω)ψ (ω)dσ +

k=1 σ

Ω



α(ω)γ(ω)ψ 2 (ω)dσ.

∂Ω

k

(4.2.2) By (4.1.3), the Friedrichs-Wirtinger inequality can be written in the form     λ(λ + n − 2) aψ 2 (ω)dΩ ≤ a|∇ω ψ|2 dΩ + β(ω)ψ 2 (ω)dσ + α(ω)γ(ω)ψ 2 (ω)dσ, Ω

for all ψ(ω) ∈ W1 (Ω).

Ω

σ

∂Ω

(4.2.3)

74

Chapter 4. Laplace operator with N different media Similarly as in §§2.2–2.3 we obtain

Corollary 4.6.  arα−4 u2 dx ≤ Gd 0

1 λ(λ + n − 2)





arα−2 |∇u|2 dx +

Gd 0

 +

rα−3 β(ω)u2 (x)ds

Σd 0

. α(x)rα−3 γ(ω)u2 (x)ds ,

∀α,

(4.2.4)

Γd 0 ◦ 1

for all u ∈ Wα−2 (Gd0 ). Lemma 4.7. Let v ∈ C0 (G) ∩ W1 (G), v(0) = 0 and β(ω) > 0, γ(ω) > 0. Then for any ε > 0,   - arεα−4 v 2 dx ≤ H(λ, n, α) arεα−2 |∇v|2 dx + rεα−3 β(ω)v 2 (x)ds Gd 0

Gd 0

 +

Σd 0

. α(x)rεα−3 γ(ω)v 2 (x)ds , (4.2.5)

Γd 0

where H(λ, n, α) is determined by (2.3.3) and α ≤ 4 − n. Lemma 4.8. Let v ∈ C0 (G) ∩ W1 (G) and β(ω) > 0, γ(ω) > 0. Then for any ε > 0,    1 α−2 −2 2 α−2 2 arε r v dx ≤ arε |∇v| dx + r−1 rεα−2 β(ω)v 2 (x)ds λ(λ + n − 2) Gd 0

Gd 0

 +

Σd 0

 α(x)r−1 rεα−2 γ(ω)v 2 ds . (4.2.6)

Γd 0

4.2.2 The comparison principle We consider the second-order linear degenerate operator Q of the form   . ∂u auxi ηxi + (f (x) + pu(x)) η(x) dx − a η(x)dΩd Q(u, η) ≡ ∂r Gd 0



−

(4.2.7)

Ωd

    1 1 h(x) − β(ω)u(x) η(x)ds α(x) g(x) − γ(ω)u(x) η(x)ds − r r 

Γd 0

Σd 0 ◦ 1

for u(x) ∈ C0 (Gd0 ) ∩ W0 (Gd0 ) and for all non-negative η belonging to C0 (Gd0 ) ∩ ◦ 1

W0 (Gd0 ).

4.3. The barrier function. The preliminary estimate of the solution modulus 75 Proposition 4.9. Let β(ω), γ(ω) be positive piecewise functions, 0 < a∗ ≤ a ≤ a∗ , ◦ 1

p ≥ 0 and d ≪ 1. Let functions u, w ∈ C0 (Gd0 ) ∩ W0 (Gd0 ) satisfy the inequality Q(u, η) ≤ Q(w, η)

(4.2.8)

◦ 1

for all non-negative η ∈ C0 (Gd0 ) ∩ W0 (Gd0 ) and also the inequality   u(x) ≤ w(x), x ∈ Ωd ∪ Γd0 ∩ D

(4.2.9)

hold in the weak sense. Then u(x) ≤ w(x) in Gd0 . Proof. Let us define z(x) = u(x) − w(x). Then we have  . 0 ≥ Q(u, η) − Q(w, η) = azxi ηxi + pz(x)η(x) dx  +

Gd 0



1 α(x) γ(ω)z(x)η(x)ds + r

Γd 0

1 β(ω)z(x)η(x)ds − r

Σd 0

 a Ωd

∂z η(x)dΩd ∂r

(4.2.10)

◦ 1

for all non-negative η ∈ C0 (Gd0 ) ∩ W0 (Gd0 ). We define the sets (Gd0 )+ := {x ∈ Gd0 | u(x) > w(x)} ⊂ Gd0 , (Σd0 )+ := {x ∈ Σd0 | u(x) > w(x)} ⊂ Σd0 , (Γd0 )+ := {x ∈ Γd0 | u(x) > w(x)} ⊂ Γd0 and assume that (Gd0 )+ = ∅. As the test function in the integral inequality (4.2.10), we choose η = max{(u − w), 0}. Then it follows from (4.2.9) and (4.2.10) that      1 1 β(ω)z 2 (x)ds ≤ 0. a|∇z|2 + pz 2 (x) dx + α(x) γ(ω)z 2 (x)ds + r r + (Gd 0)

+ (Γd 0)

+ (Σd 0)

This implies that z(x) = 0 almost everywhere in (Gd0 )+ . Thus, we have finished with a contradiction to our assumption about the set (Gd0 )+ = ∅. By this fact, Proposition 4.9 is proved. 

4.3 The barrier function. The preliminary estimate of the solution modulus Let us define the linear operators Li ≡ ai , i = 1, . . . , N and let us use numbers (4.1.2).

76

Chapter 4. Laplace operator with N different media

a tan ω20 , Lemma 4.10 (Existence of the barrier function). Let us fix numbers β0 > / ω0 γ0 > / a tan 2 , δ > 0, g0 ≥ 0, h0 ≥ 0 and let γ(ω) ≥ γ0 on ∂G and βk (ω) ≥ β0 on Σk , k = 1, . . . , N − 1. Then there exist m > 0, depending only on ω0 , a number κ0 ∈ (0, δ/a0 cot ω20 − 1) (where δ0 = min(β0 , γ0 )), numbers B > 0, d ∈ (0, 1) and a function w(x) ∈ C1 (G0 ) ∩ C2 (G0 ) that depend only on ω0 , the ellipticity constants a∗ , a∗ of the operators Li and the quantities γ0 , β0 , δ0 , g0 , h0 , ω0 such that for any κ ∈ (0; min(δ, κ0 )) the following inequalities hold: Li [w(x)] ≤ −a∗ m2 |x|κ−1 ; x ∈ Gd0 ∩ Gi ; i = 1, . . . , N ;

(4.3.1)

B[w(x)] ≥ g0 |x|δ ; x ∈ Γd0 ;

(4.3.2)

Sk [w(x)] ≥ h0 |x|δ ; x ∈ Σdk ; k = 1, . . . , N − 1;

(4.3.3)

κ+1

0 ≤ w(x) ≤ c0 (κ0 , B, ω0 )|x|

; x ∈ Gd0 ;

(4.3.4)

|∇w(x)| ≤ c1 (κ0 , B, ω0 )|x|κ ; x ∈ Gd0 ,

(4.3.5)

where operators B, Sk were defined in problem (LN ). Proof. Let (x, y, x ) ∈ Rn , where x = x1 , y = x2 , x = (x3 , . . . , xn ). In {x1 ≥ 0} we consider the cone K with the vertex in O such that K ⊃ Gd0 (we recall that Gd0 ⊂ {x1 ≥ 0}). Let ∂K be the lateral surface of K and let ∂K ∩ yOx = Γ± be x = ±my, where m = cot ω20 , 0 < ω0 < π such that in the interior of K the inequality x > m|y| holds. We shall consider the function: w(x; y, x ) ≡ xκ−1 (x2 − m2 y 2 ) + Bxκ+1

with some κ ∈ (0; 1), B > 0. (4.3.6)

Let us calculate the operator Li on the function (4.3.6). For t = obtain:

y x,

|t|
/ a tan ω20 on ∂G, it follows that   B[w]

mκ

=

Γd ±

1 + m2 mκ

≥

1 + m2

Since m >   B[w]

a∗ γ0

r  κ+1 2 r  κ+1 2

κ

!

 " Bmγ(ω) − α(x)a B(1 + κ) + 2(1 + m2 )

κ

!

" Bmγ0 − Ba∗ (1 + κ) − 2a∗ (1 + m2 ) .

and κ0 < m aγ∗0 − 1, for κ ≤ κ0 we obtain

! " mκ 0 r κ ∗ ∗ ∗ 2 δ  κ02+1 B(mγ0 − a − a κ0 ) − 2a (1 + m ) ≥ g0 r , Γd 2 ± 1+m (4.3.12) 0 < r < d < 1, if we choose κ ≤ δ =⇒ rκ ≥ rδ . Hence it follows that ≥ 

κ +1 . - g 1 + m2  02 1 0 ∗ 2 . + 2a (1 + m ) · B≥ κ 0 m mγ0 − a∗ (1 + κ0 )

(4.3.13)

Thus, (4.3.2) is proved. Now we prove (4.3.3). At first, we have on Σk : 

ω 

ω 0 0 − θk , y = r sin − θk =⇒ x = r cos 2 2 

ω 0 y = mk x, where mk = tan − θk . 2 Therefore 

ω   0 − θk w = (B + 1 − m2 m2k )x1+κ = (B + 1 − m2 m2k )r1+κ cos1+κ 2 Σk 2 2 B + 1 − m mk 1+κ = r . 1+κ (1 + m2k ) 2

78

Chapter 4. Laplace operator with N different media

Further,



ω  ω0 mk 0 + θk = sin − θk = , , 2 2 2 1 + m2k 

ω 

1 ω0 0 + θk = − cos − θk = − , cos(n k , y) = cos π − 2 2 1 + m2k

cos(n k , x) = cos

π

−

on Σk , k = 1, . . . , N − 1 and by virtue of  (1 + κ)(1 + B) + (1 − κ)m2 m2k κ  wx  = ·r ; κ Σk (1 + m2k ) 2 we obtain

  wy 

Σk

=−

2m2 m2k κ κ · r , (1 + m2k ) 2

∂w  (1 + κ)(1 + B) + (1 − κ)m2 m2k + 2m2 κ ·r .  = mk · 1+κ ∂nk Σk (1 + m2k ) 2

Thus, Sk [w] =

βk (ω)(B + 1 − m2 m2k ) + mk [a]Σk {(1 + κ)(1 + B) + (1 − κ)m2 m2k + 2m2 } (1 + m2k )

1+κ 2

· rκ .

1 Since 0 < ω1 ≤ θk < ω0 for all k = 1, . . . , N − 1, then |mk | < tan ω20 = m for all ω0 k = 1, . . . , N − 1. Now, by virtue of βk (ω) ≥ β0 > / a tan 2 on Σk , k = 1, . . . , N − 1 and κ0 < m βa00 − 1, for κ ≤ κ0 , we get

  Sk [w] ≥ {β0 Bm − a0 (1 + κ0 )B + 2(1 + m2 ) }

mκ 0 (1 +

m2 )

1+κ0 2

· r κ ≥ r δ · h0 ,

if we choose κ ≤ δ =⇒ rκ ≥ rδ . Hence it follows that κ +1 - h 1 + m2  02 . 1 0 2 . B≥ + 2a (1 + m ) · 0 κ 0 m mβ0 − a0 (1 + κ0 )

(4.3.14)

(4.3.15)

Thus, (4.3.3) is proved. Now we will show (4.3.4). Let us rewrite the function (4.3.6) in spherical coordinates. Recalling that m = cot ω20 we obtain w(x; y, x ) = (1 + B)(r cos ω)1+κ − m2 r2 sin2 ω(r cos ω)κ−1    ω0 ω0 χ(ω) , ∀ω ∈ − ; = r1+κ cosκ−1 ω B cos2 ω + 2 ω0 , 2 2 sin 2     where χ(ω) = sin ω20 − ω · sin ω20 + ω . We find χ (ω) = − sin 2ω and χ (ω) = 0 for ω = 0. Now we see that χ (0) = −2 cos 0 = −2 < 0. In this way we have max χ(ω) = χ(0) = sin2 ω20 and therefore ω∈[−ω0 /2,ω0 /2]

 w(x; y, x ) ≤ r1+κ cosκ−1 ω(B cos2 ω + 1) ≤ r1+κ cosκ+1 ω B +

1 cos2 ω



4.3. The barrier function. The preliminary estimate of the solution modulus 79 ≤r

1+κ

 B+

1 cos2 ω20

 .

Hence (4.3.4) follows. Finally, (4.3.5) follows in the same way, by virtue of (4.3.11).  Now we can estimate |u(x)| for (LN ) in a neighborhood of a conical point. Theorem 4.11. Let u(x) be a weak solution of the problem (LN) and satisfy assumptions (a)–(c). Then there exist numbers d ∈ (0, 1) and κ > 0 depending only on a∗ , n, ω0 , f0 , h0 , g0 , β0 , γ0 , s and the domain G such that |u(x) − u(0)| ≤ C0 |x|κ+1 , x ∈ Gd0 ,

(4.3.16)

where the positive constant C0 depends only on a∗ , n, ω0 , f0 , h0 , g0 , β0 , γ0 , s, M0 and the domain G, and does not depend on u(x). Proof. Without loss of generality we may suppose that u(0) ≥ 0. Let us take the barrier function w(x) defined by (4.3.6) with κ ∈ (0, κ0 ) and the function v(x) = u(x) − u(0). For them we shall verify Proposition 4.9. Let us calculate operator Q on these functions. Because of the definition (II)loc , by Lemma 4.2 and by integrating by parts, we have:   . ∂w η(x)dΩd Aawxi ηxi + (f (x) + Apw(x)) η(x) dx − Aa Q(Aw, η) ≡ ∂r  − 

Ωd

Gd 0

    1 1 h(x) − β(ω)Aw(x) η(x)ds α(x) g(x) − γ(ω)Aw(x) η(x)ds − r r 

Γd 0

Σd 0

(f (x) + Apw(x) − Aaw) η(x)dx

= Gd 0



 (B[Aw] − g(x)) η(x)ds +

+ Γd 0

(S[Aw] − h(x)) η(x)ds Σd 0

with any A > 0. Hence, by Lemma 4.10 and conditions (b), (c), we obtain   .   2 κ−1 f (x) + Aa∗ m r η(x)dx + Ag0 rδ − g(x) η(x)ds Q(Aw, η) ≥ Gd 0



+ Σd 0

Γd 0



 Ah0 rδ − h(x) η(x)ds

80

Chapter 4. Laplace operator with N different media  . rκ−1 Aa∗ m2 − f0 rβ+1−κ η(x)dx + +g0 (Arδ − rs−1 )η(x)ds

 ≥ Gd 0

Γd 0

 (Arδ − rs−1 )η(x)ds ≥ 0,

+ h0 Σd 0

because of 0 < κ < κ0 , if numbers κ0 , δ, A are chosen such that κ0 ≤ β + 1, δ ≤ s − 1, A ≥ max{1,

f0 }. a∗ m 2

(4.3.17)

On the other hand, we have 

 pu(0)η(x)dx −

Q(v, η) ≡ −

γ(ω) u(0)η(x)ds − r

Γd 0

Gd 0



σ(ω) u(0)η(x)ds ≤ 0 r

Σd 0

and thus we get Q(v, η) ≤ Q(Aw, η)

(4.3.18)

◦ 1

for all non-negative η ∈ C0 (Gd0 ) ∩ W0 (Gd0 ). Now we compare v(x) and w(x) on Ωd . Since x2 ≥ h2 y 2 in K from (4.3.6) we have  ω0  w(x) . (4.3.19) ≥ Bd1+κ cosκ+1 2 r=d On the other hand,     v(x) = (u(x) − u(0)) ≤ M0 (4.3.20) Ωd

Ωd

and therefore from (4.3.19)–(4.3.20) we obtain   Aw(x)

Ωd

≥ ABd1+κ cosκ+1

 ω0 m1+κ0  ≥ ABd1+κ 1+κ0 ≥ M0 ≥ v  2 2 Ωd 2 (1 + m )

if we choose A possibly greatest, 1+κ0

M0 (1 + m2 ) 2 A≥ , B(md)1+κ0

(4.3.21)

where B satisfies inequalities (4.3.13) and (4.3.15). Finally, if Γd0 ∩ D = ∅ and u(x) = g(x), x ∈ Γd0 ∩ D, where |g(x)| ≤ g0 |x|s , then we have   Aw

v(x) = u(x) − u(0) ≤ u(x) = g(x) ≤ g0 |x|s , x ∈ Γd0 ∩ D; Γd 0 ∩D

= ABr1+κ cosκ+1

 ω0 m1+κ0  s ≥ ABr1+κ 1+κ0 ≥ g0 r ≥ v  d 2 2 Γ0 ∩D (1 + m ) 2

4.3. The barrier function. The preliminary estimate of the solution modulus 81 if κ0 ≤ s − 1 and if we choose A possibly greatest, 1+κ0

(1 + m2 ) 2 g0 , (4.3.22) Bm1+κ0 where B satisfies inequalities (4.3.13) and (4.3.15). Thus, if we choose large numbers B > 0, A ≥ 1 according to (4.3.13), (4.3.15), (4.3.17), (4.3.21) and (4.3.22), we provide the validity of Proposition 4.9. Therefore, by the comparison principle (Proposition 4.9), we have: A≥

u(x) − u(0) ≤ Aw(x), x ∈ Gd0 .

(4.3.23)

Similarly, we derive the estimate u(x) − u(0) ≥ −Aw(x) if we consider the auxiliary function v(x) = u(0) − u(x). By virtue of (4.3.4), our theorem is proved.  Now we will estimate the gradient modulus of the problem (LN ) solution near a conical point. Theorem 4.12. Let u(x) be a weak solution of the problem (LN ) and assumptions (b)–(c) be satisfied. Let κ > 0 be a number defined by Lemma 4.10 with κ0 ≤ s− 1. Then there exists a number d > 0 such that |∇u(x)| < C1 |x|κ , x ∈ Gd0 ,

(4.3.24)

where the constant C1 does not depend on u, but depends only on a∗ , a∗ , n, ω0 , f0 , h0 , g0 , β0 , γ0 , s, M0 and the domain G. Proof. Let us consider the set G/2 ⊂ G, 0 < ρ < d. We make the transformation x = x ; v(x ) = −1−κ u(x ). The function v(x ) satisfies the problem ⎧ N −1 ⎪ a v − p2 v(x ) = 1−κ f (x ), x ∈ G11/2 \ ∪ (Σk )11/2 ; ⎪ ⎪ k=1 ⎪ ⎪ ⎨

∂v 1  −κ   1 a ∂n + |x | βk (ω)v(x ) =  hk (x ), x ∈ (Σk )1/2 , k = 1, . . . , N − 1; ⎪ k (Σk )1 ⎪ 1/2 ⎪ ⎪ ⎪ ⎩ ∂v 1  −κ α(x ) · a ∂n g(ρx ), x ∈ Γ11/2 .  + |x | γ(ω)v(x ) =  (LN ) Now we apply Theorem 2.1 §2 [6] for α = 1 and Theorem 16.2 chapter III [43] for α = 0; according to these Theorems max |∇ v(x )| ≤ M1 .

x ∈G11/2

(4.3.25)

Returning to the variable x and the function u(x) we obtain from (4.3.25) |∇u(x)| ≤ M1 ρκ , x ∈ Gρρ/2 , 0 < ρ < d. Putting now |x| = 23 ρ we obtain the desired estimate (4.3.24).



82

Chapter 4. Laplace operator with N different media

Corollary 4.13. Let u(x) be a weak solution of problem (LN ) and assumptions of Theorem 4.12 be satisfied. Then u(0) = 0 and therefore the inequality (4.3.16) takes the form |u(x)| ≤ C0 |x|κ+1 , x ∈ Gd0 . (4.3.26) Proof. From the problem boundary condition it follows that γ(ω)u(x) = |x|g(x) − α(x) · a|x|

∂u , ∂n

x ∈ ∂G \ O.

By the assumption (b)–(c) and the estimate (4.3.24), we obtain γ0 |u(x)| ≤ γ(ω)|u(x)| ≤ |x||g(x)| + a∗ |x||∇u| ≤ g0 |x|s + C1 a∗ |x|κ+1 . By letting |x| tend to 0 we get, because of the continuity of u(x), that γ0 |u(0)| ≤ 0 and taking into account γ0 > 0, we obtain that u(0) = 0. 

4.4 Local estimate at the boundary Here we formulate a result asserting the local boundedness (near the conical point) of the weak solution of problem (LN ). This result is a generalization of Theorem 3.4 in the case of a domain with N different media and it proves verbatim as Theorem 3.4. Theorem 4.14. Let u(x) be a weak solution of problem (LN) and assumptions (a)–(c) be satisfied. If, in addition, h(x) ∈ L∞ (Σ0 ), g(x) ∈ L∞ (∂G), then the inequality .

sup |u(x)| ≤ C −n/t ut,G0 + 2(1−n/p) f p/2,G0 +  g∞,Γ0 + h∞,Σ0 Gκ 0

(4.4.1) holds for any t > 0, κ ∈ (0, 1) and  ∈ (0, d), where C = C (n, a∗ , a∗ , t, p, κ, G).

4.5 Global integral estimates First we will obtain a global estimate for the Dirichlet integral. Theorem 4.15. Let u(x) be a weak solution of problem (LN) and assumptions (a)– (c) be satisfied. Suppose, in addition, that h(x) ∈ L2 (Σ), g(x) ∈ L2 (∂G). Then the inequality    β(ω) 2 γ(ω) 2 u (x)ds + u (x)ds a|∇u|2 dx + r r G Σ ∂G ⎧ ⎫    ⎨  ⎬  1 1 ≤C u2 (x) + f 2 (x) dx + h2 (x)ds + g 2 (x)ds (4.5.1) ⎩ ⎭ β0 γ0 G

Σ

∂G

4.5. Global integral estimates

83

holds, where constant C depends only on p∗ , diam G. Proof. Setting in (II) η(x) = u(x) and using the classical Hölder inequality, by assumptions (a), (c), we get 

 β(ω) 2 γ(ω) 2 u (x)ds + u (x)ds r r Σ ∂G    ≤ |u||h(x)|ds + |u||g(x)|ds + |u||f (x)|dx.

9 : a|∇u|2 + pu2 (x) dx +

G



Σ

(4.5.2)

G

∂G

By the Cauchy inequality, and by virtue of the assumption (c), we obtain       β(ω) r |u| |h(x)| ds |u||h(x)|ds = r β(ω) Σ Σ   β(ω) 2 1 1 u (x)ds + ≤ rh2 (x)ds; 2 r 2β0 Σ Σ       γ(ω) r |u| |g(x)| ds |u||g(x)|ds = r γ(ω) ∂G ∂G   1 1 γ(ω) 2 u (x)ds + ≤ rg 2 (x)ds; 2 r 2γ0 ∂G ∂G    1 1 2 2 |u||f (x)|dx ≤ |u| dx + |f | dx. 2 2 G

G

G



Hence we get the desired inequality (4.5.1).

