Boundary Value Problems and Markov Processes

Preface to the Second Edition This monograph is an expanded and revised version of a set of lecture notes for the grad...

Author: Kazuaki Taira

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Preface to the Second Edition

This monograph is an expanded and revised version of a set of lecture notes for the graduate courses given by the author both at Hiroshima University (1995– 1997) and at the University of Tsukuba (1998–2000) which were addressed to the advanced undergraduates and beginning-graduate students with interest in functional analysis, partial differential equations and probability. The first edition of this monograph, which was based on the lecture notes given at the University of Tsukuba (1988–1990), was published in 1991. This edition was found useful by a number of people, but it went out of print after a few years. This second edition has been revised to streamline some of the analysis and to give better coverage of important examples and applications. The errors in the first printing are corrected thanks to kind remarks of many friends. In order to make the monograph more up-to-date, additional references have been included in the bibliography. This second edition may be considered as a short introduction to the more advanced book “Semigroups, boundary value problems and Markov processes” which was published in the Springer Monographs in Mathematics series in 2004. For graduate students working in functional analysis, partial differential equations and probability, it may serve as an effective introduction to these three interrelated fields of analysis. For graduate students about to major in the subject and mathematicians in the field looking for a coherent overview, it will provide a method for the analysis of elliptic boundary value problems in the framework of Lp spaces. My special thanks go to the editorial staffs of Springer-Verlag for their unfailing helpfulness and cooperation during the production of this second edition. This research was partially supported by Grant-in-Aid for General Scientific Research (No. 19540162), Ministry of Education, Culture, Sports, Science and Technology, Japan.

VII

VIII

Preface to the Second Edition

Last but not least, I owe a great debt of gratitude to my family who gave me moral support during the preparation of this book. Tsukuba, March 2009

Kazuaki Taira

Preface to the First Edition

This monograph is devoted to the functional analytic approach to a class of degenerate boundary value problems for second-order elliptic differential operators which includes as particular cases the Dirichlet and Neumann problems. We prove that this class of boundary value problems provides a new example of analytic semigroups both in the Lp topology and in the topology of uniform convergence. As an application, we show that there exists a strong Markov process corresponding to such a diffusion phenomenon that either absorption or reflection phenomenon occurs at each point of the boundary. Furthermore, we study a class of initial-boundary value problems for semilinear parabolic differential equations. This monograph is an expanded version of a set of lecture notes for the graduate courses given by the author at the University of Tsukuba between 1988 and 1990. We confined ourselves to the simple boundary condition. This makes it possible to develop our basic machinery with a minimum of bother and the principal ideas can be presented concretely and explicitly. I hope that this monograph will lead to a better insight into the study of three interrelated subjects: elliptic boundary value problems, analytic semigroups and Markov processes. For additional information on many of the topics discussed here, I would like to call attention to my previous book Diffusion Processes and Partial Differential Equations, Academic Press, 1988. I would like to express my hearty thanks to many colleagues and graduate students in my courses, whose helpful criticisms of my lectures resulted in a number of improvements. The manuscript is typeset in a camera-ready form using AMS-TEX. Tsukuba, April 1991

Kazuaki Taira

IX

Contents

1

Introduction and Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

2

Semigroup Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Analytic Semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Generation of Analytic Semigroups . . . . . . . . . . . . . . . . . . 2.1.2 Fractional Powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3 The Semilinear Cauchy Problem . . . . . . . . . . . . . . . . . . . . 2.2 Markov Processes and Feller Semigroups . . . . . . . . . . . . . . . . . . . 2.2.1 Markov Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Markov Transition Functions . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Path Functions of Markov Processes . . . . . . . . . . . . . . . . . 2.2.4 Strong Markov Processes and Transition Functions . . . . 2.2.5 Markov Transition Functions and Feller Semigroups . . . 2.2.6 Generation Theorems of Feller Semigroups . . . . . . . . . . . .

13 13 13 24 26 27 28 31 35 36 39 44

3

Lp Theory of Pseudo-Differential Operators . . . . . . . . . . . . . . . 3.1 Function Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Fourier Integral Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Symbol Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Phase Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Oscillatory Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.4 Fourier Integral Operators . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Pseudo-Differential Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55 55 62 62 64 65 67 67

4

Lp 4.1 4.2 4.3

77 77 80 81

5

Proof of Theorem 1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 5.1 Boundary Value Problem with Spectral Parameter . . . . . . . . . . . 87 5.2 Proof of Estimate (1.2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

Approach to Elliptic Boundary Value Problems . . . . . . . . The Dirichlet Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Formulation of a Boundary Value Problem . . . . . . . . . . . . . . . . . . Reduction to the Boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

XI

XII

Contents

6

A Priori Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

7

Proof of Theorem 1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 7.1 Proof of Theorem 1.2, Part (i) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 7.1.1 Proof of Proposition 7.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 7.2 Proof of Theorem 1.2, Part (ii) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

8

Proof of Theorem 1.3, Part (i) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 8.1 The Space C0 (D \ M ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 8.2 Sobolev’s Imbedding Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 8.3 Proof of Part (i) of Theorem 1.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

9

Proof of Theorem 1.3, Part (ii) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 9.1 General Existence Theorem for Feller Semigroups . . . . . . . . . . . . 125 9.2 Feller Semigroups with Reflecting Barrier . . . . . . . . . . . . . . . . . . . 140 9.3 Proof of Theorem 1.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 9.4 Proof of Part (ii) of Theorem 1.3 . . . . . . . . . . . . . . . . . . . . . . . . . . 159

10 Application to Semilinear Initial-Boundary Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 10.1 Local Existence and Uniqueness Theorems . . . . . . . . . . . . . . . . . . 161 10.2 Fractional Powers and Imbedding Theorems . . . . . . . . . . . . . . . . 163 10.3 Proof of Theorems 10.1 and 10.2 . . . . . . . . . . . . . . . . . . . . . . . . . . 165 10.3.1 Proof of Theorem 10.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 10.3.2 Proof of Theorem 10.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 11 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 A