Further, we will obtain a global estimate for the weighted Dirichlet integral. Theorem 4.16. Let u(x) be a weak solution of problem (LN) and assumptions (a)– (c) be satisfied. Let λ be as above in (4.1.3), κ0 be as above in Lemma 4.10 about the barrier function. In addition, let λ > 1 and 0 < κ ≤ min(s − 1, κ0 , λ − 1) as well as   ◦ 0 α−1 2 f (x) ∈ Wα (G), r h (x)ds < ∞, rα−1 g 2 (x)ds < ∞, Σ

∂G

where 2 − n − 2κ < α ≤ 2. ◦ 1

Then u(x) ∈ Wα−2 (G) and the inequality

(4.5.3)

84

Chapter 4. Laplace operator with N different media 



a r G

≤C

α−2

- 

2

|∇u| + r

α−4 2

u



 dx +

r Σ

 u2 + (1 + rα )f 2 (x) dx +



α−3



2

β(ω)u (x)ds +

rα−1 h2 (x)ds +

Σ

G



rα−3 γ(ω)u2 (x)ds

∂G

rα−1 g 2 (x)ds

. (4.5.4)

∂G

holds, where the constant C > 0 depends only on n, a∗ , a∗ , α, λ and the domain G. Proof. Setting in (II) η(x) = rεα−2 u(x), with regard to p ≥ 0 and ηxi = rεα−2 uxi + i (α − 2)rεα−3 xi −εl rε u(x), we obtain 

arεα−2 |∇u|2 dx +



r−1 rεα−2 β(ω)u2 (x)ds +

Σ

G

=

2−α 2



 ∂G

arεα−4 (xj − εlj )(u2 )xj dx −

G

 +

α(x)r−1 rεα−2 γ(ω)u2 (x)ds  f (x)rεα−2 u(x)dx

G rεα−2 u(x)h(x)ds + α(x)rεα−2 u(x)g(x)ds.

Σ

(4.5.5)

∂G

We transform the first integral on the right-hand side by integrating by parts: 

 ∂u2 dx = ∂xj i=1 N

arεα−4 (xj − εlj ) G



=− G



=−

ai rεα−4 (xj − εlj )

∂u2i dx ∂xj

Gi

  N   ∂ → rεα−4 (xi − εli ) dx + au2 ai u2i rεα−4 (xj − εlj ) cos(− n , xj )ds ∂xi i=1 ∂Gi

   → 2 ∂ α−4 r au (xi − εli ) dx + au2 rεα−4 (xi − εli ) cos(− n , xi )ds ∂xi ε

G

+



N −1 



[a]Σk

k=1

∂G

→, x )ds, u2 rεα−4 (xi − εli ) cos(− n k i

(4.5.6)

Σk

because of [u]Σk = 0, k = 1, . . . , N − 1. We calculate:   ∂ α−4 i 1) ∂xi rε (xi − εli ) = nrεα−4 + (α − 4)(xi − εli )rεα−5 xi −εl = (n+ α − 4)rεα−4 ; rε  →, x ) n 2) by (4.2.1), (xi − εli ) cos(− k i 1, . . . , N − 1;

Σk

 →, x  = ε cos(− n k 1

Σk

= ε sin

ω

0

2

 − θk , k =

4.5. Global integral estimates

85

3) representing ∂G = Γd0 ∪ Γd and by (4.2.1),  ω0  → =⇒ n , xi ) d = −ε sin (xi − εli ) cos(− 2 Γ0  ∂G

→ au2 rεα−4 (xi − εli ) cos(− n , xi )ds 

ω0 → au2 rεα−4 (xi − εli ) cos(− n , xi )ds − ε sin 2

= Γd



au2 rεα−4 ds.

Γd 0

Hence and from (4.5.6) it follows that   2−α ∂u2 (2 − α)(4 − n − α) arεα−4 (xi − εli ) dx = arεα−4 u2 dx 2 ∂xi 2 G

−ε

+ε

ω0 2−α sin 2 2 N −1 



[a]Σk

k=1



au2 rεα−4 ds +

Γd 0

u2 rεα−4 sin

ω

0

2

Σk

2−α 2

G



→ au2 rεα−4 (xi − εli ) cos(− n , xi )ds

Γd

 − θk ds.

(4.5.7)

From (4.5.5) and (4.5.7) we obtain the following equality:    arεα−2 |∇u|2 dx + r−1 rεα−2 β(ω)u2 (x)ds + α(x)r−1 rεα−2 γ(ω)u2 (x)ds G

+ε

=

ω0 2−α sin 2 2



Σ

au2 rεα−4 ds

Γd 0

(2 − α)(4 − n − α) 2 

∂G



arεα−4 u2 dx +

G

α(x)rεα−2 u(x)g(x)ds +

+ ∂G

+ε

N −1 



[a]Σk

k=1

Σk

u2 rεα−4 sin

2−α 2

ω

0

2



 rεα−2 u(x)h(x)ds Σ

→ au2 rεα−4 (xi − εli ) cos(− n , xi )ds

Γd

  − θk ds − f (x)rεα−2 u(x)dx.

(4.5.8)

G

Now we estimate the integral over Γd . Because on Γd : rε ≥ hr ≥ hd and α−3 (α − 3) ln rε ≤ (α − 3) ln(hd), by α ≤ 2, we have rεα−3 |Γd ≤ (hd) and therefore, by (1.5.12),

86

Chapter 4. Laplace operator with N different media 2−α 2



 2−α → au2 rεα−4 (xi − εli ) cos(− n , xi )ds ≤ arεα−3 u2 ds 2 Γd Γd    2−α α−3 (hd) au2 ds ≤ cδ u2 dx + δ |∇u|2 dx, ∀δ > 0. (4.5.9) ≤ 2 Γd

Gd

Gd

Further, by the Cauchy inequality and because γ(ω) ≥ γ0 > 0,

1  δ  1 , 1 1 1 ug ≤ r 2 , rg 2 (x), ∀δ1 > 0; |g| r− 2 γ(ω)|u| ≤ r−1 γ(ω)u2 + 2 2δ1 γ0 γ(ω) taking into account property 1) of rε we obtain  rεα−2 |u||g|ds ≤ ∂G

δ1 2

 ∂G

1 1 rεα−2 γ(ω)u2 ds + r 2δ1 γ0



rα−1 g 2 (x)ds, ∀δ1 > 0.

∂G

(4.5.10) Similarly, because β(ω) ≥ β0 > 0,  rεα−2 |u||h(x)|ds ≤ Σ

δ1 2

 Σ

1 1 rεα−2 β(ω)u2 ds + r 2δ1 β0



rα−1 h2 (x)ds, ∀δ1 > 0

Σ0

(4.5.11) and  rεα−2 uf (x)dx ≤ G

δ 2



ar−2 rεα−2 u2 dx +

G

Finally, we estimate the integral ε

N −1 k=1

[a]Σk

1 2a∗ δ  Σk



rα f 2 (x)dx, ∀δ > 0.

(4.5.12)

G

u2 rεα−4 sin

 ω0 2

 − θk ds. At first, by

(4.1.2), we get

ε

N −1 



[a]Σk

k=1

Σk

u2 rεα−4 sin

ω

0

2

N −1    − θk ds ≤ a0 ε u2 rεα−4 ds k=1 Σ

k



= a0 ε

rεα−4 u2 (x)ds.

(4.5.13)

Σ

Further, we use the representation Σ = Σε0 ∪ Σε . Then, by property 1) of rε (x) and by virtue of inequality (4.3.26) of Corollary 4.13, we get

4.5. Global integral estimates 



rεα−4 u2 (x)ds ≤

ε Σε0

87

rεα−3 u2 (x)ds ≤ C02 meas σ ·



ε

rα+2κ+n−3 dr 0

Σε0

C02 meas σ εα+2κ+n−2 , (4.5.14) α + 2κ + n − 2

=

because of α + 2κ + n − 2 > 0, by our assumption (4.5.3). Similarly, for the integral over Σε we obtain 

rεα−4 u2 (x)ds

ε

 ≤ ε · meas σ ·

R

rα+2κ+n−4 dr ε

Σε

 = meas σ ·

where R =

max

1≤k≤N −1

ε·

εRα+2κ+n−3 −εα+2κ+n−2 , α+2κ+n−3 ε ln Rε ,

if α + 2κ + n − 3 = 0, if α + 2κ + n − 3 = 0,

(4.5.15)

diam Σk . Thus, from (4.5.13)–(4.5.15) we get 

N −1 

[a]Σk

k=1

u2 rεα−4 sin

ω

0

2

Σk

 − θk ds ≤ J(ε),

(4.5.16)

where J(ε) = ε · a0 · meas σ  α+2κ+n−3

C02 R α+2κ+n−3 + α+2κ+n−2 − × C02 + ln Rε ,

1 α+2κ+n−3

(4.5.17)

 ε

α+2κ+n−3

, if α + 2κ + n − 3 = 0, if α + 2κ + n − 3 = 0,

provided that α + 2κ + n − 2 > 0. Hence follows Corollary 4.17. lim J(ε) = 0 =⇒ lim ε ·

ε→+0

ε→+0

N −1 



[a]Σk

k=1

u2 rεα−4 sin

Σk

ω

0

2

 − θk ds = 0.

As a result from (4.5.8)–(4.5.16) we obtain: 

arεα−2 |∇u|2 dx

G

+ε

ω0 2−α sin 2 2

 +

 Γd 0

r−1 rεα−2 β(ω)u2 (x)ds

Σ

au2 rεα−4 ds

 + ∂G

r−1 rεα−2 γ(ω)u2 (x)ds

88

Chapter 4. Laplace operator with N different media (2 − α)(4 − n − α) 2

≤ J(ε) + δ + 2



ar−2 rεα−2 u2 dx

 G

arεα−4 u2 dx 

+ c(a∗ , α)

  a|∇u|2 + u2 dx

G

Gd 0

⎫ ⎧   ⎬ ⎨ δ1 + r−1 rεα−2 β(ω)u2 (x)ds + r−1 rεα−2 γ(ω)u2 (x)ds (4.5.18) ⎭ 2 ⎩ Σ ∂G    1 1 1 α−1 2 α−1 2 r g (x)ds + r h (x)ds + rα f 2 (x)dx, + 2δ1 γ0 2δ1 β0 2a∗ δ Σ

∂G

G

for all δ, δ1 > 0. It is clear that it is sufficient to consider the case α < 4 − n. At first we apply Lemma 4.7. Then from (4.5.18) we get: 

 2 (2 − α) (4 − α − n) 1− (4 − n − α)2 + 4λ(λ + n − 2)     1 α−2 1 α−2 α−2 2 2 2 r r arε |∇u| dx + β(ω)u (x)ds + γ(ω)u (x)ds × r ε r ε G

δ ≤ J(ε) + 2



Σ

ar−2 rεα−2 u2 dx + c(a∗ , α)



∂G



 a|∇u|2 + u2 dx

G

Gd 0



1 rα−1 g 2 (x)ds 2δ1 γ0 ∂G    δ1 + r−1 rεα−2 β(ω)u2 (x)ds + r−1 rεα−2 γ(ω)u2 (x)ds 2 Σ ∂G   1 1 rα−1 h2 (x)ds + rα f 2 (x)dx, ∀δ, δ1 > 0, ∀ε > 0. + 2δ1 β0 2a∗ δ +

Σ

(4.5.19)

G

However, if α > 4 − n − 2λ, then 2 (2 − α) (4 − α − n) < 1. (4 − n − α)2 + 4λ(λ + n − 2)

(4.5.20)

This inequality is satisfied, by virtue of α > 2 − n − 2κ and 0 < κ ≤ λ − 1. δ Therefore (4.5.20) is true. Let us apply Lemma 4.8 and choose δ1 = λ(λ+n−2) . As a result we obtain

4.5. Global integral estimates

89

 2 (2 − α) (4 − α − n) 1− (4 − n − α)2 + 4λ(λ + n − 2)     1 α−2 1 α−2 α−2 2 2 2 r r arε |∇u| dx + β(ω)u (x)ds + γ(ω)u (x)ds × r ε r ε



G



≤ J(ε) + c(a∗ , α)

Σ

∂G



a|∇u|2 + u

 2

G

1 dx + 2δ1 γ0



rα−1 g 2 (x)ds

∂G

δ + λ(λ + n − 2)     α−2 2 −1 α−2 2 −1 α−2 2 arε |∇u| dx + r rε β(ω)u (x)ds + r rε γ(ω)u (x)ds × G

1 + 2δ1 β0

 Σ

Σ

1 rα−1 h2 (x)ds + 2a∗ δ



∂G

rα f 2 (x)dx, ∀δ > 0, ∀ε > 0.

(4.5.21)

G

We can choose

  2 (2 − α) (4 − α − n) λ(λ + n − 2) 1− . δ= 2 (4 − n − α)2 + 4λ(λ + n − 2)

Thus we get    1 α−2 1 α−2 α−2 2 2 r r arε |∇u| dx + β(ω)u (x)ds + γ(ω)u2 (x)ds r ε r ε Σ

G

∂G

    a|∇u|2 + u2 + rα f 2 (x) dx ≤ C(n, a∗ , α, λ) J(ε) + 1 + γ0

G

 r

α−1 2

g (x)ds +

∂G

1 β0



 rα−1 h2 (x)ds , ∀ε > 0.

(4.5.22)

Σ

Now we can perform the passage to the limit as ε → +0 by the Fatou Theorem. Taking into account Corollary 4.17, it follows that    arα−2 |∇u|2 dx + rα−3 β(ω)u2 (x)ds + rα−3 γ(ω)u2 (x)ds (4.5.23) Σ

G

∂G

≤ C(n, a∗ , α, λ)       1 1 × a|∇u|2 + u2 + rα f 2 (x) dx + rα−1 g 2 (x)ds + rα−1 h2 (x)ds . γ0 β0 G

∂G

Σ

Applying Theorem 4.15, by the Hardy-Friedrichs-Wirtinger inequality (2.2.3), from (4.5.23) we get the desired estimate (4.5.4). 

90

Chapter 4. Laplace operator with N different media

4.6 Local integral weighted estimates Theorem 4.18. Let u(x) be a weak solution of problem (LN), λ > 1 be as above ◦ 1

in (4.1.3) and assumptions (a)–(c) be satisfied. Then u(x) ∈ W2−n (G) and there exist d ∈ (0, 1) and a constant C > 0 depending only on n, s, λ, a∗ , G, Σ such that the inequality      a r2−n |∇u|2 + r−n u2 (x) dx + r1−n β(ω)u2 (x)ds + r1−n γ(ω)u2 (x)ds G 0

Σ 0

Γ

0 ⎧ 2λ ⎪ ⎪ ,

1 2 1 2  ⎨ 2λ 2 1  2 2 ≤ C u2,G + f0 + g0 + h0 ·  ln  , ⎪ γ0 β0 ⎪ ⎩2s ,

if s > λ, if s = λ,

(4.6.1)

if s < λ

holds for almost all  ∈ (0, d). ◦ 1

Proof. By Theorem 4.16, u(x) belongs to W2−n (G), therefore it is enough to prove the estimate (4.6.1). Putting η(x) = r2−n u(x) in (II)loc and taking into account the definition (2.4.10), we obtain  U () = 

au(x) Ω

   ∂u  2−n dΩ + α(x)r u(x)g(x)ds + r2−n u(x)h(x)ds ∂r r= Γ 0

  (n − 2)ar

+

−n

Σ 0

u(x)xi uxi − pr

2−n 2

u (x) − r

2−n

 u(x)f (x) dx.

(4.6.2)

G 0

By the divergence theorem, we find   n−2 ∂u2 (n − 2) ar−n u(x)xi uxi dx = ar−n xi dx 2 ∂xi G 0

(4.6.3)

G 0

     n−2 − au2 (x) nr−n − nr−n dx + −n au2 (x)xi cos(r, xi )dΩ = 2 G 0

+

N −1 

Ω



[a]Σk

k=1

r−n u2 (x)xi cos(nk , xi )ds +

(Σk ) 0





α(x)ar−n u2 (x)xi cos(n, xi )ds .

Γ 0

By Lemmas 1.1 and 4.4 we have   n−2 −n (n − 2) ar u(x)xi uxi dx = au2 (x)dΩ. 2 G 0

Ω

(4.6.4)

4.6. Local integral weighted estimates

91

Because of Lemma 2.12 and p ≥ 0, from (4.6.2)–(4.6.4) it follows that   U () + U () ≤ 2λ

 r

2−n

 |u(x)| · |g(x)|ds +

Γ 0

Σ 0

r2−n |u(x)| · |h(x)|ds 

+

r2−n |u(x)| · |f (x)|dx.

(4.6.5)

G 0

We shall obtain upper bounds for each integral on the right. At first, applying the Cauchy and Hardy-Friedrichs-Wirtinger inequality (2.2.3)), we have for any ∀δ > 0:  r

2−n

Γ 0

 r Σ 0

 r

2−n

G 0

2−n

  

 1−n , 3−n 1 r 2 |u||g|ds = γ(ω)|u| r 2 , |g| ds γ(ω)  Γ0   1 δ r1−n γ(ω)|u|2 ds + r3−n |g|2 ds; ≤ 2 2δγ0 Γ 0

Γ 0

Σ 0

Σ 0

  

 1−n , 3−n 1 r 2 |u||h|ds = β(ω)|u| r 2 , |g| ds β(ω)  Σ0   1 δ 1−n 2 r β(ω)|u| ds + r3−n |h|2 ds; ≤ 2 2δβ0

δ |u(x)||f (x)|dx ≤ 2a∗

 ar G 0

−n

1 |u| dx + 2δ 2

 G 0

1 δ U () + ≤ 2a∗ λ(λ + n − 2) 2δ

(4.6.6)

(4.6.7)

r4−n |f |2 dx 

r4−n |f |2 dx;

(4.6.8)

G 0

Thus, from (4.6.5)–(4.6.8) we get   U () {1 − c1 (n, λ, a∗ )δ} U () ≤ 2λ ⎫ ⎧ ⎪ ⎪    ⎬ ⎨ 1 1 1 r4−n |f |2 dx + r3−n |g|2 ds + r3−n |h|2 ds , ∀δ > 0. (4.6.9) + ⎪ 2δ ⎪ γ0 β0 ⎭ ⎩   G0

Γ0

However, by the condition (c),

Σ0

92

Chapter 4. Laplace operator with N different media  r

4−n

G 0

1 |f | dx + γ0 2

 r

3−n

Γ 0

1 |g| ds + β0 2



r3−n |h|2 ds

Σ 0

c0 (G) ≤ 2s

  1 2 1 2 2 f0 + g0 + h0 · 2s . γ0 β0

Now, from (4.6.9) we obtain the differential inequality (CP ) §1.7 with 2λ · {1 − c1 (n, λ, a∗ )δ}, ∀δ > 0; N () ≡ 0;    c0 λ 1 2 1 2 2 f0 + g0 + h0 · δ −1 2s−1 , ∀δ > 0; Q() = (4.6.10) 2s γ0 β0    -  .  U0 = C u2 + (1 + r4−n )f 2 (x) dx + r3−n h2 (x)ds + r3−n g 2 (x)ds , P() =

Σ

G

∂G

by (4.5.4) with α = 4 − n. 1) Case s > λ. Choosing δ = ε , ∀ε > 0, we get:   c0 λ 1 1 2λ − 2λc1 (n, λ, a∗ )ε−1 ; Q() = f12 + g12 + h21 · 2s−1−ε . P() =  2s γ0 β0 Now for 0 <  < τ < d, τ − 

τ  

 2λ sε  dε τ + 2λc1  ≤ ln P(s)ds = −2λ ln + 2λc1 =⇒  ε τ ε ⎛

exp ⎝−

d 



⎞

 

 2λ

 2λ dε ⎠ = K0 P(τ )dτ ≤ exp 2λc1 ; d ε d ⎞ ⎛ τ 

 2λ , exp ⎝− P(τ )dτ ⎠ ≤ K0 τ 

! ε" where K0 = exp 2λc1 dε . We have also d

 τ  Q(τ ) exp − P(σ)dσ dτ





λc0 K0 ≤ 2s

  d 1 2 1 2 2λ 2 f 0 + g 0 + h0  τ 2s−2λ−ε−1 dτ ≤ γ0 β0

λc0 K0 ≤ 2s

  s−λ d 1 1 f02 + g02 + h20 · 2λ , γ0 β0 s−λ



4.6. Local integral weighted estimates

93

since- s > λ and . we can choose ε = s − λ. Therefore we have in our case K0 = s−λ exp 2λc1 ds−λ . Now we apply Theorem 1.21. Then from (1.7.1), by virtue of the deduced inequalities and with regard to (2.2.3) for α = 4 − n, we obtain the statement of (4.6.1) for s > λ. 2) Case s = λ. Taking in (4.6.10) any function δ() > 0 instead of c1 δ > 0, we obtain the problem (CP ) with   2λ(1 − δ()) c0 1 2 1 2 2 ; N () = 0; Q() = f0 + g0 + h0 · δ −1 ()2λ−1 . P() =  2 γ0 β0 We choose δ() =

1

 , 0 <  < d, where e is the Euler number. Then we

2λ ln

ed 

obtain

⎛ ⎞

 2λ

 2λ τ ln ed  dσ ⎝

 = ln

 ⎠ =⇒ − P(σ)dσ ≤ ln + + ln τ τ σ ln ed ln ed σ τ  

 ed  τ      d 

ln 2λ 2λ   ed  .

, exp − P(σ)dσ ≤ ln · exp − P(τ )dτ ≤ ed d  τ ln τ



τ



In this case we also have d

 τ  Q(τ ) exp − P(σ)dσ dτ





  d dτ 1 2 1 2 2λ ed  2

 · ≤ c2 f1 + g1 + h1  ln ed ν0 ν0  τ δ(τ ) ln τ    

ed 1 1 . ≤ 2λc2 f12 + g12 + h21 · 2λ ln2 ν0 ν0  Now we apply Theorem 1.21, and from (1.7.1), by virtue of the deduced inequalities, we obtain U () ≤ c3 (U0 + f02 +

1 2 1 1 g0 + h20 )2λ ln2 , γ0 β0 

0 0. We perform the   change of variables x = x and u(x ) = ψ()v(x ). Then the function v(x ) satisfies the problem ⎧ 2 a v − p2 v(x ) = ψ() f (x ), x ∈ G21/4 ; ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ [v(x )] k = 1, . . . , N − 1; ⎪ (Σk )21/4 = 0, ⎨

 ∂v ⎪ ⎪ a ∂n + |x1 | βk (ω)v(x ) = ψ() hk (x ), x ∈ (Σk )21/4 , k = 1, . . . , N − 1;  ⎪ ⎪ k (Σk )2 ⎪ 1/4 ⎪ ⎪ ⎪ ⎩  ∂v  α(x ) · a ∂n + |x1 | γ(ω)v(x ) = ψ() g(ρx ), x ∈ Γ21/4 . (LN ) By the Sobolev Imbedding Theorem 1.19, sup |∇ v(x )| ≤ cvW2,p (G11/2 ) ,

x ∈G11/2

(4.7.5)

p > n.

By virtue of the local Lp a priori estimate [66, 67], for the solution to the equation of problem (LN ) inside the domain G21/4 and near the smooth portions of the boundaries Σ21/4 and Γ21/4 , we have vW2,p (G11/2 ) ≤ c

.  f Lp(G21/4 ) + hW1−1/p,p(Σ21/4 ) + gW1−1/p,p(Γ21/4 ) ψ() + cvLp (G21/4 ) . (4.7.6)

Returning to the variables x, from (4.7.5) and (4.7.6), it follows that sup |∇u|

G /2

≤ c and

2−n/p

 

−3

 uLp(G2 ) + f p,G2 + gV1−1/p (Γ2 /4

/4

p,0

/4

)

+ hV1−1/p (Σ2 p,0

/4

)

96

Chapter 4. Laplace operator with N different media

2 (G 2−n/p uVp,0 ) /2  ≤ c2−n/p −2 uLp(G2 ) + f p,G2 + gV1−1/p (Γ2 /4

/4

p,0

/4

 )

+ hV1−1/p (Σ2 p,0

/4

)

or ! sup |∇u| ≤ c−1 |u|0,G2 +f V0

2 p,2p−N (G/4 )

/4

G /2

+gV1−1/p

2 p,2p−n (Γ/4 )

"

+hV1−1/p

2 p,2p−n (Σ/4 )

and 2 uVp,2p−n (G ) /2 ! ≤ c |u|0,G2 + f V0

2 p,2p−N (G/4 )

/4

+ gV1−1/p

2 p,2p−n (Γ/4 )

+ hV1−1/p

2 p,2p−n (Σ/4 )

"

.