The Maximum Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

1 Introduction and Main Results

In this introductory chapter, our problems and results are stated in such a fashion that a broad spectrum of readers could understand, and also described how these problems can be solved, using the mathematics presented in Chapters 2 through 4. In 1828, the English botanist R. Brown observed that pollen grains suspended in water move chaotically, incessantly changing their direction of motion. The physical explanation of this phenomenon is that a single grain suffers innumerable collisions with the randomly moving molecules of the surrounding water. A mathematical theory for Brownian motion was put forward by A. Einstein in 1905 ([Ei]). Einstein derived an accurate method of measuring Avogadro’s number by observing particles undergoing Brownian motion. Einstein’s theory was experimentally tested by J. Perrin between 1906 and 1909. Brownian motion was put on a firm mathematical foundation for the first time by N. Wiener in 1923 ([Wi]). Wiener characterized the “starting afresh” property of Brownian motion that if a Brownian particle reaches a position, then it behaves subsequently as though that position had been its initial position. Markov processes are an abstraction of the idea of Brownian motion. In the first works devoted to Markov processes, the most fundamental was A. N. Kolmogorov’s work in 1931 ([Ko]) where the general concept of a Markov transition function was introduced for the first time and an analytic method of describing Markov transition functions was proposed. From the viewpoint of analysis, the transition function is something more convenient than the Markov process itself. In fact, it can be shown that the transition functions of Markov processes generate solutions of certain parabolic partial differential equations such as the classical diffusion equation; and, conversely, these differential equations can be used to construct and study the transition functions and the Markov processes themselves. In the 1950s, the theory of Markov processes entered a new period of intensive development. We can associate with each transition function in a natural K. Taira, Boundary Value Problems and Markov Processes, Lecture Notes in Mathematics 1499, DOI 10.1007/978-3-642-01677-6 1, c Springer-Verlag Berlin Heidelberg 2009

1

2


way a family of bounded linear operators acting on the space of continuous functions on the state space, and the Markov property implies that this family forms a semigroup. The Hille–Yosida theory of semigroups in functional analysis made possible further progress in the study of Markov processes. The semigroup approach to Markov processes can be traced back to the pioneering work of Feller in early 1950s ([Fe1], [Fe2]). Now let D be a bounded domain of Euclidean space RN , with smooth boundary ∂D; its closure D = D ∪ ∂D is an N -dimensional, compact smooth manifold with boundary. We let A=

N i,j=1

∂2 ∂ + bi (x) + c(x) ∂xi ∂xj i=1 ∂xi N

aij (x)

be a second-order, elliptic differential operator with real smooth coefficients on D such that: (1) aij (x) = aji (x) for all x ∈ D and 1 ≤ i, j ≤ N . (2) There exists a positive constant a0 such that N

aij (x)ξi ξj ≥ a0 |ξ|2

for all (x, ξ) ∈ D × RN .

i,j=1

(3) c(x) ≤ 0 on D. We consider the following boundary value problem with spectral parameter: Given functions f (x) and ϕ(x ) defined in D and on ∂D, respectively, find a function u(x) in D such that (A − λ)u = f in D, (1.1) ∂u Lu = μ(x ) ∂n + γ(x )u = ϕ on ∂D. Here: (4) λ is a complex parameter. (5) μ(x ) and γ(x ) are real-valued, smooth functions on the boundary ∂D. (6) n = (n1 , n2 , . . . , nN ) is the unit interior normal to the boundary ∂D (see Figure 1.1). We remark that if μ(x ) = 0 and γ(x ) ≡ 0 on ∂D (resp. μ(x ) ≡ 0 and γ(x ) = 0 on ∂D), then the boundary condition L is essentially the so-called Neumann (resp. Dirichlet) condition. It is easy to see that problem (1.1) is non-degenerate (or coercive) if and only if either μ(x ) = 0 on ∂D or μ(x ) ≡ 0 and γ(x ) = 0 on ∂D. The generation theorem of analytic semigroups is well established in the non-degenerate case both in the Lp topology and in the topology of uniform convergence (cf. Friedman [Fr1], Tanabe [Tn], Masuda [Ma], Stewart [Sw]).


3

∂D n

D

Fig. 1.1.

In this book, under the condition that μ(x ) ≥ 0 on ∂D we shall consider the problem of existence and uniqueness of solutions of problem (1.1) in the framework of Sobolev spaces of Lp type, and generalize the generation theorem of analytic semigroups to the degenerate case. First, we give a fundamental a priori estimate for problem (1.1). If 1 ≤ p < ∞, we let Lp (D) = the space of (equivalence classes of) Lebesgue measurable functions u(x) on D such that |u(x)|p is integrable on D. The space Lp (D) is a Banach space with the norm u p =

1/p |u(x)|p dx .

D

If m is a non-negative integer, we define the usual Sobolev space W m,p (D) = the space of (equivalence classes of) functions u ∈ Lp (D) whose derivatives Dα u(x), |α| ≤ m, in the sense of distributions are in Lp (D). The space W m,p (D) is a Banach space with the norm ⎛ u m,p = ⎝

|α|≤m

⎞1/p |Dα u(x)|p dx⎠

.

D

We remark that W 0,p (D) = Lp (D);

· 0,p = · p .

Furthermore, we let B m−1/p,p (∂D) = the space of the boundary values ϕ(x ) of functions u ∈ W m,p (D).

4


In the space B m−1/p,p (∂D), we introduce a norm |ϕ|m−1/p,p = inf u m,p , where the infimum is taken over all functions u ∈ W m,p (D) which equal ϕ(x ) on the boundary ∂D. The space B m−1/p,p (∂D) is a Banach space with respect to this norm | · |m−1/p,p ; more precisely, it is a Besov space (cf. [BL], [Tb], [Tr]). Then we have the following result: Theorem 1.1. Let 1 < p < ∞. Assume that the following two conditions (A) and (B) are satisfied: (A) μ(x ) ≥ 0 on ∂D. (B) γ(x ) < 0 on M = {x ∈ ∂D : μ(x ) = 0}. Then, for any solution u ∈ W 2,p (D) of problem (1.1) with f ∈ Lp (D) and ϕ ∈ B 2−1/p,p (∂D) we have the a priori estimate u 2,p ≤ C(λ) f p + |ϕ|2−1/p,p + u p , (1.2) with a positive constant C(λ) depending on λ. It should be emphasized that problem (1.1) is a degenerate elliptic boundary value problem from an analytical point of view. This is due to the fact that the so-called Lopatinskii–Shapiro complementary condition is violated at each point x of the set M . Amann [Am] studied the non-degenerate case; more precisely, he assumes that the boundary ∂D is the disjoint union of the two closed subsets M = {x ∈ ∂D : μ(x ) = 0} and ∂D \ M = {x ∈ ∂D : μ(x ) > 0}, each of which is an (N − 1) dimensional compact smooth manifold. Here it is worth while pointing out that the a priori estimate (1.2) is the same one for the Dirichlet condition: μ(x ) ≡ 0 and γ(x ) = 0 on ∂D (cf. [ADN], [LM]). Now we state a generation theorem of analytic semigroups in the Lp topology. We associate with problem (1.1) a unbounded linear operator Ap from Lp (D) into itself as follows: (a) The domain of definition D(Ap ) of Ap is the set

∂u + γ(x )u = 0 . D(Ap ) = u ∈ W 2,p (D) : Lu = μ(x ) ∂n

(1.3)

(b) Ap u = Au, u ∈ D(Ap ). The next theorem is an Lp version of Taira [Ta1, Theorem 1]: Theorem 1.2. Let 1 < p < ∞. Assume that conditions (A) and (B) are satisfied. Then we have the following two assertions:


5

(i) For every positive number ε, there exists a positive constant rp (ε) such that the resolvent set of Ap contains the set Σp (ε) = λ = r2 eiθ : r ≥ rp (ε), −π + ε ≤ θ ≤ π − ε , and that the resolvent (Ap − λI)−1 satisfies the estimate (Ap − λI)−1 ≤ cp (ε) |λ|

for all λ ∈ Σp (ε),

(1.4)

where cp (ε) is a positive constant depending on ε. (ii) The operator Ap generates a semigroup Uz on the space Lp (D) which is analytic in the sector Δε = {z = t + is : z = 0, | arg z| < π/2 − ε} for any 0 < ε < π/2 (see Figure 1.2).

Σp (ε) ⏐λ⏐= rp (ε)

ε ε

ε 2

Δε

0

0 ε

Fig. 1.2.

Secondly, we state a generation theorem of analytic semigroups in the topology of uniform convergence. Let C(D) be the space of real-valued, continuous functions f (x) on D. We equip the space C(D) with the topology of uniform convergence on the whole D; hence it is a Banach space with the maximum norm f ∞ = max |f (x)|. x∈D

We introduce a subspace of C(D) which is associated with the boundary condition L. We remark that the boundary condition Lu = μ(x )

∂u + γ(x )u = 0 ∂n

on ∂D

6


includes the condition u=0

on M = {x ∈ ∂D : μ(x ) = 0},

if γ(x ) = 0 on M . With this fact in mind, we let C0 (D \ M ) = u ∈ C(D) : u = 0 on M . The space C0 (D \M ) is a closed subspace of C(D); hence it is a Banach space. Furthermore, we introduce a unbounded linear operator A from C0 (D \ M ) into itself as follows: (a) The domain of definition D(A) of A is the set D(A) = u ∈ C0 (D \ M ) : Au ∈ C0 (D \ M ), Lu = 0 .

(1.5)

(b) Au = Au, u ∈ D(A). Here Au and Lu are taken in the sense of distributions (see Chapter 9). Then Theorem 1.2 remains valid with Lp (D) and Ap replaced by C0 (D\M ) and A, respectively. More precisely, we can prove the following: Theorem 1.3. Assume that condition (A) and the following condition (B ) (replacing condition (B)) are satisfied: (B ) γ(x ) ≤ 0 on ∂D and γ(x ) < 0 on M = {x ∈ ∂D : μ(x ) = 0}. Then we have the following two assertions: (i) For every positive number ε, there exists a positive constant r(ε) such that the resolvent set of A contains the set Σ(ε) = λ = r2 eiθ : r ≥ r(ε), −π + ε ≤ θ ≤ π − ε , and that the resolvent (A − λI)−1 satisfies the estimate (A − λI)−1 ≤

c(ε) |λ|

for all λ ∈ Σ(ε),

(1.6)

where c(ε) is a positive constant depending on ε. (ii) The operator A generates a semigroup Tz on the space C0 (D \ M ) which is analytic in the sector Δε = {z = t + is : z = 0, | arg z| < π/2 − ε} for any 0 < ε < π/2 (see Figure 1.3). Moreover, the operators Tt (t ≥ 0) are non-negative and contractive on the space C0 (D \ M ): f ∈ C0 (D \ M ), 0 ≤ f (x) ≤ 1 on D \ M

=⇒

0 ≤ Tt f (x) ≤ 1 on D \ M .


ε

Σ (ε ) ⏐λ⏐= r(ε)2

ε ε

7

0

Δε 0 ε

Fig. 1.3.

The main purpose of this book is devoted to the functional analytic approach to the problem of existence of Markov processes in probability theory. A strongly continuous, non-negative and contraction semigroup {Tt }t≥0 on the space C0 (D \ M ) is called a Feller semigroup on D \ M . Therefore, we can reformulate part (ii) of Theorem 1.3 as follows: Theorem 1.4. Assume that conditions (A) and (B ) are satisfied. Then the operator A generates a Feller semigroup {Tt }t≥0 on the space D \ M . Theorem 1.4 generalizes Bony–Courrège–Priouret [BCP, Théorème XIX] to the case where μ(x ) ≥ 0 on the boundary ∂D (cf. [Ta2, Theorem 10.1.3]). It is known (cf. [Dy2], [Ta2, Chapter 9]) that if {Tt }t≥0 is a Feller semigroup on D \ M , then there exists a unique Markov transition function pt (x, ·) on the space D \ M such that Tt f (x) = pt (x, dy)f (y), f ∈ C0 (D \ M ). D\M

Furthermore, it can be shown that the function pt (x, ·) is the transition function of some strong Markov process X ; hence the value pt (x, E) expresses the transition probability that a Markovian particle starting at position x will be found in the set E at time t. The differential operator A describes analytically a strong Markov process with continuous paths in the interior D such as Brownian motion (see Figure 1.4). The terms μ(x )∂u/∂n and γ(x )u of the boundary condition L are supposed to correspond to reflection and absorption phenomena, respectively. The situation may be represented schematically by Figure 1.5. Hence the intuitive meaning of condition (B ) is that absorption phenomenon occurs at each point of the set M = {x ∈ ∂D : μ(x ) = 0}, while reflection phenomenon occurs at each point of the set ∂D \ M = {x ∈ ∂D : μ(x ) > 0}. In other words, a Markovian particle moves in the space D \ M

8


∂D D

Fig. 1.4.

D

D

∂D

∂D

absorption

reflection Fig. 1.5.

until it “dies” at the time when it reaches the set M where the particle is definitely absorbed (see Figure1.6). Therefore, Theorem 1.4 asserts that there exists a Feller semigroup corresponding to such a diffusion phenomenon.

∂D

D

M = {¹ = 0} Fig. 1.6.

It is worth while pointing out here that the condition μ(x ) ≥ 0 and γ(x ) ≤ 0 on ∂D is necessary in order that the operator A is the infinitesimal generator of a Feller semigroup {Tt }t≥0 on D \ M (cf. [Ta2, Section 9.5]).