Hence, because of (4.1.4), (4.1.5) and the assumption (c), there follow the required results (4.1.6) and (4.1.7). Now we can make more precise the statement of Theorem 4.16. In fact, estimate (4.1.4) proved above allows us to consider in Theorem 4.16 the value κ = λ−1. As a result we obtain the last statement of our theorem and the estimate (4.1.9) with the best possible exponent that satisfies the inequality (4.1.8). 

4.8 Appendix: Eigenvalue transmission problem in a composite plane domain with an angular point We consider the eigenvalue transmission boundary value problem for (LN ) in a composite plane domain with an angular point. Let G ⊂ R2 be a bounded domain with boundary ∂G that is a smooth curve everywhere except at the origin O ∈ ∂G. Near the point O it is a fan that consists of N corners with vertexes at O. Thus G=

N ; i=1

Gi ;

∂G =

N; +1

Γj ;

Σ=

j=0

N; −1

Σk .

k=1

Here Σk , k = 1, . . . , N − 1 are the rays that divide G into angular domains Gi , i = 1, . . . , N . Let ωi be apertures at the vertex O in domains Gi , i = 1, . . . , N . We N 8 Γj be the curvilinear portion define the value θk = ω1 + ω2 + · · · + ωk . Let Γ = j=1

2 of the boundary ∂G. In this case we have  ϑ=λ .   We also assume that Γ0 = {(r, ω) r > 0, ω = 0}; ΓN +1 = {(r, ω)r > 0, ω = θN }; βk Σ = βk (θk ) = βk = const; γ(0) = γ1 = const, γ(ω0 ) = γN = const. k

4.8. Appendix: Eigenvalue transmission problem

97

x n

:k

:2 n2

nk

G2

:N nN1

6N1



:1

62

6k

GN

n1

Z0

Z0

2

2

61

G1

Z2 ZN

*N1

Tk

n

Z1

*0

y

n

O Figure 7

Eigenvalue problem (EV P N ) in this case has the form

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

ψi + λ2 ψi (ω) = 0,

ω ∈ Ωi = {ωi−1 < ω < ωi }, i = 1, . . . , N ;

ψi (θi ) = ψi+1 (θi ),

i = 1, . . . , N − 1;

 ai ψi (θi ) − ai+1 ψi+1 (θi ) + βi ψi (θi ) = 0,

i = 1, . . . , N − 1;

α1 a1 ψ1 (0)

+ γ1 ψ1 (0) = 0,

 αN aN ψN (ω0 )

+ γN ψN (ω0 ) = 0,

where a1 , αN ∈ {0, 1}. By direct calculation, we get ψi (ω) = Ai cos(λω) + Bi sin(λω), i = 1, . . . , N , where constants A1 , . . . AN ; B1 , . . . BN are determined from the algebraic homogeneous system

98

Chapter 4. Laplace operator with N different media ⎧ ⎪ λα1 a1 B1 + γ1 = 0, ⎪  ⎪ ⎪ βi ⎪ 2 ai 2 ⎪ = cos (λθ ) + sin (λθ ) − sin(λθ ) cos(λθ ) · Ai A i+1 i i i i ⎪ ai+1

⎪ λai+1

 ⎪ ⎪ β ⎪ 2 a i i ⎪ + sin(λθi ) cos(λθi ) 1 − ai+1 − λai+1 sin (λθi ) · Bi , ⎪ ⎨ 



 βi ai 2 + = sin(λθ ) cos(λθ ) 1 − cos (λθ ) · Ai B i+1 i i i ai+1 λai+1 ⎪

 ⎪ ⎪ β ⎪ 2 a 2 i i ⎪ + sin (λθ ) + cos (λθ ) + sin(λθ ) cos(λθ ) · Bi , i i i i ⎪ ai+1 λai+1 ⎪ ⎪ ⎪ ⎪ (γN cos(λω0 ) − λαN aN sin(λω0 )) · AN ⎪ ⎪ ⎪ ⎩ + (γN sin(λω0 ) + λαN aN cos(λω0 )) · BN = 0,

i = 1, . . . , N − 1. The least positive eigenvalue λ is defined from the vanishing of the determinant of this system.

4.8.1 Four-media transmission problem Our goal is the derivation of the eigenvalues equation that corresponds to our transmission problem for the case N = 4. Let S 1 be the unit circle in R2 centered at O. We write: Ωi = Gi ∩S 1 ; i = 1, 2, 3, 4. The eigenvalue problem is the following: ⎧  ψi + λ2 ψi (ω) = 0, ω ∈ Ωi ; i = 1, 2, 3, 4; ⎪ ⎪ ⎪ ⎪ ⎪ ψ2 (ω1 ) = ψ1 (ω1 ); ψ3 (θ2 ) = ψ2 (θ2 ); ψ4 (θ3 ) = ψ3 (θ3 ); ⎪ ⎪ ⎪ ⎪   ⎪ a ⎪ ⎨ 1 ψ1 (ω1 ) − a2 ψ2 (ω1 ) + β1 ψ1 (ω1 ) = 0; a2 ψ2 (θ2 ) − a3 ψ3 (θ2 ) + β2 ψ2 (θ2 ) = 0; ⎪ ⎪ ⎪ a3 ψ3 (θ3 ) − a4 ψ4 (θ3 ) + β3 ψ3 (θ3 ) = 0; ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ α1 a1 ψ1 (0) + γ1 ψ1 (0) = 0; ⎪ ⎪ ⎩ α4 a4 ψ4 (θ4 ) + γ4 ψ4 (θ4 ) = 0,

(4.8.1)

    where α1 = αΓ = αω=0 , α4 = αΓ = αω=θ , γ1 = γ(0), γ4 = γ(θ4 ); α1,4 ∈ 0 5 4 {0, 1}. We find the general solution of equation (4.8.1): ψi (ω) = Ai cos(λω) + Bi sin(λω) =⇒ ψi (ω) = −λAi sin(λω) + λBi cos(λω); for i = 1, 2, 3, 4, where Ai , Bi (i = 1, 2, 3, 4) are arbitrary constants. From the boundary condition of (4.8.1) we obtain the homogenous algebraic system of eight linear equations for the finding of Ai , Bi (i = 1, 2, 3, 4):

4.8. Appendix: Eigenvalue transmission problem

99

⎧ A2 cos λω1 + B2 sin ω1 − A1 cos λω1 − B1 sin λω1 = 0, ⎪ ⎪ ⎪ ⎪ ⎪ A3 cos λθ2 + B3 sin θ2 − A2 cos λθ2 − B2 sin λθ2 = 0, ⎪ ⎪ ⎪ ⎪ ⎪ A4 cos λθ3 + B4 sin θ3 − A3 cos λθ3 − B3 sin λθ3 = 0, ⎪ ⎪ ⎪ ⎪ ⎪ λa2 A2 sin λω1 − λa2 B2 cos λω1 − λa1 A1 sin λω1 ⎪ ⎪ ⎪ ⎪ ⎪ +λa 1 B1 cos λω1 + β1 A1 cos λω1 + β1 B1 sin λω1 = 0, ⎨ λa3 A3 sin λθ2 − λa3 B3 cos λθ2 − λa2 A2 sin λθ2 ⎪ ⎪ ⎪+λa2 B2 cos λθ2 + β2 A2 cos λθ2 + β2 B2 sin λθ2 = 0, ⎪ ⎪ ⎪ ⎪ ⎪ λa4 A4 sin λθ3 − λa4 B4 cos λθ3 − λa3 A3 sin λθ3 ⎪ ⎪ ⎪ ⎪ ⎪ +λa 3 B3 cos λθ3 + β3 A3 cos λθ3 + β3 B3 sin λθ3 = 0, ⎪ ⎪ ⎪ ⎪ ⎪ α1 a1 λB1 + γ1 A1 = 0, ⎪ ⎪ ⎩ α4 a4 λA4 sin λθ4 − α4 a4 λB4 cos λθ4 − γ4 A4 cos λθ4 − γ4 B4 sin λθ4 = 0. The determinant of this system has to be equal to zero for the nontrivial solution of this system to exist. The latter gives us the required equation for eigenvalues λ: [λ4 (α1 α4 β2 a21 a24 − α4 γ1 a22 a24 + α1 γ4 a21 a23 ) + λ2 (α4 β1 β2 a24 − α1 β2 β3 γ4 a21 + β3 γ1 γ4 a22 + γ1 γ4 β1 a23 ) − β1 β2 β3 γ1 γ4 ] · sin λω1 sin λω2 sin λω3 sin λω4 + λa2 [α4 λ2 (β2 γ1 a24 + β1 γ1 a24 + λ2 α1 a21 a24 ) − γ1 γ4 β2 β3 − γ1 γ4 β1 β3 − λ2 α1 β3 γ4 a21 + λ2 γ1 γ4 a23 ] · sin λω1 cos λω2 sin λω3 sin λω4 − λa3 [γ4 (λ2 α1 β2 a21 + γ1 β1 β2 − λ2 γ1 a22 ) − λ2 α4 β1 γ1 a24 − λ4 α1 α4 a21 a24 + γ1 γ4 β1 β3 + λ2 α1 β3 γ4 a21 ] · sin λω1 sin λω2 cos λω3 sin λω4 − λa4 [λ2 α1 β2 γ4 a21 + β1 β2 γ1 γ4 − λ2 γ1 γ4 a22 + λ2 α1 α4 β2 β3 a21 + α4 β1 β2 β3 γ1 − λ2 α4 β3 γ1 a22 − λ2 α4 a23 (γ1 β1 + λ2 α1 a21 )] · sin λω1 sin λω2 sin λω3 cos λω4 − λa1 [λ2 α1 α4 β1 β2 a24 − λ2 α4 β2 γ1 a24 − λ4 α1 α4 a22 a24 − α1 β1 β2 β3 γ4 + γ1 γ4 β2 β3 + λ2 α1 β3 γ4 a22 − λ2 γ4 a23 (γ1 − α1 β1 )] · cos λω1 sin λω2 sin λω3 sin λω4 + λ2 a3 a4 [α4 (λ2 α1 β2 a21 + γ1 β1 β2 − λ2 γ1 a22 ) + α4 β1 β3 γ1 + λ2 α1 α4 β3 a21 + β1 γ1 γ4 + λ2 α1 γ4 a21 ] · sin λω1 sin λω2 cos λω3 cos λω4 + λ2 a2 a3 [γ1 (λ2 α4 a24 − γ4 β3 ) − γ4 (γ1 β2 + β1 γ1 + λ2 α1 a21 )] × sin λω1 cos λω2 cos λω3 sin λω4 + λ2 a2 a4 [λ2 α4 γ1 a23 − γ1 β2 γ4 − β1 γ1 γ4 − λ2 α1 γ4 a21 − α4 β2 β3 γ1 − α4 β1 β3 γ1 − λ2 α1 α4 β3 a21 ] · sin λω1 cos λω2 sin λω3 cos λω4 + λ2 a1 a2 [λ2 α4 γ1 a24 − λ2 α1 α4 β2 a24 − λ2 α1 α4 β1 a24 − β3 γ1 γ4 + α1 β2 β3 γ4 + α1 β1 β3 γ4 − λ2 α1 γ4 a23 ] · cos λω1 cos λω2 sin λω3 sin λω4 + λ2 a1 a4 [α1 β1 β2 γ4 − β2 γ1 γ4 − λ2 α1 γ4 a22 + α1 α4 β1 β2 β3 − α4 β2 β3 γ1 − λ2 α1 α4 β3 a22 + λ2 α4 a23 (γ1 − α1 β1 )] · cos λω1 sin λω2 sin λω3 cos λω4 + λ2 a1 a3 [γ4 (α1 β1 β2 − γ1 β2 − λ2 α1 a22 ) + λ2 α4 γ1 a24 − λ2 α1 α4 β1 a24

100

Chapter 4. Laplace operator with N different media

− γ1 γ4 β3 + α1 β1 β3 γ4 ] · cos λω1 sin λω2 cos λω3 sin λω4 − λ3 a2 a3 a4 [α4 (β2 γ1 + β1 γ1 + λ2 α1 a21 ) + γ1 (α4 β3 + γ4 )] × sin λω1 cos λω2 cos λω3 cos λω4 − λ3 a1 a3 a4 [α4 β3 γ1 − α1 α4 β1 β3 + γ1 γ4 − α1 β1 γ4 − α4 (α1 β1 β2 − γ1 β2 − λ2 α1 a22 )] · cos λω1 sin λω2 cos λω3 cos λω4 − λ3 a1 a2 a4 [γ1 γ4 − α1 β2 γ4 − α1 β1 γ4 + α4 β3 γ1 − α1 α4 β2 β3 − α1 α4 β1 β3 + λ2 α1 α4 a23 ] · cos λω1 cos λω2 sin λω3 cos λω4 − λ3 a1 a2 a3 [γ4 (γ1 − α1 β2 − α1 β1 ) + α1 (λ2 α4 a24 − γ4 β1 )] × cos λω1 cos λω2 cos λω3 sin λω4 + λ4 a1 a2 a3 a4 [α1 (α4 β3 + γ4 ) − α4 (γ1 − α1 β2 − α1 β1 )] × cos λω1 cos λω2 cos λω3 cos λω4 = 0. We consider the following particular cases of boundary conditions. 1) The Dirichlet problem: α1 = α4 = β1 = β2 = β3 = 0; γ1 = γ4 = 1. λ3 a2 a23 sin λω1 cos λω2 sin λω3 sin λω4 + a22 a3 sin λω1 sin λω2 cos λω3 sin λω4 + a22 a4 sin λω1 sin λω2 sin λω3 cos λω4 + a1 a23 cos λω1 sin λω2 sin λω3 sin λω4 − a2 a3 a4 sin λω1 cos λω2 cos λω3 cos λω4 − a1 a3 a4 cos λω1 sin λω2 cos λω3 cos λω4 − a1 a2 a4 cos λω1 cos λω2 sin λω3 cos λω4 − a1 a2 a3 cos λω1 cos λω2 cos λω3 sin λω4 = 0. In the isotropic case (a1 = a2 = a3 = a4 ) we hence obtain the following well-known result: sin(λθ4 ) = 0 ⇒ λn = πn θ4 , n = 1, 2 . . .. Corollary. λ = ωπ0 > 1, if ω0 < π. 2) The Neumann problem: α1 = α4 = 1; β1 = β2 = β3 = 0; γ1 = γ4 = 0. − a21 a2 a24 sin λω1 cos λω2 sin λω3 sin λω4 − a21 a3 a24 sin λω1 sin λω2 cos λω3 sin λω4 − a21 a23 a4 sin λω1 sin λω2 sin λω3 cos λω4 − a1 a22 a24 cos λω1 sin λω2 sin λω3 sin λω4 + a21 a2 a3 a4 sin λω1 cos λω2 cos λω3 cos λω4 + a1 a22 a3 a4 cos λω1 sin λω2 cos λω3 cos λω4 + a1 a2 a23 a4 cos λω1 cos λω2 sin λω3 cos λω4 + a1 a2 a3 a24 cos λω1 cos λω2 cos λω3 sin λω4 = 0. In the isotropic case (a1 = a2 = a3 = a4 ) we hence obtain the following well-known result: sin(λθ3 ) = 0 ⇒ λn = πn θ4 , n = 0, 1, 2 . . .. Corollary. λ = ωπ0 > 1, if ω0 < π.

4.8. Appendix: Eigenvalue transmission problem

101

3) The mixed problem: α1 = γ4 = 1, α4 = β1 = β2 = β3 = 0; γ1 = 0. a21 a23 sin λω1 sin λω2 sin λω3 sin λω4 − a21 a3 a4 sin λω1 sin λω2 cos λω3 cos λω4 − a21 a2 a3 sin λω1 cos λω2 cos λω3 sin λω4 − a21 a2 a4 sin λω1 cos λω2 sin λω3 cos λω4 − a1 a2 a23 cos λω1 cos λω2 sin λω3 sin λω4 − a1 a22 a4 cos λω1 sin λω2 sin λω3 cos λω4 − a1 a22 a4 cos λω1 sin λω2 cos λω3 sin λω4 + a1 a2 a3 a4 cos λω1 cos λω2 cos λω3 cos λω4 = 0. In the isotropic case (a1 = a2 = a3 = a4 ) we hence obtain the following well-known result: cos(λθ4 ) = 0 ⇒ λn = π(2n−1) , n = 1, 2 . . .. 2θ4 π π Corollary. λ = 2ω > 1, if ω < . 0 2 0 4) The Robin problem: α1 = α4 = 1. In the isotropic case (a1 = a2 = a3 = a4 = 1; β1 = β2 = β3 = 0) we obtain: 4 −γ1 ) tan(λω0 ) = λ(γ λ2 +γ1 γ4 .

4.8.2 Three-media transmission problem In this subsection we consider the eigenvalue problem corresponding to our transmission problem for the case N = 3. Let S 1 be the unit circle in R2 centered at O. We write: Ωi = Gi ∩ S 1 ; i = 1, 2, 3. The eigenvalue problem is the following: ⎧ ψi + λ2 ψi (ω) = 0, ω ∈ Ωi ; (i = 1, 2, 3); ⎪ ⎪ ⎪ ⎪ ⎪ψ1 (ω1 ) = ψ2 (ω1 ); ψ3 (θ2 ) = ψ2 (θ2 ); ⎪ ⎪ ⎪ ⎨a ψ  (ω ) − a ψ  (ω ) + β ψ (ω ) = 0; 2 2 1 1 1 1 1 1 1 (4.8.2)   ⎪ a ψ (θ ) − a ψ (θ ) + β ψ (θ ) = 0; 3 2 2 2 2 2 2 ⎪ 3 2 ⎪ ⎪  ⎪ ⎪ ⎪α1 a1 ψ1 (0) + γ1 ψ1 (0) = 0; ⎪ ⎩ α3 a3 ψ3 (θ3 ) + γ3 ψ3 (θ3 ) = 0. We find the general solution of equation (4.8.2): ψi (ω) = Ai cos(λω) + Bi sin(λω) =⇒ ψi (ω) = −λAi sin(λω) + λBi cos(λω); (i = 1, 2, 3), where Ai , Bi (i = 1, 2, 3) are arbitrary constants. From the boundary condition of (4.8.2) we obtain a homogenous algebraic system of six linear equations for the determination of Ai , Bi (i = 1, 2, 3). The determinant of the system must be equal to zero for the nontrivial solution of this system to exist. The latter gives us the required equation for eigenvalues λ: [λ2 α3 a23 (β1 γ1 + λ2 α1 a21 ) − γ3 (β1 β2 γ1 + λ2 α1 β2 a21 − λ2 γ1 a22 )] · sin(λω1 ) sin(λω2 ) sin(λω3 ) +λa1 · [λ2 α3 a23 (γ1 − β1 α1 ) + γ3 (β1 β2 α1 − γ1 β2 − λ2 α1 a22 )] · cos(λω1 ) sin(λω2 ) sin(λω3 ) −λa3 · [γ3 (β1 γ1 + λ2 α1 a21 ) + α3 (β1 β2 γ1 + λ2 α1 β2 a21 − λ2 γ1 a22 )] · sin(λω1 ) sin(λω2 ) cos(λω3 )

102

Chapter 4. Laplace operator with N different media +λ2 a1 a3 · [γ3 (β1 α1 − γ1 ) + α3 (β1 β2 α1 − γ1 β2 − λ2 α1 a22 )] · cos(λω1 ) sin(λω2 ) cos(λω3 ) −λa2 · [γ3 (β2 γ1 + λ2 α1 a21 + β1 γ1 ) − λ2 α3 γ1 a23 ] · sin(λω1 ) cos(λω2 ) sin(λω3 ) +λ2 a1 a2 · [γ3 (β2 α1 + α1 β1 − γ1 ) − λ2 α3 α1 a23 ] · cos(λω1 ) cos(λω2 ) sin(λω3 ) −λ2 a2 a3 · [γ1 γ3 + α3 (β2 γ1 + λ2 α1 a21 + β1 γ1 )] · sin(λω1 ) cos(λω2 ) cos(λω3 ) +λ3 a1 a2 a3 · [α1 γ3 + α3 (β2 α1 + α1 β1 − γ1 )] · cos(λω1 ) cos(λω2 ) cos(λω3 ) = 0.

(4.8.3)

We consider the following particular cases of boundary conditions. 1) The Dirichlet problem: α1 = α3 = β1 = β2 = 0; γ1 = γ3 = 1. a1 a3 · cos(λω1 ) sin(λω2 ) cos(λω3 ) + a1 a2 · cos(λω1 ) cos(λω2 ) sin(λω3 ) + a2 a3 · sin(λω1 ) cos(λω2 ) cos(λω3 ) − a22 · sin(λω1 ) sin(λω2 ) sin(λω3 ) = 0. In the isotropic case (a1 = a2 = a3 ) we hence obtain the following well-known result: sin(λθ3 ) = 0 =⇒ λn = πn θ3 , n = 1, 2, . . . . Corollary. λ = θπ3 > 1, if ω1 + ω2 + ω3 < π. 2) The Neumann problem: α0 = α3 = 1; β1 = β2 = γ0 = γ3 = 0. a22 · cos(λω1 ) sin(λω2 ) cos(λω3 ) + a2 a3 · cos(λω1 ) cos(λω2 ) sin(λω3 ) + a1 a2 · sin(λω1 ) cos(λω2 ) cos(λω3 ) − a1 a3 · sin(λω1 ) sin(λω2 ) sin(λω3 ) = 0. In the isotropic case (a1 = a2 = a3 ) we hence obtain the following well-known result: sin(λθ3 ) = 0 =⇒ λn = πn θ3 , n = 0, 1, 2, . . . . Corollary. λ = θπ3 > 1, if ω1 + ω2 + ω3 < π. 3) The mixed problem: α0 = γ3 = 1; α3 = β1 = β2 = γ0 = 0. a22 · cos(λω1 ) sin(λω2 ) sin(λω3 ) + a1 a3 · sin(λω1 ) sin(λω2 ) cos(λω3 ) + a1 a2 · sin(λω1 ) cos(λω2 ) sin(λω3 ) − a2 a3 · cos(λω1 ) cos(λω2 ) cos(λω3 ) = 0. In the isotropic case (a1 = a2 = a3 ) we hence obtain the following well-known result: cos(λθ3 ) = 0 =⇒ λn = π(2n−1) , n = 1, 2, . . . . 2θ3 Corollary. λ = 2θπ3 > 1, if ω1 + ω2 + ω3 < π2 . 4) The Robin problem: α1 = 1, α3 = 1; β1 = β2 = 0. (λ2 a21 a23 + γ1 γ3 a22 ) · sin(λω1 ) sin(λω2 ) sin(λω3 ) − λ · (γ3 a1 a22 − γ1 a1 a23 ) · cos(λω1 ) sin(λω2 ) sin(λω3 ) − λa3 · (γ3 a21 − γ1 a22 ) · sin(λω1 ) sin(λω2 ) cos(λω3 ) − a1 a3 (γ1 γ3 + λ2 a22 ) · cos(λω1 ) sin(λω2 ) cos(λω3 ) − λa2 · (γ3 a21 − γ1 a23 ) · sin(λω1 ) cos(λω2 ) sin(λω3 ) − a1 a2 · (γ1 γ3 + λ2 a23 ) · cos(λω1 ) cos(λω2 ) sin(λω3 )

4.8. Appendix: Eigenvalue transmission problem

103

− a2 a3 · (γ1 γ3 + λ2 a21 ) · sin(λω1 ) cos(λω2 ) cos(λω3 ) + λa1 a2 a3 · (γ3 − γ1 ) · cos(λω1 ) cos(λω2 ) cos(λω3 ) = 0. In the isotropic case (a1 = a2 = a3 = 1) we hence obtain the following 3 −γ1 ) well-known result (see Example 1 in §10.1.7 of [14]): tan(λθ3 ) = λ(γ λ2 +γ1 γ3 .