9

As an application of Theorem 1.2, we consider the following semilinear initial-boundary value problem: Given functions f (x, t, u, ξ) and u0 (x) defined in D×[0, T )×R×RN and in D, respectively, find a function u(x, t) in D×[0, T ) such that ⎧ ∂ ⎨ ∂t − A u(x, t) = f (x, t, u, grad u) in D × (0, T ), ∂u (1.7) Lu(x , t) = μ(x ) ∂n + γ(x )u = 0 on ∂D × [0, T ), ⎩ u(x, 0) = u0 (x) in D. By making use of the operator Ap , we can formulate problem (1.7) in terms of the abstract Cauchy problem in the Banach space Lp (D) as follows: du 0 < t < T, dt = Ap u(t) + F (t, u(t)) , (1.8) u|t=0 = u0 . Here u(t) = u(·, t) and F (t, u(t)) = f (·, t, u(t), grad u(t)) are functions defined on the interval [0, T ), taking values in the space Lp (D). We can prove local existence and uniqueness theorems for problem (1.8) (Theorems 10.1 and 10.2), by using the theory of fractional powers of analytic semigroups. Our semigroup approach here can be traced back to the pioneering work of Fujita–Kato [FK] on the Navier–Stokes equation in fluid dynamics. Theorem 1.5. Assume that conditions (A) and (B) are satisfied. If the nonlinear term f (x, t, u, ξ) is a locally Lipschitz continuous function with respect to all its variables (x, t, u, ξ) ∈ D × [0, T ) × R × RN with the possible exception of the x variables, then we have the following two assertions: (i) If N < p < ∞, then, for every function u0 (x) of D(Ap ), problem (1.8) has a unique local solution u(x, t) ∈ C ([0, T1 ]; Lp (D)) ∩ C 1 ((0, T1 ); Lp (D)) where T1 = T1 (p, u0 ) is a positive constant. (ii) If N/2 < p < N , we assume that there exist a non-negative continuous function ρ(t, r) on R × R and a constant 1 ≤ γ < N/(N − p) such that the following four conditions are satisfied: (a) |f (x, t, u, ξ)| ≤ ρ(t, |u|)(1 + |ξ|γ ). (b) |f (x, t, u, ξ) − f (x, s, u, ξ)| ≤ ρ(t, |u|) (1 + |ξ|γ ) |t − s|. γ−1

(c) |f (x, t, u, ξ) − f (x, t, u, η)| ≤ ρ(t, |u|) 1 + |ξ|γ−1 + |η| |ξ − η|. γ (d) |f (x, t, u, ξ) − f (x, t, v, ξ)| ≤ ρ(t, |u| + |v|) (1 + |ξ| ) |u − v|. Then, for every function u0 (x) of D(Ap ), problem (1.8) has a unique local solution u(x, t) ∈ C ([0, T2 ]; Lp (D)) ∩ C 1 ((0, T2 ); Lp (D)) where T2 = T2 (p, u0 ) is a positive constant.

Here C ([0, T ]; Lp (D)) denotes the space of continuous functions on the closed interval [0, T ] taking values in Lp (D), and C 1 ((0, T ); Lp (D)) denotes the space of continuously differentiable functions on the open interval (0, T ) taking values in Lp (D), respectively. The rest of this monograph is organized as follows.

10


Chapter 2 is devoted to a review of standard topics from the theory of semigroups. Section 2.1 provides a brief description of the basic results about analytic semigroups (Theorem 2.2) which forms a functional analytic background for the proof of Theorems 1.2 and 1.3. Moreover, Subsection 2.1.3 is devoted to the semigroup approach to a class of initial-boundary value problems for semilinear parabolic differential equations. By making use of fractional powers of analytic semigroups, we formulate a local existence and uniqueness theorem for semilinear initial-boundary value problems (Theorem 2.8). On the other hand, Section 2.2 provides a brief description of basic definitions and results about Markov processes and Feller semigroups. In Subsection 2.2.6 we prove various generation theorems of Feller semigroups by using the Hille–Yosida theory of semigroups (Theorems 2.16 and 2.18) which form a functional analytic background for the proof of Theorem 1.4. In Chapter 3 we present a brief description of the basic concepts and results of the Lp theory of pseudo-differential operators which may be considered as a modern theory of the classical potential theory. In particular, we formulate the Besov space boundedness theorem due to Bourdaud [Bo] (Theorem 3.15) and a useful criterion for hypoellipticity due to H¨ ormander [Ho2] (Theorem 3.16) which play an essential role in the proof of our main results. In Chapter 4 we study the boundary value problem (1.1) in the framework of Sobolev spaces of Lp type, by using the Lp theory of pseudo-differential operators. The idea of our approach is stated as follows: First, we consider the following Neumann problem: (A − λ)v = f in D, (1.9) ∂v on ∂D. ∂n = 0 The existence and uniqueness theorem for problem (1.9) is well established in the framework of Sobolev spaces of Lp type (cf. Agmon–Douglis–Nirenberg [ADN]). We let v = GN (λ)f. The operator GN (λ) is the Green operator for the Neumann problem. Then it follows that a function u(x) is a solution of problem (1.1) if and only if the function w(x) = u(x) − v(x) is a solution of the problem (A − λ)w = 0 in D, ∂v Lw = −Lv = −μ(x ) ∂n − γ(x )v = −γ(x )v on ∂D. However, we know that every solution w of the homogeneous equation (A − λ)w = 0 in D can be expressed by means of a single layer potential as follows: w = P (λ)ψ.


11

The operator P (λ) is the Poisson operator for the Dirichlet problem. Thus, by using the operators GN (λ) and P (λ) we can reduce the study of problem (1.1) to that of the equation T (λ)ψ = LP (λ)ψ = −γ(x )v,

v = GN (λ)f.

This is a generalization of the classical Fredholm integral equation. It is well known (cf. [Ho1], [Ho3], [Se2], [Ty]) that the operator T (λ) = LP (λ) is a pseudo-differential operator of first order on the boundary ∂D. We can prove that the a priori estimate (1.2) of Theorem 1.1 is entirely equivalent to the corresponding a priori estimate for the operator T (λ) (Theorem 4.10). Chapter 5 is devoted to the proof of Theorem 1.1. We study the pseudodifferential operator T (λ) in question, and prove that conditions (A) and (B) are sufficient for the validity of the a priori estimate (1.2) (Lemma 5.1). More precisely, we construct a parametrix S(λ) for T (λ) in the H¨ ormander class L01,1/2 (∂D) (Lemma 5.2), and then apply a Besov space boundedness theorem (Theorem 3.15) to the parametrix S(λ) to obtain the a priori estimate (1.2) for problem (1.1). Here it should be emphasized that if we use instead of the Neumann problem (1.9) the Dirichlet problem as usual, then we have the following a priori estimate for problem (1.1): u 1,p ≤ C(λ) f p + |ϕ|1−1/p,p + u p . In other words, we can not use this estimate to prove the generation theorem of analytic semigroups in the Lp topology. In Chapter 6 we study the operator Ap , and prove fundamental a priori estimates for Ap − λI (Theorem 6.3) which is an essential step in the proof of Theorem 1.2. We make good use of Agmon’s method (Proposition 6.4). This is a technique of treating a spectral parameter λ as a second-order, elliptic differential operator of an extra variable and relating the old problem to a new one with the additional variable. Chapter 7 is devoted to the proof of Theorem 1.2 (Theorems 7.1 and 7.9). Once again we make use of Agmon’s method in the proof of Theorems 7.1 and 7.9. In particular, Agmon’s method plays a fundamental role in the proof of the surjectivity of the operator Ap − λI (Proposition 7.2). Chapter 8 and Chapter 9 are devoted to the proof of Theorem 1.3 and Theorem 1.4. In Chapter 8 we prove part (i) of Theorem 1.3. Part (i) of Theorem 1.3 follows from Theorem 1.2 by using Sobolev’s imbedding theorems (Theorems 8.1 and 8.2) and a λ-dependent localization argument essentially due to Masuda [Ma] (Lemma 8.4). In Chapter 9 we prove Theorem 1.4 and part (ii) of Theorem 1.3. This chapter is the heart of the subject. General existence theorems for Feller semigroups are formulated in terms of elliptic boundary value problems with spectral parameter (Theorem 9.12). First, we study Feller semigroups with reflecting barrier (Theorem 9.14) and then, by using these Feller semigroups

12


we construct Feller semigroups corresponding to such a diffusion phenomenon that either absorption or reflection phenomenon occurs at each point of the boundary (Theorem 9.18). Our proof is based on the generation theorems of Feller semigroups discussed in Section 2.2. In Chapter 10 we study problem (1.8), and prove Theorem 1.5 by using the theory of fractional powers of analytic semigroups (Theorems 10.1 and 10.2). To do this, we verify that all the conditions of Theorem 2.8 are satisfied. We remark that Theorem 1.5 is a generalization of Pazy [Pa, Section 8.4, Theorems 4.4 and 4.5] to the degenerate case. In the final Chapter 11, as concluding remarks, we give an overview for general results on generation theorems for Feller semigroups proved mainly by the author using the theory of pseudo-differential operators ([Ho1], [Se1], [Se2]) and the Calder´ on–Zygmund theory of singular integral operators ([CZ]). In Appendix A, we formulate various maximum principles for secondorder elliptic differential operators such as the weak maximum principle (Theorem A.1) and the Hopf boundary point lemma (Lemma A.3) which play an important role in Chapter 9. The following diagram gives a bird’s eye view of Markov processes, Feller semigroups and boundary value problems and how these relate to each other: Probability

Functional Analysis

Partial Differential Equations

Markov process X

Feller semigroup {Tt }

infinitesimal generator A

Markov transition function pt (·, dy)

Tt f =

D

pt (·, dy)f (y)

Tt = exp[tA]

Chapman and Kolmogorov equation

semigroup property Tt+s = Tt · Ts

differential operator A

absorption and reflection phenomena

function space C0 (D \ M )

boundary condition L

2 Semigroup Theory

This chapter is devoted to a review of standard topics from the theory of semigroups which forms a functional analytic background for the proof of Theorems 1.2, 1.3, 1.4 and 1.5.

2.1 Analytic Semigroups This section provides a brief description of the basic results of the theory of analytic semigroups which forms a functional analytic background for the proof of Theorems 1.2 and 1.3. Moreover, Subsection 2.1.3 is devoted to the semigroup approach to a class of initial-boundary value problems for semilinear parabolic differential equations (Theorem 2.8). Theorem 1.5 follows by verifying all the conditions of Theorem 2.8. For more leisurely treatments of analytic semigroups, the reader is referred to Friedman [Fr1], Pazy [Pa], Tanabe [Tn], Yosida [Yo] and also Taira [Ta4]. 2.1.1 Generation of Analytic Semigroups Let E be a Banach space over the real or complex number field, and let A : E → E be a densely defined, closed linear operator with domain D(A). Assume that the operator A satisfies the following two conditions (see Figure 2.1 below): (1) The resolvent set of A contains the region Σω = {λ ∈ C : λ = 0, | arg λ| < π/2 + ω},

0 < ω < π/2.

(2) For each ε > 0, there exists a positive constant Mε such that the resolvent R(λ) = (A − λI)−1 satisfies the estimate R(λ) ≤

Mε for all λ ∈ Σεω = {λ ∈ C : λ = 0, | arg λ| ≤ π/2 + ω − ε}. |λ| (2.1)

K. Taira, Boundary Value Problems and Markov Processes, Lecture Notes in Mathematics 1499, DOI 10.1007/978-3-642-01677-6 2, c Springer-Verlag Berlin Heidelberg 2009

13

14

2 Semigroup Theory

Σω 0

Fig. 2.1.

Then we let U (t) = −

1 2πi

eλt R(λ) dλ.

(2.2)

Γ

Here Γ is a path in the set Σεω consisting of the following three curves (see Figure 2.2): Γ (1) = re−i(π/2+ω−ε) : 1 ≤ r < ∞ , Γ (2) = eiθ : −(π/2 + ω − ε) ≤ θ ≤ π/2 + ω − ε , Γ (3) = rei(π/2+ω−ε) : 1 ≤ r < ∞ .

Γ (3) Γ (2) 0

Γ (1) Fig. 2.2.

It is easy to see that the integral 3

U (t) = −

1 2πi k=1

eλt R(λ) dλ Γ (k)

2.1 Analytic Semigroups

15

converges in the uniform operator topology of the Banach space L(E, E) for all t > 0, and thus defines a bounded linear operator on E. Here L(E, E) denotes the space of bounded linear operators on E. Furthermore, we have the following: Proposition 2.1. The operators U (t), defined by formula (2.2), form a semigroup on E, that is, they enjoy the semigroup property U (t + s) = U (t) · U (s)

for all t, s > 0.

Proof. By Cauchy’s theorem, we may assume that 1 eμs R(μ) dμ, s > 0. U (s) = − 2πi Γ Here Γ is a path obtained from the path Γ by translating each point of Γ to the right by a fixed small positive distance (see Figure 2.3). Γ

Γ

0

Fig. 2.3.