4.8.3 Two-media transmission problem The two-media transmission problem was considered in detail in Chapter 3.

Chapter 5

Transmission problem for weak quasi-linear elliptic equations in a conical domain 5.1 Introduction In this chapter we investigate the behavior of weak solutions to the transmission problem for weak nonlinear equations ⎧  d  q ij ⎪ ⎪ − |u| a (x)uxj + b(x, u, ∇u) = 0, q ≥ 0, x ∈ G \ Σ0 ; ⎪ ⎪ dx i ⎪ ⎪ ⎪ ⎪ ⎨

  ∂u 1 x q = 0, S[u] ≡ + σ [u] Σ0 ∂ν Σ0 |x| |x| u · |u| = h(x, u), x ∈ Σ0 ; ⎪ ⎪ ⎪ ⎪ ⎪ ⎪

 ⎪ ⎪ ⎩B[u] ≡ ∂u + 1 γ x u · |u|q = g(x, u), x ∈ ∂G \ {Σ0 ∪ O}; ∂ν |x| |x| (WQL) (summation over repeated indices from 1 to n is understood); here: •

∂ = |u|q aij (x)ni ∂x , j  ∂u • ∂ν Σ0 denotes the saltus of the co-normal derivative of the function u(x) on crossing Σ0 , i.e.,   ∂u ∂u+  ∂u−  q ij = |u|q aij (x) n −|u| a (x) ni  .  i + − ∂ν Σ0 ∂xj ∂xj Σ0 Σ0 ∂ ∂ν

Definition 5.1. The function u(x) is called a weak solution of the problem (WQL) ◦ 1

provided that u(x) ∈ C0 (G) ∩ W0 (G) and satisfies the integral identity

M. Borsuk, Transmission Problems for Elliptic Second-Order Equations in Non-Smooth Domains, 105 Frontiers in Mathematics, DOI 10.1007/978-3-0346-0477-2_6, © Springer Basel AG 2010

106

Chapter 5. Transmission problem for weak quasi-linear elliptic equations



!

" |u|q aij (x)uxj ηxi + b(x, u, ux )η(x) dx +

G

 +

γ(ω) u|u|q η(x)ds = r

∂G

 Σ0



σ(ω) u|u|q η(x)ds r 

g(x, u)η(x)ds +

h(x, u)η(x)ds

(II)

Σ0

∂G ◦ 1

for all functions η(x) ∈ C0 (G) ∩ W0 (G). Lemma 5.2. Let u(x) be a weak solution of (W QL). For any function η(x) ∈ ◦ 1

C0 (G) ∩ W0 (G) the equality  -

 . |u|q aij (x)uxj ηxi + b(x, u, ux )η(x) dx = |u|q aij (x)uxj cos(r, xi )η(x)dΩ Ω

G 0

      γ(ω) σ(ω) u|u|q η(x)ds + u|u|q η(x)ds g(x, u) − h(x, u) − + r r Γ 0

(II)loc

Σ 0

holds for a.e.  ∈ (0, d). Proof. The proof is analogous to the proof of Lemma 3.2 Chapter 3.



Assumptions. Let q ≥ 0, 0 ≤ μ < q + 1, s > 1, f1 ≥ 0, g1 ≥ 0, h1 ≥ 0 be given; (a) the condition of the uniform ellipticity: 2 a± ξ 2 ≤ aij ± (x)ξi ξj ≤ A± ξ ,

∀ξ ∈ Rn ; a± , A± = const > 0,  a+ , x ∈ G+ , j j ij we write a (0) = aδi , where δi is the Kronecker symbol; a = a− , x ∈ G− ; a∗ = min{a+ , a− } > 0,

∀x ∈ G± ,

a∗ = max{a+ , a− } > 0,

A∗ = max(A− , A+ );

(b) aij (x) ∈ C0 (G) and the inequality  n

|aij ± (x)

−

2 aij ± (y)|

 12

≤ A(|x − y|)

i,j=1

holds for x, y ∈ G, where A(r) is a monotonically increasing, non-negative function, continuous at 0, A(0) = 0; (c) |b(x, u, ux )| ≤ aμ|u|q−1 |∇u|2 + b0 (x); b0 (x) ∈ Lp/2 (G), n < p < 2n; (d) σ(ω) ≥ ν0 > 0 on σ0 ; γ(ω) ≥ ν0 > 0 on ∂G;

5.1. Introduction (e)

∂h(x,u) ∂u

≤ 0,

107 ∂g(x,u) ∂u s−2

(f) |b0 (x)| ≤ f1 |x|

≤ 0;

, |g(x, 0)| ≤ g1 |x|s−1 , |h(x, 0)| ≤ h1 |x|s−1 .

We make the function change u = v|v|ς−1 with

ς=

1 . q+1

(5.1.1)

Then identities (II) and (II)loc can be presented in the form     γ(ω) σ(ω) ij vη(x)ds + vη(x)ds ςa (x)vxj ηxi + B(x, v, vx )η dx + r r G

 G(x, v)η(x)ds +

=

G 0



ςaij (x)vxj ηxi + B(x, v, vx )η dx +





Ω

Γ 0

γ(ω) vη(x)ds + r

 Σ 0



G(x, v)η(x)ds +

ςaij (x)vxj cos(r, xi )η(x)dΩ +

=

< (II)

H(x, v)η(x)ds; Σ0

∂G

 

Σ0

∂G



Γ 0

σ(ω) vη(x)ds r

< loc H(x, v)η(x)ds (II)

Σ 0

◦ 1

◦ 1

for a.e.  ∈ (0, d), v(x) ∈ C0 (G) ∩ W0 (G) and any η(x) ∈ C0 (G) ∩ W0 (G), where B(x, v, vx ) ≡ b(x, v|v|ς−1 , ς|v|ς−1 vx ),

G(x, v) ≡ g(x, v|v|ς−1 ),

H(x, v) ≡ h(x, v|v|ς−1 ).

(5.1.2)

We assume without loss of generality that there exists d > 0 such that Gd0 is a rotational cone with the vertex at O and the aperture ω0 , thus   n   ω0  2 d 2 ω0 2 , ω0 ∈ (0, 2π) . (5.1.3) x ; r ∈ (0, d), ω1 = Γ0 = (r, ω)x1 = cot 2 i=2 i 2 Theorem 5.3. Let u be a weak solution of the problem (W QL) and assumptions (a)–(f) be satisfied where A(r) is Dini-continuous at zero. Let us assume that M0 = max |u(x)| is known and λ is as above in (2.2.1). Then there exist d ∈ (0, 1) x∈G

and a constant C0 > 0 depending only on n, a∗ , A∗ , p, q, λ, μ, f1 , h1 , g1 , ν0 , s, M0 , 1 meas G, diam G and on the quantity A(r) r dr such that the inequality 0

⎧ λ(1+q−μ) ⎪|x| (q+1)2 , ⎨ λ(1+q−μ)

 ⎪

 1 1 |u(x)| ≤ C0 u2(q+1),G + f1 + g1 + h1 · |x| (q+1)2 ln q+1 |x| , ⎪ ⎪ s ⎩ q+1 |x| ,

if s > λ 1+q−μ 1+q , if s = λ 1+q−μ 1+q ,

if s < λ 1+q−μ 1+q (5.1.4)

108

Chapter 5. Transmission problem for weak quasi-linear elliptic equations

holds for all x ∈ Gd0 . In addition, if coefficients of the problem (W QL) satisfy such conditions, which guarantee the local a-priori estimate |∇u|0,G ≤ M1 for any smooth G ⊂⊂ G \ {O} (see for example §4 in [6] or [64], [47]), then the inequality ⎧ λ(1+q−μ) −1 (q+1)2 ⎪ ⎪ ⎨|x| λ(1+q−μ) ,

 1 1 |∇u(x)| ≤ C1 · |x| (q+1)2 −1 ln q+1 |x| , ⎪ ⎪ s ⎩ q+1 |x| −1 ,

if s > λ 1+q−μ 1+q , if s = λ 1+q−μ 1+q ,

(5.1.5)

if s < λ 1+q−μ 1+q

  holds for all x ∈ Gd0 with C1 = c1 u2(q+1),G + f1 + g1 + h1 , where c1 depends on M0 , M1 and C0 from above.

5.2 Local estimate at the boundary We formulate here a result asserting the local boundedness (near the conical point) of the weak solution of problem (W QL). Theorem 5.4. Let u(x) be a weak solution of the problem (WQL). Let assumptions (a), (c)–(e) be satisfied. Suppose, in addition, that h(x, 0) ∈ L∞ (Σ0 ), g(x, 0) ∈ L∞ (∂G). Then the inequality 1 2 q+1 sup |u(x)| ≤ C −n/t(q+1) ut(q+1),G0 +  q+1 (1−n/p) b0 p/2,G 

Gκ 0

1

+  q+1

 . 1 1 q+1 q+1 g(x, 0)∞,Γ  + h(x, 0)  ∞,Σ 0

0

(5.2.1)

0

holds for any t > 0, κ ∈ (0, 1) and  ∈ (0, d), where C = const(n, a∗ , A∗ , t, p, q, κ, μ, G) and d ∈ (0, 1). Proof. See the proof of Theorem 3.4 or Theorem 6.6 for m = 2.



5.3 Global integral estimate In this section we estimate the weighted Dirichlet integral. Theorem 5.5. 1 Let u be a weak solution of the problem (W QL) and assumptions (a)–(e) are satisfied where A(r) is Dini-continuous at zero. In addition, let us satisfy   ◦ 0 b0 (x) ∈ Wα (G), rα−1 h2 (x, 0)ds < ∞, rα−1 g 2 (x, 0)ds < ∞, 4−n ≤ α ≤ 2. Σ0 1 See

also Subsection 5.5.2.

∂G

5.3. Global integral estimate

109

◦ 1

Then |u(x)|q+1 ∈ Wα−2 (G) and there exists a constant C > 0 depending only on a∗ , α, μ, q, n, λ from (2.2.1) and the domain G such that the inequality  

 α−2 2q 2 α−4 2(q+1) dx + rα−3 σ(ω)|u|2(q+1) ds a r |u| |∇u| + r |u| G

Σ0



rα−3 γ(ω)|u|2(q+1) ds

+ ∂G

≤C

-

2(q+1)

|u|

(5.3.1)

+ (1 + r

α





)b20 (x)

dx +

r



α−1 2

h (x, 0)ds +

Σ0

G

rα−1 g 2 (x, 0)ds

.

∂G

holds. Proof. At first we make the function change (5.1.1) and consider the integral < for the function v(x). Putting in this identity η(x) = rεα−2 v(x) we identity (II) obtain    ς arεα−2 |∇v|2 dx + r−1 rεα−2 σ(ω)v 2 (x)ds + r−1 rεα−2 γ(ω)v 2 (x)ds G

=ς

2−α 2



Σ0

arεα−4 (xi − εli )(v 2 )xi dx

G



+ ς(2 − α)



 aij (x) − aij (0) rεα−4 (xi − εli )vxj v(x)dx

G





−ς 

∂G

 a (x) − a (0) rεα−2 vxi vxj dx − ij

G

B(x, v, vx )rεα−2 v(x)dx G



rεα−2 v(x)H(x, v)ds +

+



ij

Σ0

rεα−2 v(x)G(x, v)ds.

(5.3.2)

∂G

Integrating by parts, we have  arεα−4 (xi G

∂v 2 − εli ) dx = ∂xi

a− rεα−4 (xi − εli ) G−



=− G

a+ rεα−4 (xi − εli )

2 ∂v+ dx ∂xi

G+

 +



av 2

2 ∂v− dx ∂xi

(5.3.3)

   ∂ → 2 α−4 rεα−4 (xi − εli ) dx + a+ v+ rε (xi − εli ) cos(− n , xi )ds ∂xi ∂G+

110

Chapter 5. Transmission problem for weak quasi-linear elliptic equations 

→ 2 α−4 a− v− rε (xi − εli ) cos(− n , xi )ds = −

+ ∂G−



av 2

  ∂ rεα−4 (xi − εli ) dx ∂xi

G

→ av 2 rεα−4 (xi − εli ) cos(− n , xi )ds + [a]Σ0

+





→ v 2 rεα−4 (xi − εli ) cos(− n , xi )ds,

Σ0

∂G

because of [v]Σ0 = 0. Now we calculate:  α−4  ∂ i rε (xi − εli ) = nrεα−4 + (α − 4)(xi − εli )rεα−5 xi −εl = (n + α − 4)rεα−4 ; 1) ∂x rε i   → 2) because of cos(− n , xi ) = cos(xn , xi ) = δin , Σ0

  → (xi − εli ) cos(− n , xi )

Σ0

  = δin (xi − εli )

Σ0

  = (xn − εln )

Σ0

  = xn 

Σ0

= 0,

since Σ0 = {xn = 0} ∩ G and ln = 0; 3) from the representation ∂G = Γd0 ∪ Γd and by (4.2.1),  ω0  → =⇒ (xi − εli ) cos(− n , xi ) d = −ε sin 2 Γ0  → av 2 rεα−4 (xi − εli ) cos(− n , xi )ds ω0 = −ε sin 2



∂G



av 2 rεα−4 ds +

→ av 2 rεα−4 (xi − εli ) cos(− n , xi )ds.

Γd

Γd 0

Then from (5.3.3) it follows that 2−α 2 −ε

 arεα−4 (xi − εli ) G

ω0 2−α sin 2 2



∂v 2 (2 − α)(4 − n − α) dx = ∂xi 2

av 2 rεα−4 ds +

Γd 0

2−α 2





arεα−4 v 2 dx

(5.3.4)

G

→ av 2 rεα−4 (xi − εli ) cos(− n , xi )ds.

Γd

Then we can rewrite (5.3.2) in the following form:  ς

arεα−2 |∇v|2 dx + ες

G

 + ∂G

ω0 2−α sin 2 2

r−1 rεα−2 γ(ω)v 2 (x)ds =

 Γd 0

2−α ς 2



Γd

av 2 rεα−4 ds +



r−1 rεα−2 σ(ω)v 2 (x)ds

Σ0

→ av 2 rεα−4 (xi − εli ) cos(− n , xi )ds

5.3. Global integral estimate

+ς

(2 − α)(4 − n − α) 2 

−ς

arεα−4 v 2 dx

G



 aij (x) − aij (0) rεα−4 (xi − εli )vxj v(x)dx

+ ς(2 − α) 



111

G



 a (x) − a (0) rεα−2 vxi vxj dx − ij



ij

B(x, v, vx )rεα−2 v(x)dx

G  G + rεα−2 v(x)H(x, v)ds + rεα−2 v(x)G(x, v)ds. Σ0

(5.3.5)

∂G

Now we estimate the integral over Γd . Because on Γd : rε ≥ hr ≥ hd ⇒ (α − 3) ln rε ≤ (α − 3) ln(hd), α−3

by α ≤ 2, we have rεα−3 |Γd ≤ (hd) 2−α ς 2

 Γd

and therefore:

 2−α → ς arεα−3 v 2 ds av 2 rεα−4 (xi − εli ) cos(− n , xi )ds ≤ 2 Γd   2−α α−3 2 ς (hd) av ds ≤ c (v 2 + |∇v|2 )dx, ≤ 2 Γd

(5.3.6)

Gd

by (1.5.12). By virtue of assumption (c) and the function change (5.1.1), we have |v · B(x, v, vx )| ≤ aμς 2 |∇v|2 + b0 (x)|v|. Using the Cauchy inequality we deduce the following:    B(x, v, vx )rεα−2 v(x)dx ≤ μς 2 arεα−2 |∇v|2 dx + rεα−2 |v|b0 (x)dx G

≤ μς

2



arεα−2 |∇v|2 dx

G

δ + 2

 G

G

ar−2 rεα−2 v 2 dx

G

1 + 2δa∗



(5.3.7)

(5.3.8)

r2 rεα−2 b20 (x)dx, ∀δ > 0.

G

Now we use the representation G = Gd0 ∪ Gd . For the estimating integrals over Gd0 , by assumption (b) and the Cauchy inequality, we obtain   ij   a (x) − aij (0) rεα−2 vxi vxj + rεα−4 (xi − εli )v(x)vxj dx (5.3.9) Gd 0



≤ A(d) Gd 0

 a rεα−2 |∇v|2 + rεα−3 |∇v| · |v(x)| dx

112

Chapter 5. Transmission problem for weak quasi-linear elliptic equations 3 ≤ A(d) 2



  a rεα−2 |∇v|2 + rεα−4 v 2 dx.

Gd 0

Now we estimate integrals over Gd . By assumptions (a), the Cauchy inequality and taking into account that rε ≥ hd for r ≥ d, we get   ij .  a (x) − aij (0) rεα−2 vxi vxj + rεα−4 (xi − εli )v(x)vxj dx Gd ∗

 

≤A

Gd

(5.3.10)

    3 α−2 2 α−4 2 ∗ r |∇v|2 + v 2 dx. |∇v| + rε |v| dx ≤ C(A , h, α, d) 2 ε Gd

Further, because of assumption (e), 2

1

vG(x, v) = vG(x, 0) + v · 0

∂G(x, τ v) dτ ≤ |g(x, 0)| · |v|. ∂(τ v)

(5.3.11)

But, by the Cauchy inequality and γ(ω) ≥ ν0 > 0,   1 ,

1 1 |g(x, 0)| r− 2 γ(ω)|v| |g(x, 0)| · |v| = r 2 , γ(ω) 1 δ −1 rg 2 (x, 0), ≤ r γ(ω)v 2 + 2 2δν0 for all δ > 0; taking into account that rε ≥ hr (see Subsection 1.3) we obtain    1 δ 1 rεα−2 vG(x, v)ds ≤ rεα−2 γ(ω)v 2 ds + rα−1 g 2 (x, 0)ds, (5.3.12) 2 r 2δν0 ∂G

∂G

∂G

for all δ > 0. Similarly,    1 δ 1 rεα−2 vH(x, v)ds ≤ rεα−2 σ(ω)v 2 ds + rα−1 h2 (x, 0)ds, 2 r 2δν0 Σ0

Σ0

(5.3.13)

Σ0

for all δ > 0. As a result, with regard to ς ≤ 1, μς < 1 and 4 − n ≤ α ≤ 2 from (5.3.5)–(5.3.13), we obtain 

 1 α−2 1 α−2 rε σ(ω)v 2 (x)ds + r γ(ω)v 2 (x)ds r r ε G Σ0 ∂G     α−2 δ 3 ≤ A(d) a rε |∇v|2 + rεα−4 v 2 dx + ar−2 rεα−2 v 2 dx 2 2

ς(1 − μς)

Gd 0

arεα−2 |∇v|2 dx +



G

5.3. Global integral estimate 



+C Gd

 |∇v|2 + v 2 dx +

113 

1 2a∗ δ



rα b20 (x)dx

G

 1 rα−1 g 2 (x, 0)ds + rα−1 h2 (x, 0)ds 2δν0 Σ0 ∂G    1 1 δ rεα−2 σ(ω)v 2 ds + rεα−2 γ(ω)v 2 ds , + 2 r r 

+

Σ0

(5.3.14)

∂G

for all δ > 0. By the inequality rε ≥ hr, we have rεα−4 ≤ h−2 r−2 rεα−2 . Hence, by Lemma 2.5, from (5.3.14) it follows that     1 α−2 1 α−2 ς(1 − μς) rε σ(ω)v 2 (x)ds + rε γ(ω)v 2 (x)ds arεα−2 |∇v|2 dx + r r G Σ0 ∂G   arεα−2 |∇v|2 dx + r−1 rεα−2 σ(ω)v 2 (x)ds (5.3.15) ≤ c(λ, ω0 ) (δ + A(d))  +

G



r−1 rεα−2 γ(ω)v 2 ds + C

∂G

1 + 2δν0

 ∂G



Σ0

  |∇v|2 + v 2 dx +

G

1 rα−1 g 2 (x, 0)ds + 2δν0



1 2a∗ δ

rα−1 h2 (x, 0)ds,



rα b20 (x)dx

G

∀δ > 0, ∀ε > 0.

Σ0

ς Because of 0 ≤ μ < 1 + q, we can choose δ = 4c(λ,ω (1 − μς) and next d > 0 0) such that, by the continuity of A(r) at zero, c(λ, ω0 )A(d) ≤ 4ς (1 − μς). Thus, from (5.3.15) we get    1 α−2 1 α−2 α−2 2 2 rε σ(ω)v (x)ds + r arε |∇v| dx + γ(ω)v 2 (x)ds r r ε Σ0

G

∂G

  ≤ C(a∗ , α, λ, μ, q, n, d) (|∇v|2 + v 2 )dx + rα b20 (x)dx 1 + ν0

 ∂G

G

1 rα−1 g 2 (x, 0)ds + ν0

G

 r

α−1 2



h (x, 0)ds , ∀ε > 0.

(5.3.16)

Σ0

We can observe that the right-hand side of (5.3.16) does not depend on ε. Therefore, performing the passage to the limit as ε → +0, by the Fatou Theorem, we ◦ 1

derive v(x) ∈ Wα−2 (G) and    arα−2 |∇v|2 dx + rα−3 σ(ω)v 2 (x)ds + rα−3 γ(ω)v 2 (x)ds G

Σ0

∂G

114

Chapter 5. Transmission problem for weak quasi-linear elliptic equations   (|∇v|2 + v 2 )dx + rα b20 (x)dx ≤ C(a∗ , α, λ, μ, q, n, d) 1 + ν0

 ∂G

G

1 rα−1 g 2 (x, 0)ds + ν0

G

 r

α−1 2

 (5.3.17)

h (x, 0)ds .

Σ0

< and setting in it η(x) = v(x), we get Returning to the integral identity (II)     γ(ω) 2 σ(ω) 2 v ds + v ds ςaij (x)vxj vxi + B(x, v, vx )v dx + r r G

 G(x, v)vds +

=

Σ0

∂G

 H(x, v)vds. Σ0

∂G

From the ellipticity condition (a), inequalities (5.3.7)and (5.3.12)–(5.3.13) for α = 2, δ = 2, it follows that  ς(1 − μς) a|∇v|2 dx (5.3.18) G

  -  .  2 2 α−1 2 |v| + b0 (x) dx+ r g (x, 0)ds+ rα−1 h2 (x, 0)ds . ≤ c(a∗ , ν0 , diamG) G

Σ0

∂G

Now, using the inequality (2.2.3) and returning to the function u(x), by means of the function change (5.1.1), from (5.3.17)–(5.3.18) we get the desired estimate (5.3.1). 

5.4 Local integral weighted estimates In this section we will derive a local estimate for the weighted Dirichlet integral. Theorem 5.6. Let u be a weak solution of the problem (W QL), λ be as above in (2.2.1) and assumptions (a)–(f) be satisfied where A(r) is Dini-continuous at ◦ 1

zero. Then |u(x)|q+1 ∈ W2−n (G) and there exist d ∈ (0, 1) and a constant C > 0 1 depending only on n, s, λ, q, μ, ν0 , a∗ , G, Σ0 and on A(r) r dr such that the inequality  G 0

0

 

2−n 2q 2 −n 2(q+1) dx + r1−n σ(ω)|u|2(q+1) ds a r |u| |∇u| + r |u| 

+ Γ 0

Σ 0

r1−n γ(ω)|u|2(q+1) ds

5.4. Local integral weighted estimates

≤C

 G

⎧ ⎪2λ(1−μς) , ⎨  ⎪

 1 1 |u|2(q+1) dx+f12 + g12 + h21 · 2λ(1−μς) ln2 1 , ⎪ ν0 ν0 ⎪ ⎩2s , 1 1+q

and ς =

115 if s > λ(1 − μς), if s = λ(1 − μς), if s < λ(1 − μς) (5.4.1)

holds for all  ∈ (0, d).