Then we have, by Fubini’s theorem, 1 eλt eμs R(λ) R(μ) dλ dμ U (t) · U (s) = (2πi)2 Γ Γ 1 R(λ) − R(μ) = eλt eμs dλ dμ (2πi)2 Γ Γ λ−μ 1 eμs 1 dμ dλ eλt R(λ) = 2πi Γ 2πi Γ λ − μ 1 1 eλt μs e R(μ) − dλ dμ. 2πi Γ 2πi Γ λ − μ We calculate the two terms in the last part. (a) We let eμs , μ ∈ C. f (μ) = λ−μ

16

2 Semigroup Theory

|¹|= r ¸ 0

Fig. 2.4.

Then, by applying the residue theorem we obtain that (see Figure 2.4) f (μ) dμ + f (μ) dμ + f (μ) dμ Γ (1) ∩{|μ|≤r}

Γ (2)

π/2+ω−ε

Γ (3) ∩{|μ|≤r}

f (reiθ )rieiθ dθ

+ −(π/2+ω−ε)

= 2πi Res [f (μ)]μ=λ = −2πieλs . However, we have, as r → ∞,

Γ (1) ∩{|μ|≤r}

f (μ) dμ −→

Γ (3) ∩{|μ|≤r}

f (μ) dμ, Γ (1)

f (μ) dμ −→

f (μ) dμ, Γ (3)

and π/2+ω−ε π/2+ω−ε dθ iθ iθ −rs·sin(ω−ε) λ −→ 0. f (re )rie dθ ≤ e iθ −(π/2+ω−ε) −(π/2+ω−ε) r − e Therefore, we find that 1 2πi

Γ

eμs dμ = −eλs . λ−μ

(b) Similarly, since the path Γ lies to the left of the path Γ , we find that 1 eλt dλ = 0. 2πi Γ λ − μ


17

Summing up, we obtain that 1 U (t) · U (s) = − eλ(t+s) R(λ) dλ = U (t + s) for all t, s > 0. 2πi Γ The proof of Proposition 2.1 is complete. The next theorem states that the semigroup U (t) can be extended to an analytic semigroup in some sector containing the positive real axis. Theorem 2.2. The semigroup U (t), defined by formula (2.2), can be extended to a semigroup U (z) which is analytic in the sector Δω = {z = t + is : z = 0, | arg z| < ω} , and enjoys the following properties: (a) The operators AU (z) and dU dz (z) are bounded operators on E for each z ∈ Δω , and satisfy the relation dU (z) = AU (z) dz

for all z ∈ Δω .

(2.3)

1 (ε) 0 (ε) and M (b) For each 0 < ε < ω/2, there exist positive constants M such that 0 (ε) U (z) ≤ M 1 (ε) M AU (z) ≤ |z|

for all z ∈ Δ2ε ω ,

(2.4)

for all z ∈ Δ2ε ω ,

(2.5)

where (see Figure 2.5) Δ2ε ω = {z ∈ C : z = 0, | arg z| ≤ ω − 2ε} . (c) For each x ∈ E, we have, as z → 0, z ∈ Δ2ε ω , U (z)x −→ x

in E.

Proof. (i) The analyticity of U (z): If λ ∈ Γ (3) and z ∈ Δ2ε ω , that is, if we have the formulas λ = |λ|eiθ , z = |z|eiϕ ,

θ = π/2 + ω − ε, |ϕ| ≤ ω − 2ε,

then it follows that λz = |λ| |z|ei(θ+ϕ) , with π/2 + ε ≤ θ + ϕ ≤ π/2 + 2ω − 3ε < 3π/2 − 3ε.

18

2 Semigroup Theory

Δ2ε ω 0

Fig. 2.5.

Note that cos(θ + ϕ) ≤ cos(π/2 + ε) = − sin ε. Hence we have the inequality |eλz | ≤ e−|λ| |z| sin ε

for all λ ∈ Γ (3) and z ∈ Δ2ε ω .

(2.6)

for all λ ∈ Γ (1) and z ∈ Δ2ε ω .

(2.7)

Similarly, we have the inequality |eλz | ≤ e−|λ| |z| sin ε

For each small ε > 0, we let Kωε = Δ2ε {z ∈ C : |z| ≥ ε} = {z ∈ C : |z| ≥ ε, | arg z| ≤ ω − 2ε} . ω Then, by combining estimates (2.1), (2.6) and (2.7) we obtain that λz e R(λ) ≤ Mε e−ε sin ε·|λ| |λ|

for all λ ∈ Γ (1)

Γ (3) and z ∈ Kωε .

On the other hand, we have the estimate λz e R(λ) ≤ Mε e|z| for all λ ∈ Γ (2) and z ∈ Kωε .

(2.8)

(2.9)

Therefore, we find that the integral 1 U (z) = − 2πi

3

1 e R(λ) dλ = − 2πi

λz

Γ

k=1

eλz R(λ) dλ

(2.10)

Γ (k)

converges in the Banach space L(E, E), uniformly in z ∈ Kωε , for every ε > 0. This proves that the operator U (z) is analytic in the domain Δω = ε>0 Kωε . By the analyticity of U (z), it follows that the operators U (z) also enjoy the semigroup property


U (z + w) = U (z) · U (w)

19

for all z, w ∈ Δω .

(ii) We prove that the operators U (z) enjoy properties (a), (b) and (c). (b) First, by using Cauchy’s theorem we obtain that 1 1 λz U (z) = − e R(λ) dλ = − eλz R(λ) dλ, 2πi Γ 2πi Γ|z| where Γ|z| is a path consisting of the following three curves (see Figure 2.6):

1 (1) ≤r 0, (−A)−α (−A)−β = (−A)−(α+β) . (ii) If α is a positive integer n, then we have the formula n (−A)−α = (−A)−1 . (iii) The fractional power (−A)−α is invertible for all α > 0. If 0 < α < 1, we have the following useful formula for the fractional power (−A)−α : Theorem 2.4. We have, for all 0 < α < 1, sin απ ∞ −α (−A)−α = − s R(s)ds. π 0

(2.23)

By Remark 2.1, we may assume that there exist positive constants M0 , M1 and a such that U (t) ≤ M0 e−at

for all t > 0.

1 AU (t) ≤ M1 e−at t

for all t > 0.

Then we can prove still another useful formula for the fractional power (−A)−α for all 0 < α < 1.