Proof. Performing the function change (5.1.1) we consider the integral identity < loc for the function v(x). From Theorem 5.5 it follows that v(x) belongs to (II) ◦ 1

W2−n (G), so it is enough to derive the estimate (5.4.1). Using the function V () that is defined by (2.4.10) and setting η(x) = r2−n v(x) in the integral identity < loc , we obtain (II)      ∂v  ςV () ≤ ς av(x)  dΩ + ς r2−n v(x) aij (x) − aij (0) vxj cos(r, xi )dΩ ∂r r=  +

Ω

r2−n v(x)G(x, v)ds +

Γ 0

 

+

r2−n v(x)H(x, v)ds + ς(n − 2)

Σ 0



ar−n xi vvxi dx

G 0

    −ςr2−n aij (x) − aij (0) vxi vxj + ς(n − 2)r−n v(x) aij (x) − aij (0) xi vxj

G 0

−r

Ω



2−n

 v(x)B(x, v, vx ) dx.

(5.4.2)

We transform some integrals on the right. By the divergence theorem,   axi ∂v 2 n−2 −n dx (n − 2) ar v(x)xi vxi dx = 2 rn ∂xi G 0

G 0

 n − 2- 1 av 2 (x)xi cos(r, xi )dΩ = 2 n  + [a]Σ0

Ω

r



−n 2

v (x)xi cos(n, xi )ds +

Σ 0

. ar−n v 2 (x)xi cos(n, xi )ds .

Γ 0

        Since xi cos(n, xi )  = 0, xi cos(r, xi ) = ; xi cos(n, xi ) = xi cos(xn , xi ) = Γ0 Ω Σ0 Σ0   xn  = 0, we have from above Σ0

 (n − 2) G 0

ar

−n

n−2 v(x)xi vxi dx = 2

 Ω

av 2 (x)dΩ.

(5.4.3)

116

Chapter 5. Transmission problem for weak quasi-linear elliptic equations

Under Lemma 2.12, from (5.4.2)–(5.4.3) it follows that      V () +  v(x) aij (x) − aij (0) vxj cos(r, xi )dΩ V () ≤ 2λ Ω   1 1 2−n r v(x)G(x, v)ds + r2−n v(x)H(x, v)ds + ς ς Γ 0

Σ 0

     −r2−n aij (x) − aij (0) vxi vxj + (n − 2)r−n v(x) aij (x) − aij (0) xi vxj + G 0

. 1 − r2−n v(x)B(x, v, vx ) dx. ς

(5.4.4)

By virtue of assumptions (b)–(d) together with inequalities (5.3.7) and (5.3.11), from (5.4.4) it follows that     1 V () + A() a|v||∇v|dΩ + (1 − μς)V () ≤ r2−n |v(x)||g(x, 0)|ds 2λ ς +

+

1 ς 1 ς



Ω

r2−n |v(x)||h(x, 0)|ds + c1 (n)A()

Σ 0





Γ 0

  a r2−n |∇v|2 + r1−n |v||∇v| dx

G 0

r2−n |v(x)||b0 (x)|dx.

(5.4.5)

G 0

Applying the Cauchy and Friedrichs-Wirtinger inequalities we have (see (W )2 , (2.4.12))     1 a|v||∇v|dΩ ≤ a 2 |∇v|2 + |v|2 dΩ ≤ c2 (λ)V  () (5.4.6) 2 Ω

Ω

as well as, by virtue of the inequality (2.2.3) with α = 4 − n,     1−n ar |v||∇v|dx ≤ a r2−n |∇v|2 + r−n |v|2 dx ≤ c3 (λ)V (). G 0

(5.4.7)

G 0

Further, by the Cauchy inequality with ∀δ > 0,    

 1−n , 3−n 1 r 2 r2−n |v||g(x, 0)|ds = γ(ω)|v| r 2 , |g(x, 0)| ds γ(ω)   Γ0 Γ0   1 δ 1−n 2 r γ(ω)|v| ds + r3−n |g(x, 0)|2 ds; (5.4.8) ≤ 2 2δν0 Γ 0

Γ 0

5.4. Local integral weighted estimates  Σ 0



117

  

 1−n , 3−n 1 r 2 r2−n |v||h(x, 0)|ds = σ(ω)|v| r 2 , |h(x, 0)| ds σ(ω)  Σ0   1 δ 1−n 2 r σ(ω)|v| ds + r3−n |h(x, 0)|2 ds; (5.4.9) ≤ 2 2δν0 Σ 0

r2−n |v(x)||b0 (x)|dx ≤

G 0

≤

δ 2a∗



ar−n |v|2 dx +

G 0

1 δ c4 (λ)V () + 2a∗ 2δ

1 2δ





Σ 0

r4−n |b0 |2 dx

G 0

r4−n |b0 |2 dx

(5.4.10)

G 0

because of the inequality (2.2.3) with α = 4 − n. Thus, from (5.4.5)–(5.4.10) we get  {(1 − μς) − c5 (n, λ, q, a∗ )(δ + A())} V () ≤ (1 + c6 (λ)A()) V  () (5.4.11) 2λ     1 1 1 r4−n |b0 |2 dx + r3−n |g(x, 0)|2 ds + r3−n |h(x, 0)|2 ds , ∀δ > 0. + 2δ ν0 ν0 G 0

Γ 0

Σ 0

However, by the condition (f ),    1 1 r4−n |b0 |2 dx + r3−n |g(x, 0)|2 ds + r3−n |h(x, 0)|2 ds ν0 ν0 G 0

Γ 0

  c0 (G) 1 1 f12 + g12 + h21 · 2s . ≤ 2s ν0 ν0

Σ 0

From (5.4.11) we obtain the differential inequality (CP ) §2.4 with 2λ · {(1 − μς) − c5 (n, λ, a∗ )(δ + A())}, ∀δ > 0; N () ≡ 0;    λ 1 1 Q() = c0 (G) f12 + g12 + h21 · δ −1 2s−1 , ∀δ > 0; (5.4.12) s ν0 ν0      2  4−n 2 3−n 2 3−n 2 V0 = C v + (1 + r )b0 (x) dx + r h (x, 0)ds + r g (x, 0)ds , P() =

Σ0

G

∂G

by (5.3.1) with α = 4 − n. 1) Case s > λ(1 − μς). Choosing δ = ε , ∀ε > 0, we have P() =

2λ · {(1 − μς) − c5 (n, λ, a∗ )(ε + A())}; 

118

Chapter 5. Transmission problem for weak quasi-linear elliptic equations   λ 1 2 1 2 2 Q() = c0 (G) f1 + g1 + h1 · 2s−1−ε . s ν0 ν0

We can represent P() = at zero. Therefore

2λ(1−μς) 

τ −

P(s)ds = −2λ(1 − μς) ln 

⎛ exp ⎝− ⎛ exp ⎝−

d 

−

K()  ,

where K() satisfies the Dini condition

  τ

 2λ(1−μ) d K(r) K(s) τ + ds ≤ ln dr =⇒ +  s τ r 0



⎞ ⎛ d 

 2λ(1−μς)

 2λ(1−μς) K(τ ) dτ ⎠ = K0 P(τ )dτ ⎠ ≤ exp ⎝ ; d τ d ⎞

0

⎞ ⎛ d 

 2λ(1−μς)

 2λ(1−μς) K(τ ) dτ ⎠ = K0 P(τ )dτ ⎠ ≤ exp ⎝ . τ τ τ ⎞

τ

0



We have as well: d

 τ  Q(τ ) exp − P(σ)dσ dτ





 λc0 K0  2 f1 + g12 + h21 2λ(1−μςς) ≤ s

d

τ 2s−2λ(1−μς)−ε−1 dτ



 ds−λ(1−μς) 2λ(1−μς) λc0 K0  2 f1 + g12 + h21 ·  , ≤ s s − λ(1 − μς) since s > λ(1 − μς) and we can choose ε = s − λ(1 − μς). Now we apply Theorem 1.21: then from (1.7.1), by virtue of the deduced inequalities and taking into account (2.2.3) for α = 4 − n, we get      2−n 2 −n 2 1−n 2 a r |∇v| + r v dx + r σ(ω)v (x)ds + r1−n γ(ω)v 2 (x)ds G 0

Σ 0

  ≤ C v22,G + f12 + g12 + h21 2λ(1−μς) ,

Γ 0

(5.4.13)

where C = const(n, s, q, λ, μ, ν0 , G). Returning to the function u(x) by means of the function change (5.1.1), we obtain the statement of (5.4.1) for s > λ(1 − μ). 2) Case s = λ(1 − μς). Substituting in (5.4.12) an arbitrary function δ() > 0 instead of δ > 0, we have the problem (CP ) with P() =

2λ{(1 − μς) − c5 δ()} A() − c5 ;  

N () = 0;

5.4. Local integral weighted estimates

119

  1 2 1 2 2 Q() = c0 (G) f1 + g1 + h1 · δ −1 ()2λ(1−μς)−1 . ν0 ν0 Choosing δ() =

1

2λc5 ln

ed 

 , 0 <  < d, where e is the Euler number, we obtain

τ

d

 2λ(1−μς) τ dσ A(τ )

 + c5 dτ P(σ)dσ ≤ ln + ed τ τ σ ln σ   0 ⎛ ⎞ d 

 2λ(1−μς) ln ed  A(τ ) = ln dτ =⇒ + ln ⎝  ⎠ + c5 τ τ ln ed τ 0

 ed   τ  d   A(τ )  2λ(1−μς) ln  dτ , exp − P(σ)dσ ≤ ·  exp c5 τ τ ln ed −

 exp −

0

τ





d P(τ )dτ

≤

 2λ(1−μς) d



ed  exp c5 ln 



d 0

 A(τ ) dτ . τ

We also have d

 τ  Q(τ ) exp − P(σ)dσ dτ





  d dτ 1 2 1 2 2λ(1−μς) ed  2

 · ln ≤ c6 f 1 + g 1 + h 1  ed ν0 ν0  τ δ(τ ) ln τ    

ed 1 1 . ≤ 2λc5 c6 f12 + g12 + h21 · 2λ(1−μς) ln2 ν0 ν0  Now we apply Theorem 1.21: then from (1.7.1), by virtue of the deduced inequalities and with regard to (2.2.3) for α = 4 − n, we get 



a r G 0

2−n

2

|∇v| + r

−n 2

v



 dx +

r Σ 0

1−n



2

σ(ω)v (x)ds +

  ≤ C v22,G + f12 + g12 + h21 2λ(1−μς) ln2

ed  

r1−n γ(ω)v 2 (x)ds

Γ 0

,

(5.4.14)

where C = const(n, s, q, λ, μ, ν0 , G). Returning to the function u(x) by means of the function change (5.1.1) we obtain the statement of (5.4.1) for s = λ(1 − μς).

120

Chapter 5. Transmission problem for weak quasi-linear elliptic equations

3) Case 0 < s < λ(1 − μς). Analogously to case 1), taking into account (5.4.12), we have     d  d

 2λ(1−μς−δ) A(τ )  2λ(1−μς−δ) dτ = c8 exp c5 ; exp − P(τ )dτ ≤ d τ d 0



d

 τ  Q(τ ) exp − P(σ)dσ dτ





 c0 (ν0 , G)  2 f1 + g12 + h21 2λ(1−μς−δ) ≤ δ   1 1 ≤ c9 f12 + g12 + h21 · 2s , ν0 ν0

d

τ 2s−2λ(1−μς−δ)−1 dτ



if we choose δ ∈ (0, 1 − μς − λs ). Now we apply Theorem 1.21: then from (1.7.1), by virtue of the deduced inequalities and taking into account (2.2.3) for α = 4 − n, we get      2−n 2 −n 2 1−n 2 a r |∇v| + r v dx + r σ(ω)v (x)ds + r1−n γ(ω)v 2 (x)ds G 0

Σ 0

  ≤ C v22,G + f12 + g12 + h21 2s ,

Γ 0

(5.4.15)

where C = const(n, s, q, λ, μ, ν0 , G). Returning to the function u(x) by means of the function change (5.1.1) we obtain the statement of (5.4.1) for s < λ(1−μς). 

5.5 The power modulus of continuity at the conical point for weak solutions 5.5.1 Proof of Theorem 5.3. Let us introduce the function ⎧ λ(1−μς) ⎪ ,  if s > λ(1 − μς), ⎪ ⎨ λ(1−μς) ln 1 , if s = λ(1 − μς), ψ() =  ⎪ ⎪ ⎩ s , if s < λ(1 − μς)

(5.5.1)

for 0 <  < d. We perform the function change (5.1.1) and consider the function v(x). For it, by Theorem 5.4 about the local bound of the weak solution modulus, we have

5.5. Power modulus of continuity at the conical point for weak solutions

121

sup |v(x)| ≤ C −n/2 v2,G0 + 2(1−n/p) b0 p/2,G0 /2

G0

.

+  g(x, 0)∞,Γ0 + h(x, 0)∞,Σ0 ,  ∈ (0, d),

(5.5.2)

where C = C (n, a∗ , A∗ , p, μ, G) and n < p < 2n. Further, by Theorem 5.6 (see (5.4.13)–(5.4.15)), we have  1/2 

−n/2 v2,G0 ≤ r−n v 2 (x)dx ≤ C v2,G + f1 + g1 + h1 ψ(). (5.5.3) G 0

Now, by the assumption (f), 

2(1−n/p) b0 p/2,G0 +  g(x, 0)∞,Γ0 + h(x, 0)∞,Σ0 ≤ c (f1 + g1 + h1 ) ψ(). (5.5.4) From (5.5.2)–(5.5.4) it follows that 

sup |v(x)| ≤ C v2,G + f1 + g1 + h1 ψ(). /2

G/4

Setting now |x| = 13  and returning to the function u(x) by (5.1.1) we obtain finally the desired estimate (5.1.4). As well we shall estimate the gradient modulus of the problem (W QL) so 2 lution near a conical point. Let us introduce two sets G2 /4 and G/2 ⊂ G/4 ,     > 0. We apply the change of variables x = x and v(x ) = ψ()z(x ). Then the function z(x ) satisfies in the weak sense the problem

 ⎧ 2 d ij    ψ() ⎪ x ∈ G21/4 , ⎪−ς dxi a (x )zxj + ψ() B(x , ψ()z(x ),  zx ) = 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎨  ∂z     2 + σ(ω) [z(x )]Σ2 = 0, ς ∂ν  |x | z(x ) = ψ() H(x , ψ()z(x )), x ∈ Σ1/4 , Σ21/4 1/4 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ς ∂z + γ(ω) z(x ) =  G(x , ψ()z(x )), x ∈ Γ21/4 . ∂ν  |x | ψ() (W QL) Now we apply our assumption about a priori estimate of the gradient modulus of the problem (W QL) solution max |∇ z(x )| ≤ M1

x ∈G11/2

(5.5.5)

(see §4 Theorem 4.4 in [6] or §§3, 5 from [64] as well as [47]). Returning to the variable x and the function v(x) we obtain from (5.5.5) |∇v(x)| ≤ M1 −1 ψ(), x ∈ G/2 , 0 <  < d. Setting now |x| = 23  and returning to the function u(x) by (5.1.1), we obtain desired estimate (5.1.5).

122

Chapter 5. Transmission problem for weak quasi-linear elliptic equations

5.5.2 Remark to Theorem 5.5 Now we can state that Theorem 5.5 is true for α ∈ (4−n−2λ, 2], if a neighborhood of the conic point is convex and the inequality 0≤μ
0, ∀ε > 0.

Σ0

H(λ, n, α) But, by (2.3.3) and 4−n−2λ < α < 4−n, we verify that 1− (2−α)(4−n−α) 2 > 0. Then, by virtue of (5.5.6), we obtain that 1− (2−α)(4−n−α) H(λ, n, α)−μς > 0. 2 Now, choosing first δ > 0 and next d > 0, together with A(d) appropriately small,

5.6. Example

123

we guarantee the realization of (5.3.16). Further, just as in the proof of Theorem 5.5 we obtain the required statement.

5.6 Example In this section we consider the two-dimensional transmission problem for the Laplace operator with absorbtion term in an angular domain and investigate the corresponding eigenvalue problem. Let n = 2, the domain G lie inside the corner G0 = {(r, ω) |r > 0; −

ω0 ω0 0; ω = ±   and set σ(ω)

Σ0

ω0 }, 2

Σ0 = {(r, ω) | r > 0; ω = 0}

  = σ(0) = σ = const > 0, γ(ω)

consider the problem:

ω=±

ω0 2

= γ± = const > 0. Let us

⎧ d q−2 2 ⎪ ⎪ (|u|q uxi ) = a0 r−2 u|u|q − μu|u| |∇u| , x ∈ G0 \ Σ0 ; ⎪ ⎪ dxi ⎪ ⎪ ⎪ ⎨  1 ∂u [u]Σ0 = 0, a|u|q ∂n + |x| σ(0)u|u|q = 0, x ∈ Σ0 ; ⎪ Σ0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ± 1 q α± a± |u± |q ∂u x ∈ Γ± \ O ∂n + |x| γ± u± |u± | = 0,

(AL)

where a0 ≥ 0, 0 ≤ μ < 1 + q, q ≥ 0; α± ∈ {0; 1}; a± > 0. We perform the function change (5.1.1); then the problem for the function v(x) has the following form: ⎧ 1 v + μςv −1 |∇v|2 = a0 (1 + q)r−2 v; ς = 1+q , x ∈ G0 \ Σ0 ; ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨  ∂v a ∂n Σ0 + (1 + q)σ(0) v(x) x ∈ Σ0 ; [v]Σ0 = 0, |x| = 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩α a ∂v± + (1 + q)γ v± (x) = 0, x ∈ Γ± \ O. ± ± ∂n ± |x|

124

Chapter 5. Transmission problem for weak quasi-linear elliptic equations

We want to find the exact solution of this problem in the form v(r, ω) = rκ ψ(ω). For ψ(ω) we obtain the problem ⎧  ! " 2 μς ψ (ω) + ψ(ω) ψ  (ω) + (1 + μς)κ 2 − a0 (1 + q) · ψ(ω) = 0, ⎪ ⎪ ⎪     ⎪ ⎪ ω ∈ − ω20 , 0 ∪ 0, ω20 ; ⎪ ⎪ ⎪ ⎨ ⎪ [ψ]ω=0 = 0, [aψ  (0)] = (1 + q)σ(0)ψ(0); ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪  ω0    ⎩  ± 2 + (1 + q)γ± ψ± ± ω20 = 0. ±α± a± ψ± 2

(1+q) and define the value Let us assume κ 2 > a0 1+q+μ

= Υ =

κ 2 − a0

(1 + q)2 . 1+q+μ

(5.6.1)

We consider separately two cases: μ = 0 and μ = 0. Case μ = 0. We have (5.6.2)

ψ± (ω) = A cos(Υ ω) + B± sin(Υ ω),

where constants A, B± should be determined from the conjunction and boundary conditions; namely, they satisfy the system ⎧ ⎪ (1 + q)σ(0) · A − a+ Υ · B+ + a− Υ · B− =0 ⎪ ⎪ !  ω0   ω0 " ⎪ ⎪ ⎪ + q)γ+ cos Υ 2 − α+ a+ Υ sin Υ 2 ·A ⎨ (1 !  "   + (1 + q)γ+ sin Υ ω20 + α+ a+ Υ cos Υ ω20 · B+ = 0 ⎪ !  "   ⎪ ⎪ + q)γ− cos Υ ω20 − α− a− Υ sin Υ ω20 · A ⎪ ⎪ (1 !    ⎪ ⎩− (1 + q)γ sin Υ ω0 + α a Υ cos Υ ω0 " · B − − − − = 0. 2 2 The Dirichlet problem: α± = 0, γ± = 0. Direct calculations give

ω  0 · sin(Υ ω), ψ± (ω) = cos(Υ ω) ∓ cot Υ 2

 Υ =

π ω0 , ∗

if σ(0) = 0; Υ , if σ(0) = 0,

where Υ ∗ is the least positive root of the transcendence equation

ω  1+q 0 =− σ(0) Υ · cot Υ 2 a + + a− and from the graphic solution (see Figure 2 §3.6) we obtain

π ω0

< Υ∗
0; a = b; x > 0

y = a tan x − b cot x

y

y=

x0

x∗

π

π 2

 arctan

c x

b π < x∗ < a 2

Figure 8: The graphic solution

where T =

(1 + q)σ(0) +

,

(1 + q)2 σ 2 (0) + 4(Υ ∗ )2 a+ a− . 2Υ ∗ a+

The Robin problem: α± = 1, γ± = 0. Direct calculations of the above system give: a+ γ+ = . In this case we get either 1) γ− a− ψ± (ω) = a∓ sin(Υ ∗ ω), where Υ ∗ is the least positive root of the transcendence equation

ω  γ+ 0 = −(1 + q) Υ · cot Υ 2 a+

x

5.6. Example

127

and from the graphic solution (see Figure 2 §3.6) we obtain π 2π < Υ∗ < , ω0 ω0 or ψ± (ω) = cos(Υ ∗ ω) ±

1+q σ(0) · ∗ sin(Υ ∗ ω), a + + a− Υ

where Υ ∗ is the least positive root of the transcendence equation

ω  a+ σ(0) + γ+ (a+ + a− ) 0 = (1 + q)Υ . tan Υ 2 a+ (a+ + a− )Υ 2 − (1 + q)2 γ+ σ(0)

(5.6.3)

In particular, for σ(0) = 0 we have ψ± (ω) = cos(Υ ω),

ω  1+q γ 0 + = · tan Υ 2 Υ a+

and from the graphic solution (see Figure 3 §3.6) 0 < Υ ∗
0 a 2

y = tan x

y

x ; a2 x2 − b2

y= x∗

π

π 2

x

y=

x a2 x2 − b2

b a

x a2 x2 − b2

b π < x∗ < a 2

Figure 9 a+ γ+ = . In this case we get A = 0 and from (5.6.2) it follows that γ− a− ψ± (0) = 0; further see below the general case μ = 0. 2)

Case μ = 0. We observe that in this case ψ(0) = 0. By setting y(ω) =

ψ  (ω) ψ(ω) ,

we arrive at the

5.6. Example

129 a, b > 0;

b π < < π; x > 0 2 a

y = tan x

y

x ; − b2

a2 x2

y = tan x

tan x =

π 2

x∗

y=

b a 3π 2

x

y=

x a2 x2 − b2

π

x a2 x2 − b2

π b < x∗ < 2 a

Figure 10 problem for y(ω): ⎧     ⎪ y  + (1 + μς)y 2 (ω) + (1 + μς)κ 2 − a0 (1 + q) = 0, ω ∈ − ω20 , 0 ∪ 0, ω20 ; ⎪ ⎪ ⎨ a+ y+ (0) − a− y− (0) = (1 + q)σ(0); ⎪ ⎪ ⎪   ⎩ ±α± a± y± ± ω20 + (1 + q)γ± = 0.