26

2 Semigroup Theory

Theorem 2.5. We have, for all 0 < α < 1, ∞ 1 tα−1 U (t) dt. (−A)−α = Γ (α) 0 In view of part (iii) of Proposition 2.3, we can define the fractional power (−A)α for all α > 0 as follows: (−A)α = the inverse of (−A)−α ,

α > 0.

The next theorem states that the domain D((−A)α ) of (−A)α is bigger than the domain D(A) of A when 0 < α < 1. Theorem 2.6. We have, for all 0 < α < 1, D(A) ⊂ D ((−A)α ) . We can give an explicit formula for the fractional power (−A)α (0 < α < 1) on the domain D(A): Theorem 2.7. Let 0 < α < 1. Then we have, for any x ∈ D(A), sin απ ∞ α−1 (−A)α x = s R(s)Ax ds. π 0 2.1.3 The Semilinear Cauchy Problem This subsection is devoted to the semigroup approach to a class of initialboundary value problems for semilinear parabolic differential equations. By making good use of fractional powers of analytic semigroups, we formulate a local existence and uniqueness theorem for semilinear initial-boundary value problems (Theorem 2.8). Our semigroup approach can be traced back to the pioneering work of Fujita–Kato [FK] for the Navier–Stokes equation in fluid dynamics. Assume that the operator A satisfies condition (2.21). Then we can define the fractional power (−A)α for all 0 < α < 1 (see Subsection 2.1.2). The operator (−A)α is a closed linear, invertible operator with domain D((−A)α ) ⊃ D(A). We let Eα = the space D((−A)α ) endowed with the graph norm · α of (−A)α , where

1/2 , x α = x 2 + (−A)α x 2

Then we have the following three assertions: (i) The space Eα is a Banach space.

x ∈ D((−A)α ).

2.2 Markov Processes and Feller Semigroups

27

(ii) The graph norm x α is equivalent to the norm (−A)α x . (iii) If 0 < α < β < 1, then we have Eβ ⊂ Eα with continuous injection. Now we consider the following semilinear Cauchy problem: du t0 < t < t1 , dt = Au(t) + f (t, u(t)), u(t0 ) = x0 .

(2.24)

Here f (t, x) is a function defined on an open subset U of [0, ∞) × Eα (0 < α < 1), taking values in E. We assume that f (t, x) is locally Hölder continuous in t and locally Lipschitz continuous in x. That is, for each point (t, x) of U there exist a neighborhood V ⊂ U , constants L = L(t, x, V ) > 0 and 0 < γ ≤ 1 such that f (s1 , y1 ) − f (s2 , y2 ) ≤ L (|s1 − s2 |γ + y1 − y2 α ) , (s1 , y1 ), (s2 , y2 ) ∈ V. A function u(t) : [t0 , t1 ) → E is called a solution of problem (2.24) if it satisfies the following three conditions: (1) u(t) ∈ C([t0 , t1 ); E) ∩ C 1 ((t0 , t1 ); E) and u(t0 ) = x0 . (2) u(t) ∈ D(A) and (t, u(t)) ∈ U for all t0 < t < t1 . (3) du dt = Au(t) + f (t, u(t)) for all t0 < t < t1 . Here C([t0 , t1 ); E) denotes the space of continuous functions on [t0 , t1 ) taking values in E, and C 1 ((t0 , t1 ); E) denotes the space of continuously differentiable functions on (t0 , t1 ) taking values in E, respectively. Our main result is the following local existence and uniqueness theorem for problem (2.24): Theorem 2.8. Let f (t, x) be a function defined on an open subset U of [0, ∞) × Eα (0 < α < 1), taking values in E. Assume that f (t, x) is locally H¨ older continuous in t and locally Lipschitz continuous in x. Then, for every (t0 , x0 ) ∈ U , problem (2.24) has a unique local solution u(t) ∈ C([t0 , t1 ]; E) ∩ C 1 ((t0 , t1 ); E) where t1 = t1 (t0 , x0 ) > t0 . For a proof of Theorem 2.8, the reader is referred to [He, Theorem 3.3.3], Pazy [Pa, Chapter 6, Theorem 3.1] and also Taira [Ta4, Theorem 1.18].

2.2 Markov Processes and Feller Semigroups This section provides a brief description of basic definitions and results about Markov processes and a class of semigroups (Feller semigroups) associated with Markov processes. In Subsection 2.2.6 we prove various generation theorems of Feller semigroups by using the Hille–Yosida theory of semigroups (Theorems 2.16 and 2.18) which form a functional analytic background for the proof of Theorem 1.4. The results discussed here are adapted from

28

2 Semigroup Theory

Blumenthal–Getoor [BG], Dynkin [Dy2], Lamperti [La], Revuz–Yor [RY] and also Taira [Ta2, Chapter 9]. The semigroup approach to Markov processes can be traced back to the pioneering work of Feller [Fe1] and [Fe2] in early 1950s (cf. [BCP], [SU], [Ta3]). 2.2.1 Markov Processes In 1828 the English botanist R. Brown observed that pollen grains suspended in water move chaotically, incessantly changing their direction of motion. The physical explanation of this phenomenon is that a single grain suffers innumerable collisions with the randomly moving molecules of the surrounding water. A mathematical theory for Brownian motion was put forward by A. Einstein in 1905 (cf. [Ei]). Let p(t, x, y) be the probability density function that a onedimensional Brownian particle starting at position x will be found at position y at time t. Einstein derived the following formula from statistical mechanical considerations: 1 (y − x)2 p(t, x, y) = √ exp − . 2Dt 2πDt Here D is a positive constant determined by the radius of the particle, the interaction of the particle with surrounding molecules, temperature and the Boltzmann constant. This gives an accurate method of measuring Avogadro’s number by observing particles. Einstein’s theory was experimentally tested by J. Perrin between 1906 and 1909. Brownian motion was put on a firm mathematical foundation for the first time by N. Wiener in 1923 ([Wi]). Let Ω be the space of continuous functions ω : [0, ∞) → R with coordinates xt (ω) = ω(t) and let F be the smallest σ-algebra in Ω which contains all sets of the form {ω ∈ Ω : a ≤ xt (ω) < b}, t ≥ 0, a < b. Wiener constructed probability measures Px , x ∈ R, on F for which the following formula holds: Px {ω ∈ Ω : a1 ≤ xt1 (ω) < b1 , a2 ≤ xt2 (ω) < b2 , . . . , an ≤ xtn (ω) < bn } b b b = a11 a22 . . . ann p(t1 , x, y1 )p(t2 − t1 , y1 , y2 ) . . . p(tn − tn−1 , yn−1 , yn ) dy1 dy2 . . . dyn , 0 < t1 < t2 < . . . < tn < ∞. This formula expresses the “starting afresh” property of Brownian motion that if a Brownian particle reaches a position, then it behaves subsequently as though that position had been its initial position. The measure Px is called the Wiener measure starting at x. P. Lévy found another construction of Brownian motion, and gave a profound description of qualitative properties of the individual Brownian path in his book: Processus stochastiques et mouvement brownien (1948). Markov processes are an abstraction of the idea of Brownian motion. Let K be a locally compact, separable metric space and let B be the σ-algebra


29

of all Borel sets in K, that is, the smallest σ-algebra containing all open sets in K. Let (Ω, F , P ) be a probability space. A function X(ω) defined on Ω taking values in K is called a random variable if it satisfies the condition X −1 (E) = {ω ∈ Ω : X(ω) ∈ E} ∈ F

for all E ∈ B.