130

Chapter 5. Transmission problem for weak quasi-linear elliptic equations a, b > 0;

π b = ;x>0 a 2

y = tan x

y

x ; a2 x2 − b2

y = tan x

tan x =

b a

=

y=

π 2

x∗

3π 2

x

y=

x a2 x2 − b2

π

x a2 x2 − b2

π < x∗
0, a∗ = Assumptions. Let a = a− , x ∈ G− , max{a+ , a− } > 0; let 1 < m < n, mn > p > n > m, q ≥ 0, 0 ≤ μ < q+m−1 m−1 be given numbers; a0 (x), α(x) and b0 (x) be non-negative measurable functions; let ai (x, u, ξ), i = 1, . . . , n; b(x, u, ξ) be Caratheodory functions G × R × Rn → R and continuously differentiable with respect to xi ; h(x, u) be Caratheodory function Σ0 ×R → R and continuously differentiable with respect to variable u, while g(x, u) be Caratheodory function ∂G × R → R and continuously differentiable with respect to variable u. We assume the following properties: 1) ai (x, u, ξ)ξi ≥ a|u|q |ξ|m − a0 (x); a0 (x) ∈ Lp/m (G);

6.1. Introduction

137

2 2 3 n  3 n  3  ∂ai (x, u, ξ) 2 3 2 4 4   ≤ a|u|q |ξ|m−1 + α(x); ai (x, u, ξ) + 2)   ∂x i i=1 i=1 p (G); α(x) ∈ L m−1

3a) |b(x, u, ξ)| ≤ aμ|u|q−1 |ξ|m + b0 (x); b0 (x) ∈ L mp (G); 3b) b(x, u, ξ) = β(x, u) + /b(x, u, ξ), u · β(x, u) ≥ a|u|q+m ; |/b(x, u, ξ)| ≤ aμ|u|q−1 |ξ|m + b0 (x), b0 (x) ∈ L mp (G); ∂h(x,u) ∂u

4)

≤ 0,

∂g(x,u) ∂u

≤ 0;

5) σ(ω) ≥ ν0 ≥ 0 on σ0 ; γ(ω) ≥ ν0 ≥ 0 on ∂G. The functions ai (x, u, ξ) are continuously differentiable with respect to variables u, ξ in Md,M0 = Gd0 × [−M0 , M0 ] × Rn and satisfy, in Md,M0 , the following conditions: 6) (m − 1)u ∂ai (x,u,ξ) = q ∂ai (x,u,ξ) ξj ; i = 1, . . . , n; ∂u ∂ξj = n  2 7) |ai (x, u, ux ) − a|u|q |∇u|m−2 uxi | ≤ aA(|x|)|u|q |∇u|m−1 , x ∈ Gd0 , where i=1

A(r) is a function which is Dini-continuous at zero. Let us consider the function change u = v|v|ς−1 with

ς=

m−1 . q+m−1

(6.1.1)

By virtue of the assumption 6), the identity (II)loc takes the form   G 0

 Ai (x, vx )ηxi + B(x, v, vx )η dx +



+

Γ 0

σ(ω) v|v|m−2 η(x)ds rm−1

Σ 0





Ai (x, vx ) cos(r, xi )η(x)dΩ +

=

γ(ω) v|v|m−2 η(x)ds rm−1

Ω

 G(x, v)η(x)ds +

Γ 0

H(x, v)η(x)ds

(6.1.2)

Σ 0

1 1 for a.e.  ∈ (0, d), v(x) ∈ C0 (G) ∩ Vm,0 (G) and any η(x) ∈ C0 (G) ∩ Vm,0 (G), where

Ai (x, vx ) ≡ ai (x, v|v|ς−1 , ς|v|ς−1 vx ), B(x, v, vx ) ≡ b(x, v|v|ς−1 , ς|v|ς−1 vx ), G(x, v) ≡ g(x, v|v|ς−1 ),

H(x, v) ≡ h(x, v|v|ς−1 ).

(6.1.3)

138

Chapter 6. Transmission problem for strong quasi-linear elliptic equations

We show that coefficients Ai , i = 1, . . . , n do not depend on v explicitly. In fact, by the change (6.1.1) and the assumption 6), we calculate   ∂a (x, u, ξ) ∂Ai ∂ 2 ς−1 ∂ai (x, u, ξ) ∂ 2 ς−1 i = · |v | 2 · v + |v | 2 · ςvxj ∂v ∂u ∂v ∂ξj ∂v ∂ai ξj ∂ai ∂ai u ∂ai + ς(ς − 1)vxj v|v|ς−3 · + (ς − 1) · · = ς|v|ς−1 =ς· · ∂u ∂ξj v ∂u v ∂ξj     ∂ai m − 1 ∂ai u ∂ai m−1 1 ςu · + (ς − 1) · u = · · ς + (ς − 1) · = 0, = v ∂u q ∂u v ∂u q because of (6.1.1). Our assumptions can be rewritten as follows: 1) Ai (x, vx )vxi ≥ aς m−1 |∇v|m − 1ς |v|1−ς a0 (x); a0 (x) ∈ Lp/m (G); 2 2 3 n  3 n  3  ∂Ai (x, vx ) 2 3  4 2 4   ≤ aς m−1 |∇v|m−1 + α(x); Ai (x, vx ) + 2)   ∂x i i=1 i=1 p (G); α(x) ∈ L m−1

3a) |B(x, v, vx )| ≤ aμς m |v|−1 |∇v|m + b0 (x); b0 (x) ∈ L mp (G), 4)

∂H(x,v) ∂v

≤ 0,

∂G(x,v) ∂v

≤ 0; 2 3 n 3 2  4 |Ai (x, vx ) − aς m−1 |∇v|m−2 vxi | ≤ aς m−1 A(|x|)|∇v|m−1 , x ∈ Gd0 . 7) i=1

The main result in this chapter is the following statement: Theorem 6.3. Let u be a weak solution of the problem (QL), assumptions 1)–7) be satisfied and ϑ be the smallest positive eigenvalue of the problem (N EV P ) (see §2.1). Let us assume that M0 = max |u(x)| is known. In addition, let h(x, 0) ∈ x∈G

L∞ (Σ0 ), g(x, 0) ∈ L∞ (∂G) and let there exist real numbers ks ≥ 0, K ≥ 0 such that   m(q+m−1) q 1 m ks =: sup −ms r q+m−1 |a0 (x)| (m−1)(q+m) dx + r m−1 |b0 (x)| m−1 dx >0

G 0





m

|h(x, 0)| m−1 ds +

+ Σ 0



G 0

m

|g(x, 0)| m−1 ds , s > 1;

(6.1.4)

Γ 0

 q+m−1 1 q+m−1 n n  m(1− p ) (m−1)(q+m) a0  (m−1)(q+m) K =: sup + 1− p α(x) m−1  p p , G , G 0 0 m m−1 >0 ψ()   1 1 1 n m m−1 m−1 , (6.1.5) + (1− p ) m−1 b0 (x) m−1 +  g(x, 0)∞, + h(x, 0)∞, p , G Γ Σ n m −1

m

0

0

0

6.2. Comparison principle where

ψ() =

139

⎧ 1 ϑ m (m) q+(m−1)(1−μ) ⎪ ⎪ ⎪  Ξ(m) · q+m−1 , ⎪ ⎨ 1 ϑ m (m) q+(m−1)(1−μ) · q+m−1

 Ξ(m) ⎪ ⎪ ⎪ ⎪ ⎩ s  ,

1

s>

ϑ m (m) Ξ(m) 1

ln d ,

s= s
0 independent of u such that m−1

 q+m−1 n , ∀x ∈ Gd0 . (6.1.7) |u(x)| ≤ C0 |x|1− m ψ(|x|) Furthermore, if coefficients of the problem (QL) satisfy such conditions which guarantee the local a priori estimate |∇u|0,G ≤ M1 for any smooth G ⊂⊂ G\{O} (see for example §4 in [6] or [64], [47]), then there is a constant C1 > 0 independent of u such that |∇u(x)| ≤ C1 |x|−

n(m−1)+qm m(q+m−1)

m−1

ψ q+m−1 (|x|),

∀x ∈ Gd0 .

(6.1.8)

6.2 Comparison principle Let us consider the second-order quasi-linear degenerate operator Q of the form    γ(ω) Ai (x, vx )ηxi + B(x, v, vx )η dx + v|v|m−2 η(x)ds Q(v, η) ≡ rm−1 Gd 0



 H(x, v)η(x)ds +

− Σd 0



Γd 0

σ(ω) v|v|m−2 η(x)ds rm−1

Σd 0



Ai (x, vx ) cos(r, xi )η(x)dΩd −

− Ωd

G(x, v)η(x)ds

(6.2.1)

Γd 0

1 (G) and for all non-negative η belonging to C0 (G) ∩ for v(x) ∈ C0 (G) ∩ Vm,0 1 Vm,0 (G) under the following assumptions: The functions Ai (x, ξ), B(x, v, ξ), G(x, v), H(x, v) are Caratheodory, continuously differentiable with respect to the v, ξ variables in M = Ω×R×RN and satisfy in M the following inequalities:

∂Ai (x, ξ) pi pj ≥ aγm |ξ|m−2 p2 , ∀p ∈ Rn \ {0}; ∂ξj 2 3N  3 ∂B(x, v, ξ) 2  ≤ a|v|−1 |ξ|m−1 ; ∂B(x, v, ξ) ≥ a|v|−2 |ξ|m ; (ii) 4   ∂ξi ∂v i=1 (i)

140 (iii)

Chapter 6. Transmission problem for strong quasi-linear elliptic equations ∂H(x, v) ∂G(x, v) ≤ 0, ≤ 0, γ(ω) ≥ 0, ∂v ∂v

σ(ω) ≥ 0.

Here: m > 1, γm > 0 and a > 0. Proposition 6.4. Let operator Q satisfy assumptions (i)–(iii) and functions v, w ∈ 1 (Gd0 ) (d ≪ 1) satisfy the inequality C0 (Gd0 ) ∩ Vm,0 Q(v, η) ≤ Q(w, η)

(6.2.2)

1 for all non-negative η ∈ C0 (Gd0 ) ∩ Vm,0 (Gd0 ) and also the inequality

v(x) ≤ w(x) on Ωd

(6.2.3)

hold. Then v(x) ≤ w(x) in Gd0 . Proof. Let us define z = v − w and v τ = τ v + (1 − τ )w, τ ∈ [0, 1]. Then we have  * 0 ≥ Q(v, η) − Q(w, η) =

 ηxi zxj

Gd 0

1

0

∂Ai (x, vxτ ) dτ ∂vxτ j

+  1 ∂B(x, v τ , vxτ ) ∂B(x, v τ , vxτ ) + ηzxi dτ + ηz dτ dx ∂vxτ i ∂v τ 0 0  1   γ(ω) ∂(v τ |v τ |m−2 ) + dτ z(x)η(x)ds rm−1 ∂v τ 0 

1

Γd 0



+

σ(ω) rm−1

Σd 0

 

− Ωd

0

 

− Γd 0

1

0

1



1

0

∂(v τ |v τ |m−2 ) dτ ∂v τ

∂Ai (x, vxτ ) dτ ∂vxτ j ∂G(x, v τ ) dτ ∂v τ

 z(x)η(x)ds

 cos(r, xi ) · zxj η(x)dΩd  

 z(x)η(x)ds −

Σd 0

0

1

∂H(x, v τ ) dτ ∂v τ

1 (Gd0 ). for all non-negative η ∈ C0 (Gd0 ) ∩ Vm,0 Now we introduce the sets

(Gd0 )+ := {x ∈ Gd0 | v(x) > w(x)} ⊂ Gd0 , (Σd0 )+ := {x ∈ Σd0 | v(x) > w(x)} ⊂ Σd0 , (Γd0 )+ := {x ∈ Γd0 | v(x) > w(x)} ⊂ Γd0

(6.2.4)  z(x)η(x)ds

6.3. Maximum principle

141

and assume that (Gd0 )+ = ∅. Let k ≥ 1 be any odd number. We choose η = max{(v − w)k , 0} as test function in the integral inequality (6.2.4). We have  1  1 ∂(v τ |v τ |m−2 ) dτ = (m − 1) |v τ |m−2 dτ > 0. ∂v τ 0 0  Then, by assumptions (i)–(iii) and η Ω = 0, we obtain from (6.2.4) 



kγm az

k−1



d

1

0

+ (Gd 0)

|∇v |

τ m−2

 ≤

az

k

τ −1

z |∇z||v | k

|∇v |

τ m−1

2

dτ |∇z| dx + az



1

0

+ (Gd 0)

By the Cauchy inequality



≤



1 0

τ −2

|v |

 |∇v | dτ dx τ m

 |v τ |−1 |∇v τ |m−1 dτ |∇z|dx.

τ −1

|v |

z

k+1 2

|∇v |

τ m/2

·



(6.2.5)

 



=

k+1



z

k−1 2

|∇z||∇v |

τ m/2−1

ε τ −2 k+1 1 |v | z |∇v τ |m + z k−1 |∇z|2 |∇v τ |m−2 , ∀ε > 0. 2 2ε

Hence, taking ε = 2, we obtain from (6.2.5) the inequality   1    1 k−1 2 τ m−2 z a kγm − |∇z| |∇v | dτ dx ≤ 0. 4 0

(6.2.6)

+ (Gd 0)

  Now choosing the odd number k ≥ max 1; 2γ1m , in view of z(x) ≡ 0 on ∂(Gd0 )+ , we get from (6.2.6) z(x) ≡ 0 in (Gd0 )+ . We have arrived at a contradiction to our definition of the set (Gd0 )+ . By this fact, the proposition is proved. 

6.3 Maximum principle In this section we derive an L∞ (G)-a priori-estimate of the weak solution to problem (QL). Theorem 6.5. Let u(x) be a weak solution of (QL) and assumptions 1), 3b), 4) and 5) hold. In addition, let h(x, 0) ∈ L

j j−1

(Σ0 ), g(x, 0) ∈ L

j j−1

(∂G),

1≤j
0, depending only on meas G, meas ∂G, meas Σ0 , n, m, μ, q, h(x, 0)L j (Σ0 ) , g(x, 0)L j (∂G) , a0 (x)L p (G) , b0 (x)L p (G) , j−1

such that uL∞ (G) ≤ M0 .

j−1

m

m

142

Chapter 6. Transmission problem for strong quasi-linear elliptic equations

Proof. Let us define the set A(k) = {x ∈ G, |u(x)| > k} and let χA(k) be the characteristic function of the set A(k). We observe that A(k + d) ⊆ A(k) for all d > 0. Putting η((|u| − k)+ )χA(k) · signu as test function in (II), where η is defined by Lemma 1.23 and k ≥ k0 (without loss of generality we can assume k0 ≥ 1), under assumptions 1), 3b), 5) we obtain the inequality  ! " a|u|q |∇u|m η  ((|u| − k)+ ) + a|u|q+m−1 η((|u| − k)+ ) dx (6.3.1) A(k)



+ Σ0 ∩A(k)



!

≤



σ(ω) q+m−1 |u| η((|u| − k)+ )ds + rm−1

γ(ω) q+m−1 |u| η((|u| − k)+ )ds rm−1

∂G∩A(k)

" aμ|u|q−1 |∇u|m η((|u|−k)+ ) + a0 (x)η  ((|u|−k)+ ) + b0 (x)η((|u|−k)+ ) dx

A(k)



 h(x, u)sign u · η((|u| − k)+ )ds +

+ Σ0 ∩A(k)

g(x, u)sign u · η((|u| − k)+ )ds. ∂G∩A(k)

By virtue of g(x, u) − g(x, 0) =

1 0

d dτ g(x, τ u)dτ

= u·

1 0

∂g(x,τ u) ∂(τ u) dτ

and the assump-

tion 4),  g(x, u) · sign u · η((|u| − k)+ )ds ∂G∩A(k)

⎛



|u(x)| ⎝

=

1 0

∂G∩A(k)



⎞ ∂g(x, τ u) ⎠ dτ η((|u| − k)+ )ds ∂(τ u)

g(x, 0) · sign u · η((|u| − k)+ )ds

+ ∂G∩A(k)



|g(x, 0)| · η((|u| − k)+ )ds,

≤ ∂G∩A(k)

as well 

 h(x, u) · sign u · η((|u| − k)+ )ds ≤

Σ0 ∩A(k)

|h(x, 0)| · η((|u| − k)+ )ds. Σ0 ∩A(k)

Therefore from (6.3.1), by assumption 5), it follows that

6.3. Maximum principle 

143

! " a|u|q |∇u|m η  ((|u| − k)+ ) + a|u|q+m−1 η((|u| − k)+ ) dx

A(k)



! " aμk0−1 |u|q |∇u|m η((|u| − k)+ ) + a0 (x)η  ((|u| − k)+ ) + b0 (x)η((|u| − k)+ ) dx

≤ A(k)



 |h(x, 0)|η((|u| − k)+ )ds +

+ Σ0 ∩A(k)

|g(x, 0)|η((|u| − k)+ )ds.

(6.3.2)

∂G∩A(k)

 (|u| − k)+ . By (1.8.7) from Lemma m

 Now we introduce the function wk (x) := η 1.23, we have 

 |g(x, 0)|η((|u| − k)+ )ds ≤ M ·

∂G∩A(k)

|g(x, 0)||wk |m ds

∂G∩A(k+d)

+e

κd



·

|g(x, 0)|ds.

(6.3.3)

∂G∩{A(k)\A(k+d)}

By the Hölder inequality  |g(x, 0)| · |wk |m ds ≤ |wk |m Lj (∂G∩A(k)) · g(x, 0)L

j j−1

(∂G)

∂G∩A(k+d)

= wk m Lmj (∂G∩A(k)) · g(x, 0)L

j j−1

(∂G) ,

∀j ≥ 1

and the Sobolev boundary trace embedding theorem (1.6.8), we derive  |g(x, 0)| · |wk |m ds ∂G∩A(k+d)



≤ Cg(x, 0)L

j j−1

(∂G)

·

  n−1 . |∇wk |m + |wk |m dx, 1 ≤ j < n−m

A(k)

In the same way  |h(x, 0)| · |wk |m ds Σ0 ∩A(k+d)

≤ Ch(x, 0)L

 j j−1

(Σ0 )

· A(k)

  n−1 . |∇wk |m + |wk |m dx, 1 ≤ j < n−m

144

Chapter 6. Transmission problem for strong quasi-linear elliptic equations

Now from (6.3.2) and (6.3.3) it follows that  ! 9 : a|u|q |∇u|m η  ((|u| − k)+ ) − μk0−1 η((|u| − k)+ ) A(k)

" + a|u|q+m−1 η((|u| − k)+ ) dx  ! " ≤ a0 (x)η  ((|u| − k)+ ) + b0 (x)η((|u| − k)+ ) dx A(k)

 + CM h(x, 0)L

+e

κd



 j j−1

(Σ0 )

+ g(x, 0)L





 |∇wk |m + |wk |m dx

·

(∂G)



A(k)



 |g(x, 0)|ds ,

|h(x, 0)|ds + Σ0 ∩{A(k)\A(k+d)}

1≤j
0. e κ(|u|−k)+ |∇u|m = κ

(6.3.5)

Therefore, choosing κ > m+ 2μ k0 according to Lemma 1.23 and using (1.8.5)–(1.8.7), from (6.3.4), we obtain  

m m  k0q a|∇wk |m dx + k0q+m−1 a|wk |m dx ≤ c1 M B0 (x)|wk |m dx κ A(k)

 h(x, 0)L c + M a−1 2 ∗

+ c3 e

κd



A(k)

j j−1

(Σ0 )

+ g(x, 0)L



j j−1

(∂G)





 a|∇wk |m + a|wk |m dx

Σ0 ∩{A(k)\A(k+d)}

|g(x, 0)|ds ,

+

·

|h(x, 0)|ds

B0 (x)dx + {A(k)\A(k+d)}



A(k)





A(k+d)



(6.3.6)

∂G∩{A(k)\A(k+d)}

where 1 ≤ j
1. Under assumptions 1) and 3b) we get that B0 (x) ∈ Ls (G), where s > m 1 1  Using the Hölder inequality with exponents s and s , where s + s = 1, we obtain ⎛ ⎞ 1 s    ⎜ ⎟ B0 (x)|wk |m dx ≤ B0 (x)Ls (G) · ⎝ |wk |ms dx⎠ . (6.3.8) A(k+d)

A(k)

6.3. Maximum principle

145

 # = From the inequality 1s < m n it follows that ms < m interpolation inequality (1.5.9) gives

⎛ ⎜ ⎝

⎞ 1



⎛

s

ms

|wk |

⎟ dx⎠

⎜ ≤⎝

A(k)

⎞θ ⎛



⎟ ⎜ |wk | dx⎠ · ⎝

and then the

⎞ (1−θ)m #



m

m#

|wk |

m

A(k)

mn n−m

⎟ dx⎠

A(k)

1 θ 1−θ n with θ ∈ (0, 1), which is defined by the equality ms =⇒ θ = 1 − ms .  = m + m# 1 1 Now, by using the Young inequality with exponents θ and (1−θ) , from (6.3.8) we obtain



B0 (x)|wk |m dx ≤ a−1 ∗ θε

θ−1 θ

1



B0 (x)Lθ s (G)

A(k+d)

⎛ ⎜ + (1 − θ)ε · ⎝

a|wk |m dx

A(k)



⎞

m m#

# ⎟ |wk |m dx⎠

∀ε > 0. (6.3.9)

,

A(k)

From (6.3.6), (6.3.9) it follows that

k0q

m m

≤ c6

κ ⎧ ⎪ ⎨ ⎪ ⎩

− c4

 

 

θ−1 a|∇wk |m dx + k0q+m−1 − c4 − c5 ε θ a|wk |m dx

A(k)

A(k)

 |h(x, 0)|ds +

B0 (x)dx +

A(k)

⎛

⎜ + c1 M (1 − θ)ε · ⎝



Σ0 ∩A(k)



⎞

|g(x, 0)|ds ∂G∩A(k)

⎫ ⎪ ⎬ ⎪ ⎭

m m#

# ⎟ |wk |m dx⎠

, ∀ε > 0, ∀k ≥ k0 ,

(6.3.10)

A(k)

where  c4 = M a−1 c 2 h(x, 0)L ∗ 1

 j j−1

θ c5 = M a−1 ∗ c1 θB0 (x)Ls (G) ,

(Σ0 ) + g(x, 0)L

c6 = c 3 e

κd

j j−1

(∂G)

,

.

Now we apply the Sobolev embedding Theorem 1.15; as a result we get from (6.3.10) the inequality

146

Chapter 6. Transmission problem for strong quasi-linear elliptic equations

k0q

m m κ

− c4 − c7 ε

 

a|∇wk |m dx

A(k)

  θ−1 + k0q+m−1 − c4 − c5 ε θ − c7 ε a|wk |m dx  

A(k)



≤ c6





|h(x, 0)|ds +

B0 (x)dx + Σ0 ∩A(k)

A(k)

(6.3.11)

|g(x, 0)|ds , ∀ε > 0, ∀k ≥ k0 , ∂G∩A(k)

cm a−1 where c7 = / ∗ c1 M (1 − θ). Let us choose c5 ε and



θ−1 θ

 ⇒

= c7 ε

ε=

c5 c7

θ (6.3.12)

 m ≥ 2(c4 + c7 ε) = 2(c4 + c1−θ cθ5 ); k0q m 7 κ q+m−1 k0 ≥ 2(c4 + 2c7 ε) = 2c4 + 4c1−θ cθ5 7

=⇒

 

κ  mq  1   1 1 1−θ θ q+m−1 θ q k0 ≥ max 1; 2 q . · c4 + c1−θ c ; 2c + 4c c 4 5 5 7 7 m

(6.3.13)

Thus, from (6.3.10)–(6.3.13) we obtain the inequality  (a|∇wk |m + a|wk |m ) dx A(k)

≤

c6 c4 + c1−θ cθ5 7

 

 Σ0 ∩A(k)

A(k)



 |h(x, 0)|ds +

B0 (x)dx +

|g(x, 0)|ds ∂G∩A(k)

for all k ≥ k0 . Applying the Sobolev embedding Theorems 1.15 and 1.20, we have ⎛ ⎜ ⎝

⎞



# ⎟ |wk |m dx⎠

m m#

⎛ ⎜ +⎝

∗ ⎟ |wk |j ds⎠

Σ0 ∩A(k)

A(k)

 ≤/ c

⎞ jm∗



(a|∇wk |m + a|wk |m ) dx ≤

A(k)



 |h(x, 0)|ds +

+ Σ0 ∩A(k)

c/c6 c4 + c1−θ cθ5 7 

|g(x, 0)|ds , ∂G∩A(k)

⎛



⎜ +⎝ 

⎞ jm∗ ∗ ⎟ |wk |j ds⎠

∂G∩A(k)

B0 (x)dx

(6.3.14)

A(k)

j∗ =

m(n − 1) , ∀k ≥ k0 . n−m

6.3. Maximum principle

147

At last, by the Hölder inequality, we get  B0 (x)dx ≤ B0 (x)Ls (G) meas

1− 1s

A(k);

A(k)

 |h(x, 0)|ds ≤ h(x, 0)L

Σ0 ∩A(k)

(Σ0 )

j j−1

 1 meas(Σ0 ∩ A(k)) j ;



|g(x, 0)|ds ≤ g(x, 0)L

(∂G)

j j−1

 1 · meas(∂G ∩ A(k)) j ,

1 n−m < ≤ 1. n−1 j

∂G∩A(k)

Next from (6.3.14) it follows that ⎛ ⎞ m# ⎛ m   # ⎜ ⎟ ⎜ m |wk | dx⎠ +⎝ ⎝ A(k)

≤

⎞ jm∗ ∗ ⎟ |wk |j ds⎠

Σ0 ∩A(k)

-

⎛ ⎜ +⎝



⎞ jm∗ ∗ ⎟ |wk |j ds⎠

∂G∩A(k)

1 / cc6 B0 (x)Ls (G) meas1− s A(k) 1−θ θ c 4 + c7 c 5  1 + h(x, 0)L j (Σ0 ) · meas(Σ0 ∩ A(k)) j j−1

+ g(x, 0)L

j j−1

(∂G)

(6.3.15)

 1 . · meas(∂G ∩ A(k)) j ,

< 1j ≤ 1, j ∗ = m(n−1) n−m , for all k ≥ k0 . Let now l > k > k0 . By (1.8.8) and 1 the definition of the function wk (x), we have |wk | ≥ m (|u| − k)+ and therefore n−m n−1



m#

|wk |

 dx ≥

A(l)





j∗

|wk | ds ≥ Σ0 ∩A(l)



j∗

l−k m

l−k m



|wk | ds ≥

m#

l−k m

· measA(l); j ∗ · meas (Σ0 ∩ A(l)); j ∗ · meas (∂G ∩ A(l)).

∂G∩A(l)

From (6.3.15) it follows:  m#  m# meas A(l) + meas(Σ0 ∩ A(l)) j∗ + meas(∂G ∩ A(l)) j∗ ⎛ ⎞ mj∗# ⎛ m#      # ∗ m ⎜ ⎟ ⎜ ≤ |wk |m dx + ⎝ |wk |j ds⎠ + ⎝ l−k A(k)

Σ0 ∩A(k)

⎞ mj∗# ⎟ |wk | ds⎠

∂G∩A(k)

j∗



148

Chapter 6. Transmission problem for strong quasi-linear elliptic equations 

≤ c8  ×

m l−k

meas

m#  · B0 (x)Ls (G) + h(x, 0)L

m# m

j j−1

(Σ0 )

+ |g(x, 0)L

 mm# j j−1

(∂G)

 m# 1 m# 1 (1− 1s ) A(k) + meas(Σ ∩ A(k)) m · j + meas(∂G ∩ A(k)) m · j , 0

n−m 1 < ≤ 1, n−1 j

j∗ =

m(n − 1) , ∀l > k ≥ k0 . n−m

(6.3.16)

Let us introduce m#  m#  ψ(k) = meas A(k) + meas(Σ0 ∩ A(k)) j∗ + meas(∂G ∩ A(k)) j∗ . Then from (6.3.16) it follows that + m# *  j∗ 1 m# 1 m · 1− ( ) s + [ψ(k)] m j , ∀l > k ≥ k0 ; · [ψ(k)] m ψ(l) ≤ c9 l−k

(6.3.17)

n−m 1 m(n − 1) < ≤ 1, j ∗ = . (6.3.18) n−1 j n−m     j∗ 1 m# 1 > 1. Then from By (6.3.18), we observe that γ = min m 1 − s , m · j m# =

mn , n−m

s>

n > 1, m

c10 ψ γ (k), γ > 1; ∀l > k ≥ k0 and therefore, (l − k)m# because of the Stampacchia Lemma, we have that ψ(k0 + δ) = 0 with δ depending only on quantities given in the formulation of Theorem 6.5. This fact means that |u(x)| < k0 + δ for almost all x ∈ G. Theorem 6.5 is proved.  (6.3.17)–(6.3.18) we get ψ(l) ≤

6.4 Local estimate at the boundary In this section we derive the local boundedness (near the conical point) of the weak solution of problem (QL). Theorem 6.6. Let u(x) be a weak solution of the problem (QL) and assumptions 1), 2), 3a), 4), 5), 6), (6.1.5) be satisfied. In addition, let h(x, 0) ∈ L∞ (Σ0 ), g(x, 0) ∈ L∞ (∂G). Then there exists a constant C > 0 depending only on n, m, p, t, q, κ, a∗ , p p a∗ , m∗ , m∗ , d, a0 (x) mp , G , α(x) m−1 , G , b0 (x) m , G such that the inequality  ς mς(p−n) sup |u(x)| ≤ C −nς/t u ςt ,G0 +  p(m−1+ς) · a0 (x) m−1+ς p , G

x∈Gκ 0

m

ς m−1  p m−1 , G0

0

ς

n n m (6.4.1) + ς (1− p ) α(x) + ς (1− p ) m−1 b0 (x) m−1  p m , G0 

 ς ς m−1 m−1 , p>n>m + ς g(x, 0)∞, + h(x, 0)∞, Γ Σ 0

0

6.4. Local estimate at the boundary

149

holds for any t > 0, κ ∈ (0, 1),  ∈ (0, d) and ς =

m−1 q+m−1 .

Proof. We apply the Moser iteration method. Let us introduce the change of function (6.1.1). By the assumption 6), the identity (II) takes the form (see §1):    γ(ω) Ai (x, vx )ηxi + B(x, v, vx )η dx + v|v|m−2 η(x)ds rm−1 G



+ Σ0

σ(ω) v|v|m−2 η(x)ds = rm−1



∂G



G(x, v)η(x)ds +

< (II)

H(x, v)η(x)ds Σ0

∂G

with coefficients that are determined by (6.1.3). < and make the At first, let t ≥ m. We consider the integral identity (II) coordinate transformation x = x . Let G be the image of G, ∂G be the image of ∂G, Σ0 be the image of Σ0 , and z(x ) = v(x ). We have dx = n dx , ds = n−1 ds . < means Then (II) 

!

" Ai (x , −1 zx )ηxi + B(x , z, −1 zx )η(x ) dx

G

+



1 m−1 

=

Σ0

σ(ω) 1 z|z|m−2 η(x )ds + m−1 |x |m−1 

G(x , z)η(x )ds +



 ∂G

γ(ω) z|z|m−2 η(x )ds |x |m−1

H(x , z)η(x )ds

(II)

Σ0

∂G

1 for all η(x ) ∈ C0 (G ) ∩ Vm,0 (G ). Let us define the quantity k by m   m−1+ς 1  · a0 (x ) m−1+ς k = k() = p 1 ,G 0 m ς    1  m−1

1   p 1 · α(x ) m−1 + + b (x ) p 1 0 m ,G0 m−1 ,G0 ς  1 1 m−1 m−1  + G(x , 0)∞,Γ 1 + H(x , 0)∞,Σ1 0

and set   m−1

z(x ) = |z(x )| + k.

0

(6.4.2)

(6.4.3)

· z(x )z t−m (x )ζ m (|x |), where ζ(|x |) ∈ C∞ We choose η(x ) = ς 0 ([0, 1]) is a non-negative function to be further specified, as the test function in the integral identity (II) . By the chain and product rules, η is a valid test function in (II) and also

150

Chapter 6. Transmission problem for strong quasi-linear elliptic equations

ηxi =

  m−1   |z| · 1 + (t − m) z t−m zxi ζ m (|x |) ς z  m−1  +m · z(x )z t−m (x )ζ m−1 ζxi . ς

Then taking into account that 0 ≤ |z| ≤ z and t ≥ m, by virtue of assumptions 1) , 2) , 3a) , 4) and 5) we obtain  az

t−m

  amz t−m+1 |∇ z|m−1 ζ m−1 (|x |)|∇ ζ| |∇ z| ζ (|x |)dx ≤ 

m m

G10





G10

+ aμςz t−m |∇ z|m ζ m + (t − m + 1)

 m  · a0 (x )z t−m+1−ς (x )ζ m (|x |) ς

 m−1   m   t−m+1 m−1   t−m+1 m  +m · |α(x )| · |∇ ζ|z ζ (|x |) + m−1 b0 (x )z ζ (|x |) dx ς ς  m−1   + · z(x )G(x , z)z t−m (x )ζ m (|x |)ds ς Γ10

 m−1   + · z(x )H(x , z)z t−m (x )ζ m (|x |)ds . ς

(6.4.4)

Σ10

We estimate every term by the Young inequality with regard to z ≥ k:  t

 m−1 amz t−m+1 |∇ z|m−1 ζ m−1 |∇ ζ| = am z (t−m) m |∇ z|m−1 ζ m−1 · z m |∇ ζ| ≤ ε(m − 1)az t−m |∇ z|m ζ m + ε1−m az t |∇ ζ|m , ∀ε > 0;  m  m 1    t−m+1−ς m · a0 (x )z ζ ≤ m−1+ς · a0 (x )z t ζ m ; ς k ς  m−1  m · |α(x )| · |∇ ζ|z t−m+1 ζ m−1 ς    m−1

t  m−1    (t−m) m−1 m ζ = m · z m |∇ ζ| × · |α(x )|z ς  m m  ≤ (m − 1) · |α(x )| m−1 · z t−m ζ m ς  m m  + z t |∇ ζ|m ≤ z t |∇ ζ|m + (m − 1) · |α(x )| m−1 · z t ζ m ; kς m    m  m  t−m+1 m t m b0 (x ) b (x )z ζ = z ζ ≤ b0 (x ) · z t ζ m . 0 ς m−1 ς m−1 (kς)m−1 z m−1

6.4. Local estimate at the boundary

151

Further, z · G(x , z) = z · G(x , 0) + z · 

2



because of

0



= z · G(x , 0) + z · ∂G(x ,τ z) ∂(τ z)

1

d G(x , τ z)dτ dτ 1

0

∂G(x , τ z) dτ ≤ z · |G(x , 0)|, ∂(τ z)

≤ 0. Therefore just as above

  m−1  z · G(x , z)z t−m ζ m ds ς

Γ10



≤

ζ

m

·z

Γ10

t m

   m−1   (t−m) m−1 m ds · |G(x , 0)| ·z ς

 

≤



z + |G(x , 0)| t

Γ10

m m−1

 m   t−m ζ m ds z ς

⎛ ⎞m 1    m−1   · G(x , 0) ∞ ⎝ ⎠ 1+ ≤ z t ζ m ds . kς Γ10

In the same way   m−1  z · H(x , z)zt−m (x )ζ m (|x |)ds ς

Σ10

  ≤ Σ10

⎛

⎞m 1  m−1   · H(x , 0) ∞ ⎠ 1+⎝ z t ζ m ds . kς

Now from (6.4.4) it follows that  (1 − ςμ) az t−m |∇ z|m ζ m dx G10

  ε(m − 1)az t−m |∇ z|m ζ m + ε1−m az t |∇ ζ|m ≤ G10

 m  t m + z t |∇ ζ|m + m−1+ς · a0 (x )z t ζ m + m z t ζ m + b0 (x ) · z t ζ m k ς (kς)m−1   m m  + (m − 1) · |α(x )| m−1 · z t ζ m dx kς

152

Chapter 6. Transmission problem for strong quasi-linear elliptic equations

  m

  m m  m−1 m−1  · G(x , 0)∞,Γ + H(x , 0) + 1+ 1 ∞,Σ10 0 kς

z t ζ m ds (6.4.5)

Γ10 ∪Σ10

for all ε > 0. By inequality (1.5.11) 





z ζ ds ≤ / c t m

Γ10 ∪Σ10

 t m  z ζ + |∇ (z t ζ m )| dx .

G10

Again, by the Young inequality, we have |∇ (z t ζ m )| ≤ tz t−1 |∇ z|ζ m + mz t ζ m−1 |∇ ζ|

t−m  m−1  = t z m |∇ z| · z t m ζ m + mz t ζ m−1 |∇ ζ| ≤

1 tδ t−m  m m m − 1 1−m δ z |∇ z| ζ + t z t ζ m + (|∇ ζ|m + (m − 1)ζ m ) z t , ∀δ > 0. m m

From above we get  z t ζ m ds Γ10 ∪Σ10

 

≤ G10

(6.4.6)

 1 tδ t−m  m m m − 1 1−m δ z |∇ z| ζ + t z t ζ m + (|∇ ζ|m + mζ m ) z t dx m m

for any δ > 0. Thus, from (6.4.5)–(6.4.6) according to the definition (6.4.2) of the (1−ςμ) 1−ςμ number k and choosing ε = 4(m−1) , δ = ma∗8/ , we obtain ct 1 − ςμ 2



az t−m |∇ z|m ζ m dx

G10

 m  ≤ c1 1 + t m−1



G10

(6.4.7)

(ζ m + |∇ ζ|m ) z t dx + c2 t



F (x )z t ζ m dx ,

G10

c), c2 = const (m) and where c1 = const (m, μ, ς, a∗ , /  m  m m a0 (x )   m F (x ) = m−1+ς + |α(x )| m−1 · + b0 (x ) · . k ς kς (kς)m−1

(6.4.8)

Let us define the function  m

m t 1 t  t −1 m m w(x ) = a z =⇒ z = a wm , |∇ w|m = az t−m |∇ z|m . t m (6.4.9)

6.4. Local estimate at the boundary

153

Then, by virtue of 1 − ςμ > 0, (6.4.7) takes the form 

|∇ w|m ζ m dx

(6.4.10)

G10

 m  ≤ C1 tm 1 + t m−1



(ζ m + |∇ ζ|m ) wm dx + C2 tm+1

G10



F (x )wm ζ m dx .

G10

The required iteration process can now be developed from (6.4.10). For this we apply the Hölder inequality 

|F (x )| · wm (x )ζ m (x )dx ≤ F p/m,G10 · wζmmp

1 p−m ,G0

,

p > m,

(6.4.11)

G10

the interpolation inequality for Lp -norms n

n−p ζw mp mn ζw p−m ,G10 ≤ εζw n−m ,G10 + ε m,G10 ,

p > n > m, ∀ε > 0,

(6.4.12)

and the Sobolev embedding inequality ζwmmn

1 n−m ,G0

≤ C∗



{(|∇ ζ|m + ζ m ) |w|m + ζ m |∇ w|m } dx ,

n > m, (6.4.13)

G10

where C ∗ depends only on n, m and the domain G. From (6.4.10)–(6.4.13), we derive 1

m−1 ) · (ζ + |∇ ζ|)w mn ζw n−m ,G10 ≤ c3 t(1 + t m,G10

 m+1 n 1/m n−p ζw mn + c4 t m F p/m,G1 εwζ n−m ,G10 + ε m,G10 , p > n > m, ∀ε > 0. 0

1/m

By (6.4.2) and (6.4.8), F p/m,G1 ≤ c(p, m). If we choose ε = 0 we obtain mn ζw n−m ,G10 ≤ C(1 + t)

p m+1 m · p−n

(ζ + |∇ ζ|)wm,G10 ,

−1 1 − m+1 m F  m , 2c4 t p/m,G10

mn > p > n > m, (6.4.14)

1/m

c, n, p, F p/m,G1 and it is independent of t. where C depends only on m, μ, ς, a∗ , / 0 Let us recall the definition of w by (6.4.9) and t ≥ m > 1 by (6.4.14). Finally, we establish the inequality mn ζ · z t/m  n−m ,G10 ≤ Ct

p m+1 m · p−n

(ζ + |∇ ζ|) · z t/m m,G10 ,

mn > p > n > m. (6.4.15) This inequality can now be iterated to yield the desired estimate.

154

Chapter 6. Transmission problem for strong quasi-linear elliptic equations κ+(1−κ)2−j

Let us define sets G(j) ≡ G0 , j = 0, 1, 2, . . . for all κ ∈ (0, 1). It κ   is easy to see that G0 ≡ G(∞) ⊂ · · · ⊂ G(j+1) ⊂ G(j) ⊂ · · · ⊂ G(0) ≡ G10 . Let us introduce also the sequence of cut-off functions ζj (x ) ∈ C∞ (G(j) ) such that 0 ≤ ζj (x ) ≤ 1 in G(j)

and ζj (x ) ≡ 1 in G(j+1) ,

ζj (x ) ≡ 0

for |x | > κ + 2−j (1 − κ); |∇ ζj | ≤

2j+1 1−κ

κ + 2−j−1 (1 − κ) < |x | < κ + 2−j (1 − κ)

for



and the number sequence tj = t 

n n−m

j , j = 0, 1, 2, . . . . Now we can rewrite the

inequality (6.4.15) replacing ζ(|x |) by ζj (x ) and t by tj . Then taking the tj -th root, we obtain m/tj mj  (m+1)p 1 C ·  · 2 tj · (tj ) p−n tj ztj ,G(j) . ztj+1 ,G(j+1) ≤ 1−κ After iteration, we find  ztj+1 ,G(j+1) ≤ ∞ 

The series series

∞  j=0

j=0 1 tj

=

j tj 1 t

C 1−κ

m

∞ 

1 t j=0 j

·2

m



∞ 

j t j=0 j

·

nt n−m

 (m+1)p p−n

∞  j=0

1 tj

· zt,G10 .

is convergent according to the d’Alembert ratio test, while the ∞    n−m j j=0

n

=

n mt

as a geometric series. Hence we get

ztj+1 ,G(j+1) ≤

C zt,G10 . (1 − κ)n/t

Consequently, letting j → ∞, we have sup z(x ) ≤ x ∈Gκ 0

C zt,G10 . (1−κ)n/t

Thus, by

(6.4.3) and (6.4.2), we obtain sup |z(x )|

x ∈Gκ 0

m   m−1+ς    1 1   m−1+ς  · α(x ) m−1 ≤ C z + · a0 (x ) p ,G1 + p 1 0 m m−1 ,G0 ς ς  1  m−1

1 1 m−1 m−1  . + b0 (x ) mp ,G10 + G(x , 0)∞,Γ 1 + H(x , 0)∞,Σ1



t,G10

0

Returning to the variables x, v we obtain the estimate . sup |v(x)| ≤ C −n/t vt,G0 + K() , t ≥ m, x∈Gκ 0

0

(6.4.16)

6.5. Integral estimates

155

where 1

m(p−n)

1

+ (1− p ) m−1 b0 (x) m−1 (6.4.17) K() =  p(m−1+ς) · a0 (x) m−1+ς   p p m ,G0 m ,G0   1 1 1 n m−1 m−1 , + 1− p α(x) m−1 +  g(x, 0)∞,Γ  + h(x, 0) p ,G ∞,Σ m−1

n

m

0

0

0

p > n > m and the constant C depends on m, q, κ. Let now 0 < t < m. We put in (6.4.16) t = m: . sup |v(x)| ≤ C −n/m vm,G0 + K() . x∈Gκ 0

Using the Young inequality with s = ⎛ n n ⎜ C− m vm,G0 = C− m ⎝



m t

and s =

m m−t

(6.4.18)

we can write

⎞1/m ⎟ |v|t · |v|m−t ⎠

(6.4.19)

G 0

1−t/m

m−t n n t/m · C− m vt,G ≤ ≤ sup |v(x)| sup |v(x)| + C1 − t vt,G0 .   0 m G0 G0 Let us define the function ψ(s) = sup |v(x)|. Then from (6.4.18)–(6.4.19) it follows x∈Gs0

that ψ(κ) ≤

n m−t ψ() + C1 − t vt,G0 + CK(), m

κ ∈ (0, 1).

(6.4.20)

t Further, we apply Lemma 1.24: setting r = κ, R = , δ = 1 − m , α = nt , A = C1 vt,G0 , B = CK() from (6.4.20) we obtain the validity of estimate (6.4.16) in the case 0 < t < m. Returning to the variable u by (6.1.1), from (6.4.16)–(6.4.17) we get the estimate (6.4.1). Thus, the proof of Theorem 6.6 is complete. 

6.5 Integral estimates Now we will derive a global estimate for the Dirichlet integral. Theorem 6.7. Let u(x) be a weak solution of the problem (QL) and assumptions 1), 3a), 4) and 5) with ν0 > 0 be satisfied. In addition, we assume that M0 = max |u(x)| is known. If a0 (x) ∈ L1 (G), b0 (x) ∈ L1 (G), h(x, 0) ∈ L1 (Σ0 ), g(x, 0) ∈ x∈G

L1 (∂G), then the inequality    qm m m σ(ω) m−1 γ(ω) m−1 (q+m−1) (q+m−1) a|u| m−1 |∇u|m dx + |u| ds + |u| ds rm−1 rm−1 G

Σ0

∂G

156

Chapter 6. Transmission problem for strong quasi-linear elliptic equations ≤ c(M0 , a∗ , ν0 , q, m, μ, n, meas G)   

 (a0 (x) + b0 (x)) dx + |h(x, 0)|ds + |g(x, 0)|ds × Σ0

G

(6.5.1)

∂G

holds. Proof. Using the function change (6.1.1) and putting η(x) = v(x) in the integral < for v(x) (see §6.4), we have identity (II)   G

 Ai (x, vx )vxi + B(x, v, vx )v dx +



∂G

 G(x, v)v(x)ds +

=

γ(ω) m |v| ds + rm−1

 Σ0

σ(ω) m |v| ds rm−1

H(x, v)v(x)ds. Σ0

∂G

According to assumptions 1) , 3a) , 4) , since ς m−1 (1 − ςμ) < 1 by (6.1.1), we obtain     σ(ω) m γ(ω) m ς m−1 (1 − ςμ) a|∇v|m dx + |v| ds + |v| ds rm−1 rm−1 G Σ0 ∂G     1 1−ς |v| a0 (x)dx + |h(x, 0)| · |v|ds + |g(x, 0)| · |v|ds. ≤ |v|b0 (x)dx + ς G

Σ0

G

∂G 1

From M0 = sup |u(x)|, by the change (6.1.1), it follows that |v(x)| ≤ M0ς . ThereG

fore we get 

 a|∇v|m dx + Σ0

G

σ(ω) m |v| ds + rm−1



γ(ω) m |v| ds rm−1

∂G

≤ c(M0 , m, q, μ, meas G)   

 (a0 (x) + b0 (x)) dx + |h(x, 0)|ds + |g(x, 0)|ds . × G

Σ0

(6.5.2)

∂G

Returning to the function u(x), by (6.1.1), we obtain the desired estimate (6.5.1).  Further, we establish a local integral weighted estimate. Theorem 6.8. Let u(x) be a weak solution of the problem (QL) and ϑ(m) be the smallest positive eigenvalue of (N EV P ). Let us assume that M0 = max |u(x)| x∈G

is known and assumptions of Theorem 6.7 and assumption 7) are satisfied. In

6.5. Integral estimates

157

addition, assume that there exists a real number ks ≥ 0 defined by (6.1.4). Then there exist d ∈ (0, 1) and a constant c > 0 independent of u and depending only on m, n, s, q, d, ϑ(m), k1 , ks , meas Ω and M0 such that, for any  ∈ (0, d),    qm m m σ(ω) m−1 γ(ω) m−1 (q+m−1) (q+m−1) a|u| m−1 |∇u|m dx + |u| ds + |u| ds rm−1 rm−1 G 0

Σ 0

Γ 0

≤ cψ (), m

(6.5.3)

where ψ() is defined by (6.1.6) with (2.4.3). Proof. We perform the change (6.1.1). By virtue of Theorem 6.7, we have that    σ(ω) m γ(ω) m m |v| ds + |v| ds < ∞,  ∈ (0, d). (6.5.4) V () = a|∇v| dx + rm−1 rm−1 G 0

Σ 0

Γ 0

Therefore we can set η(x) = v(x) in the identity (6.1.2):   G 0

 Ai (x, vx )vxi + B(x, v, vx )v(x) dx + 

 Ai (x, vx ) cos(r, xi ) · v(x)dΩ +

= Ω

Σ 0

σ(ω) m |v| ds + rm−1

 Γ 0

γ(ω) m |v| ds rm−1



G(x, v) · v(x)ds + Γ 0

H(x, v) · v(x)ds. Σ 0

(6.5.5) By assumptions 1) , 3a) , 4) , 5), 7) and μς < 1, we get   1 |v|1−ς a0 (x)dx (1 − ςμ)ς m−1 V () ≤ |v|b0 (x)dx + ς G 0

 + ς m−1 A()

Σ 0

a|∇v|m−2 · v

Ω



|h(x, 0)| · |v|ds +

+



a|v| · |∇v|m−1 dΩ + ς m−1

Ω



G 0

∂v dΩ ∂r

|g(x, 0)| · |v|ds.

(6.5.6)

Γ 0

By Lemma 2.11, from (6.5.6) it follows that (1 − ςμ)V () ≤ 1 + m ς



1−ς

|v| G 0

Ξ(m) 1 m

mϑ (m)

V  () + A()

a0 (x)dx + ς

1−m



 a|v| · |∇v|m−1 dΩ

Ω

|v|b0 (x)dx + ς G 0

1−m

 |h(x, 0)| · |v|ds Σ 0

158

Chapter 6. Transmission problem for strong quasi-linear elliptic equations +ς

1−m

 |g(x, 0)| · |v|ds.

(6.5.7)

Γ 0

We estimate every term on the right-hand side in (6.5.7). By the Hölder inequality for integrals, ⎛



⎜ a|v| · |∇v|m−1 dΩ ≤ ⎝

Ω



⎞ m1 ⎛ ⎜ ·⎝

⎟ a|v|m dΩ ⎠

Ω



⎞ m−1 m ⎟ a|∇v|m dΩ ⎠

.

(6.5.8)

Ω

Because of the inequality |∇ω v| ≤ |∇v|, the inequality (W )m and formula (2.4.4), we have      n−1 a|v|m dΩ ≤ a|∇ω v|m dΩ + σ(ω)|v|m dσ + γ(ω)|v|m dσ ϑ Ω

Ω

≤

σ0

∂Ω

m

 V  (). ϑ

(6.5.9)

From (6.5.8)–(6.5.9) it follows that   a|v| · |∇v|m−1 dΩ ≤ 1 V  (). ϑm

(6.5.10)

Ω

By the Young inequality and (2.2.2),    1 (1−ς)(1−ς−m) 1−ς  (1−ς)(m+ς−1) 1 1−ς m m r · r |v| a0 (x)dx = |v| a0 (x) dx ς ς G 0

 

≤ G 0

≤

G 0

m 1 − ς 1−ς−m m m + ς − 1 1−ς r r |a0 (x)| m+ς−1 |v| + mς mς

m+ς −1 1−ς 1−ς V () + a∗ mςϑ(m) mς

 |v|b0 (x)dx = G 0

 

≤ G 0





m

G 0

G 0 1 m 1 −1 m m − 1 m−1 r |v| + r |b0 (x)| m−1 m m



1

(6.5.11)

 dx m

r m−1 |b0 (x)| m−1 dx G 0

dx

r1−ς |a0 (x)| m+ς−1 dx;

 1  1 r− m |v| r m b0 (x) dx

m 1 m−1 V () + ≤ a∗ mϑ(m) m−1



6.5. Integral estimates 

159



1 |g(x, 0)| · |v|ds ≤ m

|h(x, 0)| · |v|ds + Σ 0

m−1 m

+

 + Γ 0

≤

Γ 0





m

|v| ds + Σ 0

γ(ω) m m−1 |v| ds + rm−1 m

m−1 m−1 V () + mν0 m



|h(x, 0)|

m m−1

Σ 0



|v| ds

≤

 Σ 0

|g(x, 0)|

ds +

|h(x, 0)| m−1 ds + Σ 0

m−1 mν0





m

m

Γ 0



|g(x, 0)| m−1 ds Γ 0





 m

m

|h(x, 0)| m−1 ds + Σ 0



σ(ω) m |v| ds rm−1 

m m−1

ds

Γ 0

 m |g(x, 0)| m−1 ds ,

(6.5.12)

Γ 0

by assumption 5) and (6.5.4). Thus, from (6.5.7)–(6.5.12) it follows that

mς 1−m + m−1 +

m−1 mς m−1

 r

1 m−1



Ξ(m)

(1 − ςμ) (1 − δ()) V () ≤

1 m

 / 1 + A() V  ()

mϑ (m)  m m+ς −1 |b0 (x)| dx + r1−ς |a0 (x)| m+ς−1 dx m mς m m−1

G 0





m

|h(x, 0)| m−1 ds + Σ 0

G 0

 m |g(x, 0)| m−1 ds ,

where δ() = const (m, μ, a∗ , q, ν0 , ϑ(m)) · m−1 + m Ξ(m) A().

(6.5.13)

Γ 0 q q q+m−1 q+m−1 



/ and A() =

We observe that  0

δ() d < ∞, 

 0

/ A() d < ∞. 

(6.5.14)

Thus, from (6.5.13), by virtue of assumption (6.1.4), we derive the Cauchy problem (CP ) from §1.7 with 1

P() =

1 1 − δ() mϑ m (m) · (1 − ςμ), Q() = kms−1 , · /  1 + A() Ξ(m) k = const (ks , m, q, ϑ(m)).

Now we have: 

 m

a|∇v| dx +

V (d) = Gd 0

Σd 0

σ(ω) m |v| ds + rm−1

 Γd 0

γ(ω) m |v| ds rm−1

N () ≡ 0; (6.5.15)

160

Chapter 6. Transmission problem for strong quasi-linear elliptic equations  ≤

 m

a|∇v| dx + Σ0

G

σ(ω) m |v| ds + rm−1



γ(ω) m |v| ds rm−1

∂G

≤ c(M0 , m, q, μ, meas G) · V0 ,    V0 ≡ (a0 (x) + b0 (x)) dx + |h(x, 0)|ds + |g(x, 0)|ds, Σ0

G

∂G

by virtue of (6.5.2). The solution of problem (CP ) is determined by (1.7.1) from Theorem 1.21. Direct calculations give: τ

1

mϑ m (m)(1 − ςμ) P(ξ)dξ = − Ξ(m)

τ

1 1 − δ(ξ) · dξ / ξ 1 + A(ξ)     τ 1 / 1 δ(ξ) + A(ξ) mϑ m (m)(1 − ςμ) dξ · 1− =− / Ξ(m) ξ 1 + A(ξ)

−



⎛

1 m

mϑ (m)(1 − ςμ) ⎝  ln + Ξ(m) τ ⎞ ⎛ τ  exp ⎝− P(ξ)dξ ⎠ ≤

 1

≤ d

(m)(1−ςμ)

  mϑ m Ξ(m)

τ ⎛ Q(τ ) exp ⎝−



τ

d 0

⎛

⎞ / δ(ξ) + A(ξ) ⎠ dξ ξ

1

mϑ m (m)(1 − ςμ) · exp ⎝ Ξ(m) ⎞

P(ξ)dξ ⎠ dτ ≤ kC1 

d 0

=⇒

⎞ / δ(ξ) + A(ξ) dξ ⎠ ; ξ

1 mϑ m (m)(1−ςμ) Ξ(m)



d

τ ms−1 τ −

1 mϑ m (m)(1−ςμ) Ξ(m)



1 mϑ m (m)(1−ςμ) Ξ(m)

= kC1  ⎧ ⎛ ⎛ ⎞ ⎞ 1 1 ϑ m (m)(1−ςμ) ⎠ ϑ m (m)(1−ςμ) ⎠ ⎪ m⎝ s− m⎝ s− ⎪ Ξ(m) Ξ(m) ⎪ ⎪ − ⎨d   , s = 1 m (m)(1−ςμ) ϑ × m s− Ξ(m) ⎪ ⎪ ⎪ ⎪ ⎩ln d , s=    1 d / m (m)(1−ςμ)  δ(ξ)+A(ξ) dξ . where C1 = exp mϑ Ξ(m) ξ ⎛ V0 · exp ⎝−

d 

⎞

1

ϑ m (m)(1−ςμ) ; Ξ(m) 1

ϑ m (m)(1−ςμ) , Ξ(m)

0

P(ξ)dξ ⎠ ≤ V0 C1

1

(m)(1−ςμ)

  mϑ m Ξ(m)

d

=⇒

dτ

6.6. The power modulus of continuity at the conical point for weak solutions161

V () ≤ cC1 (V0 + k) ·

⎧ 1 mϑ m (m)(1−ςμ) ⎪ ⎪ Ξ(m) ⎪  , ⎪ ⎨ 1 mϑ m (m)(1−ςμ) Ξ(m)

 ⎪ ⎪ ⎪ ⎪ ⎩ ms  ,

1

s>

ln d , s = s
0, we get that u(0) = 0. 

Chapter 7

Best possible estimates of solutions to the transmission problem for a quasi-linear elliptic divergence second-order equation in a domain with a boundary edge 7.1 Introduction. Assumptions This chapter is devoted to the estimate of weak solutions to the transmission problem for elliptic quasi-linear degenerate second-order equations. We investigate the behavior of weak solutions of the transmission problem with Robin boundary condition for quasi-linear second-order elliptic equations with triple degeneracy and singularity in the coefficients in a neighborhood of the boundary edge. Let G ⊂ Rn , n ≥ 3 be a domain bounded by an (n − 1)-dimensional manifold ∂G and let Γ− , Γ+ be open nonempty disjoint submanifolds such that ∂G = Γ− ∪Γ+ , where Γ− ∩Γ+ is a smooth (n−2)-dimensional submanifold that contains an edge Γ0 ⊆ Γ− ∩ Γ+ . For x = (x1 , . . . , xn ) let,us introduce the cylindrical coordinates (r, ω, x ), where x = (x3 , . . . , xn ), r = x21 + x22 , ω = arctan xx21 . We assume that G is divided into two subdomains G− and G+ by an (n − 1) open n−2    dimensional hyperplane {(r, 0, x ) x ∈ R , r > 0}, thus G = G+ ∪ G− . We set  Σ0 = G ∩ {(r, 0, x ) x ∈ Rn−2 , r > 0} and assume that Γ0 ⊂ Σ0 .

M. Borsuk, Transmission Problems for Elliptic Second-Order Equations in Non-Smooth Domains, 171 Frontiers in Mathematics, DOI 10.1007/978-3-0346-0477-2_8, © Springer Basel AG 2010

172

Chapter 7. Estimates of Solutions to the Transmission Problem

We derive the exact estimate of the weak solution boundary edge for the problem ⎧ d i − dxi a (x, u, ∇u) + b(x, u, ∇u) + a0 c(x, u) = f (x), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ [u]Σ0 = 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎨  S[u] ≡ aai (x, u, ∇u)ni Σ0 + β0 rτ −m+1 u|u|q+m−2 ⎪ ⎪ ⎪ = h(x, u), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪B[u] ≡ aai (x, u, ∇u)n + γ(ω)rτ −m+1 u|u|q+m−2 ⎪ ⎪ i ⎪ ⎩ = g(x, u),

in a neighborhood of the x ∈ G \ Σ0 ; x ∈ Σ0 ;

x ∈ Σ0 ;

x ∈ ∂G \ {Σ0 ∪ Γ0 }, (TDQL) where q ≥ 0, m > 1, a± > 0, a0 ≥ 0, β0 > 0, τ ≥ m − 2 are given numbers. For a sufficiently small number d > 0 we also define the sets  Gd0 = G ∩ {(r, ω, x ) 0 < r < d, ω ∈ (−ω0 /2, ω0 /2), x ∈ Rn−2 }; Γd = Γd+ ∪ Γd− , where Γd± = Γ± ∩ Gd0 ⊂ ∂Gd0 ;  Ωd = G ∩ {(r, ω, x ) r = d, ω ∈ (−ω0 /2, ω0 /2), x ∈ Rn−2 }.

We will assume the following: • ∂G \ Γ0 is a smooth submanifold in Rn ; • there exists a number d > 0 such that Γd0 = {(0, 0, x )| |x | < d} ⊂ Γ0 is the straight edge with the center in the origin; • Gd0 is locally diffeomorphic to the dihedral cone  Dd = {(r, ω) 0 < r < d, ω ∈ (−ω0 /2, ω0 /2)} × Rn−2 ;

0 < ω0 < 2π;

thus we assume that Gd0 ⊂ G and, consequently, the domain G is a “wedge” in some vicinity of the edge; • ω |Γd± = ±ω0 /2. We denote by N1m,q,τ (G) the set of functions u(x) ∈ C0 (G) having first weak derivatives with the finite integral 



 arτ |u|q |∇u|m + a0 rτ −m |u|q+m dx < ∞,

G

q ≥ 0, m > 1, a > 0, a0 ≥ 0, τ ≥ m − 2.

(7.1.1)

Regarding the equation we assume that the following conditions are satisfied:

7.1. Introduction. Assumptions

173 x

Γ−

Γ0 Σ0

x1

x2

Γ+

Figure 12

ai (x, u, ξ), i = 1, . . . , n; b(x, u, ξ) are the Caratheodory functions G × R × Rn → R and continuously differentiable with respect to xi , u, ξi ; c(x, u) is the Caratheodory function G × R → R and continuously differentiable with respect to the variable u, h(x, u) is a function Σ0 × R → R continuously differentiable with respect to the variable u, while g(x, u) is a function ∂G × R → R continuously differentiable with respect to variable u, as well as f (x) ∈ L1 (G) and h(x, u(x)) ∈ L1 (Σ0 ), g(x, u(x)) ∈ L1 (∂G) ∀u(x) ∈ C0 (G), possessing the properties: 1)

∂ai (x, u, ξ) pi pj ≥ rτ |u|q |ξ|m−2 p2 , ∀p ∈ Rn \ {0}; ∂ξj

∂ai (x, u, ξ) ∂ai (x, u, ξ) =q ξj ; i = 1, . . . , n; ∂u ∂ξj   i  ∂a (x, u, ξ)   − rτ |u|q |ξ|m−4 δij |ξ|2 + (m − 2)ξi ξj  ≤ A(r)rτ |u|q |ξ|m−2 ; 3)  ∂ξj   i   ∂a (x, u, ξ) − τ rτ −2 |u|q |ξ|m−2 xi ξi  ≤ A(r)rτ −1 |u|q |ξ|m−1 ; 4)  ∂xi 2 3 n 3 2 |ai (x, u, ξ) − rτ |u|q |ξ|m−2 ξi | ≤ A(r)rτ |u|q |ξ|m−1 ; 5) 4 2) (m − 1)u

i=1

174

Chapter 7. Estimates of Solutions to the Transmission Problem

2 3 n   3  ∂b(x, u, ξ) 2 τ q−1 m 4   ≤ μ0 rτ |u|q−1 |ξ|m−1 , 0 ≤ 6) |b(x, u, ξ)| ≤ μ0 r |u| |ξ| ,   ∂ξi i=1

μ0
0; 7)

10) |f (x)| ≤ f1 rτ −m+(q+m−1)λ , |h(x, 0)| ≤ h1 rτ −m+1+(q+m−1)λ , |g(x, 0)| ≤ g1 rτ −m+1+(q+m−1)λ , where γm > 0, f1 ≥ 0, h1 ≥ 0, g1 ≥ 0 and A(r) is a nonnegative non-decreasing function continuous at zero with A(0) = 0. Definition 7.1. The function u(x) is called a weak solution of the problem (T DQL) provided that u(x) ∈ C0 (G) ∩ N1m,q,τ (G) and satisfies the integral identity 

! i " aa (x, u, ∇u)ηxi + ab(x, u, ux )η(x) + aa0 c(x, u)η(x) dx

G

 Σ0

 =



rτ −m+1 u|u|q+m−2 η(x)ds +

+ β0

∂G



af (x)η(x)dx + G

γ(ω)rτ −m+1 u|u|q+m−2 η(x)ds 

g(x, u)η(x)ds +

h(x, u)η(x)ds

(II)

Σ0

∂G

for all functions η(x) ∈ C0 (G) ∩ N1m,q,τ (G). Lemma 7.2. Let u(x) be a weak solution of (T DQL). For any function η(x) ∈ C0 (G) ∩ N1m,q,τ (G) the following equality holds for a.e.  ∈ (0, d):  G 0

. aai (x, u, ∇u)ηxi + ab(x, u, ux )η(x) + aa0 c(x, u)η(x) dx 



aai (x, u, ∇u) cos(r, xi )η(x)dΩ

af (x)η(x)dx +

= G 0



+ Γ

Ω



 g(x, u) − γ(ω)rτ −m+1 u|u|q+m−2 η(x)ds

7.1. Introduction. Assumptions  +

175



 h(x, u) − β0 rτ −m+1 u|u|q+m−2 η(x)ds.

(II)loc

Σ 0

Proof. The proof is similar to the proof of Lemma 3.2 Chapter 3; see also the proof of Lemma 5.2 in [14] (pp. 167–170).  We make the function change u = v|v|ς−1 with ς = (II)loc can be presented in the following form:

m−1 q+m−1 .

Then the identity

  aAi (x, vx )ηxi + aB(x, v, vx )η + aa0 C(x, v)η dx G 0

 γ(ω)r

+ 

τ −m+1

 m−2

v|v|

η(x)ds + β0

Γ



Ω



Γ

 H(x, v)η(x)ds +

+ Σ 0

Σ 0

G(x, v)η(x)ds

aAi (x, vx ) cos(r, xi )η(x)dΩ +

=

rτ −m+1 v|v|m−2 η(x)ds

af (x)η(x)dx

> loc (II)

G 0

for a.e.  ∈ (0, d), v(x) ∈ C0 (G) ∩ N1m,0,τ (G) and arbitrary η(x) ∈ C0 (G) ∩ N1m,0,τ (G), with Ai (x, vx ) ≡ ai (x, v|v|ς−1 , ς|v|ς−1 vx ), C(x, v) ≡ c(x, v|v|ς−1 ),

B(x, v, vx ) ≡ b(x, v|v|ς−1 , ς|v|ς−1 vx ),

G(x, v) ≡ g(x, v|v|ς−1 ),

H(x, v) ≡ h(x, v|v|ς−1 ).

The explicit independence of Ai from v is guaranteed by the assumption 2) (see in detail §6.1). Now our assumptions can be rewritten as follows: ∂Ai (x, η) pi pj ≥ ς m−1 rτ |η|m−2 p2 , ∀p ∈ Rn \ {0}; ∂ηj 

  ∂Ai (x, η) 3)  − ς m−1 rτ |η|m−4 δij |η|2 + (m − 2)ηi ηj  ≤ ς m−1 A(r)rτ |η|m−2 ; ∂ηj    ∂Ai (x, η)  4)  − τ ς m−1 rτ −2 |η|m−2 xi ηi  ≤ ς m−1 A(r)rτ −1 |η|m−1 ; ∂xi 2 3 n 3 2  4 5) |Ai (x, η) − ς m−1 rτ |η|m−2 ηi | ≤ ς m−1 A(r)rτ |η|m−1 ;

1)

i=1

176

Chapter 7. Estimates of Solutions to the Transmission Problem

2 3 n   3  ∂B(x, v, η) 2  m τ −1 m 4   ≤ μ0 ς m rτ |v|−1 |η|m−1 , 6) |B(x, v, η)| ≤ μ0 ς r v |η| ,   ∂ηi i=1

0 ≤ μ0
0 and A(r) is a non-negative continuous at zero function with A(0) = 0. 7)

7.2 The comparison principle We consider the second-order quasi-linear degenerate operator Q defined as Q(v, η) ≡

  Gd 0

Ai (x, vx )ηxi + B(x, v, vx )η + a0 C(x, v)η dx

 γ(ω)r

+ Γd

 r

+ β0  −

τ −m+1

τ −m+1

 m−2

v|v|

 m−2

v|v|

Σd 0



G(x, v)η(x)ds − Γd

η(x)ds −

H(x, v)η(x)ds Σd 0

η(x)ds −

Ai (x, vx ) cos(r, xi )η(x)dΩd Ωd

af (x)η(x)dx

(7.2.1)

Gd 0

for v(x) ∈ C0 (G) ∩ N1m,0,τ (G) and for all non-negative η belonging to C0 (G) ∩ N1m,0,τ (G) under the following assumptions: Ai (x, ξ), B(x, v, ξ), C(x, v), G(x, v), H(x, v) are Caratheodory functions in M = Gd0 × R × Rn , continuously differentiable with respect to the variables v, ξ, f (x) ∈ L1 (Gd0 ) and satisfy in M the following inequalities: ∂Ai (x, ξ) pi pj ≥ aγm rτ |ξ|m−2 |p|2 , ∀p ∈ Rn \ {0}; ∂ξj 2 3N  3 ∂B(x, v, ξ) 2  ≤ arτ |v|−1 |ξ|m−1 ; ∂B(x, v, ξ) ≥ arτ |v|−2 |ξ|m ; (ii) 4   ∂ξi ∂v i=1 (i)

7.3. Construction of the barrier function

177

∂C(x, v) ≥ aγm rτ −m |v|m−2 ; ∂v ∂H(x, v) ∂G(x, v) ≤ 0, ≤ 0, γ(ω) ≥ 0, β0 ≥ 0, a0 ≥ 0. (iv) ∂v ∂v (here: τ ≥ m − 2, m > 1, γm > 0 and a > 0). (iii)

Proposition 7.3. Let operator Q satisfy assumptions (i)–(iv) and d ≪ 1. We also assume that functions v, w ∈ C0 (Gd0 ) ∩ N1m,0,τ (Gd0 ) and satisfy the inequality Q(v, η) ≤ Q(w, η)

(7.2.2)

for all non-negative η ∈ C0 (Gd0 ) ∩ N1m,0,τ (Gd0 ) and also the inequality v(x) ≤ w(x) on Ωd

(7.2.3)

holds. Then v(x) ≤ w(x) in Gd0 . Proof. For the proof see word for word Proposition 6.4.



7.3 Construction of the barrier function In this section, for an n-dimensional infinite dihedral cone .  ω0 ω0    n−2