We express this by saying that X is F /B-measurable. A family {xt }t≥0 of random variables is called a stochastic process, and it may be thought of as the motion in time of a physical particle. The space K is called the state space and Ω the sample space. For a fixed ω ∈ Ω, the function xt (ω), t ≥ 0, defines in the state space K a trajectory or path of the process corresponding to the sample point ω. In this generality the notion of a stochastic process is of course not so interesting. The most important class of stochastic processes is the class of Markov processes which is characterized by the Markov property. Intuitively, this is the principle of the lack of any “memory” in the system. More precisely, (temporally homogeneous) Markov property is that the prediction of subsequent motion of a particle, knowing its position at time t, depends neither on the value of t nor on what has been observed during the time interval [0, t); that is, a particle “starts afresh”. Now we introduce a class of Markov processes which we will deal with in this book (cf. [Dy2], [BG], [RY]). Assume that we are given the following: (1) A locally compact, separable metric space K and the σ-algebra B of all Borel sets in K. A point ∂ is adjoined to K as the point at infinity if K is not compact, and as an isolated point if K is compact (see Figure 2.10). We let K∂ = K ∪ {∂}, B∂ = the σ-algebra in K∂ generated by B.

K

K∂

∂ Fig. 2.10.

(2) The space Ω of all mappings ω : [0, ∞] → K∂ such that ω(∞) = ∂ and that if ω(t) = ∂ then ω(s) = ∂ for all s ≥ t. Let ω∂ be the constant map ω∂ (t) = ∂ for all t ∈ [0, ∞].

30

2 Semigroup Theory

(3) For each t ∈ [0, ∞], the coordinate map xt defined by xt (ω) = ω(t), ω ∈ Ω. (4) For each t ∈ [0, ∞], a mapping ϕt : Ω → Ω defined by (ϕt ω)(s) = ω(t+ s), ω ∈ Ω. Note that ϕ∞ ω = ω∂ and xt ◦ ϕs = xt+s for all t, s ∈ [0, ∞]. (5) A σ-algebra F in Ω and an increasing family {Ft }0≤t≤∞ of sub-σ-algebras of F . (6) For each x ∈ K∂ , a probability measure Px on (Ω, F ). We say that these elements define a (temporally homogeneous) Markov process X = (xt , F , Ft , Px ) if the following four conditions are satisfied: (i) For each 0 ≤ t < ∞, the function xt is Ft /B∂ -measurable, that is, {xt ∈ E} = {ω ∈ Ω : xt (ω) ∈ E} ∈ Ft

for all E ∈ B∂ .

(ii) For all 0 ≤ t < ∞ and E ∈ B, the function pt (x, E) = Px {xt ∈ E}

(2.25)

is a Borel measurable function of x ∈ K. (iii) Px {ω ∈ Ω : x0 (ω) = x} = 1 for each x ∈ K∂ . (iv) For all t, h ∈ [0, ∞], x ∈ K∂ and E ∈ B∂ , we have the formula Px {xt+h ∈ E | Ft } = ph (xt , E) a. e., or equivalently, Px (A ∩ {xt+h ∈ E}) =

ph (xt (ω), E) dPx (ω) for all A ∈ Ft . A

Here is an intuitive way of thinking about the above definition of a Markov process. The sub-σ-algebra Ft may be interpreted as the collection of events which are observed during the time interval [0, t]. The value Px (A), A ∈ F, may be interpreted as the probability of the event A under the condition that a particle starts at position x; hence the value pt (x, E) expresses the transition probability that a particle starting at position x will be found in the set E at time t (see Figure 2.11). The function pt (x, ·) is called the transition function of the process X . The transition function pt (x, ·) specifies the probability structure of the process. The intuitive meaning of the crucial condition (iv) is that the future behavior of a particle, knowing its history up to time t, is the same as the behavior of a particle starting at xt (ω), that is, a particle starts afresh. A particle moves in the space K until it “dies” at the time when it reaches the point ∂; hence the point ∂ is called the terminal point. With this interpretation in mind, we let ζ(ω) = inf{t ∈ [0, ∞] : xt (ω) = ∂}. The random variable ζ is called the lifetime of the process X .


31

E

t

x Fig. 2.11.

2.2.2 Markov Transition Functions In the first works devoted to Markov processes, the most fundamental was A. N. Kolmogorov’s work ([Ko]) where the general concept of a Markov transition function was introduced for the first time and an analytic method of describing Markov transition functions was proposed. From the point of view of analysis, the transition function is something more convenient than the Markov process itself. In fact, it can be shown that the transition functions of Markov processes generate solutions of certain parabolic partial differential equations such as the classical diffusion equation; and, conversely, these differential equations can be used to construct and study the transition functions and the Markov processes themselves. In the 1950s, the theory of Markov processes entered a new period of intensive development. We can associate with each transition function in a natural way a family of bounded linear operators acting on the space of continuous functions on the state space, and the Markov property implies that this family forms a semigroup. The Hille–Yosida theory of semigroups in functional analysis made possible further progress in the study of Markov processes, as will be shown in Subsection 2.2.5. Our first job is thus to give the precise definition of a transition function adapted to the theory of semigroups: Definition 2.1. Let (K, ρ) be a locally compact, separable metric space and let B be the σ-algebra of all Borel sets in K. A function pt (x, E), defined for all t ≥ 0, x ∈ K and E ∈ B, is called a (temporally homogeneous) Markov transition function on K if it satisfies the following four conditions: (a) pt (x, ·) is a non-negative measure on B and pt (x, K) ≤ 1 for all t ≥ 0 and x ∈ K. (b) pt (·, E) is a Borel measurable function for all t ≥ 0 and E ∈ B. (c) p0 (x, {x}) = 1 for all x ∈ K.

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2 Semigroup Theory

(d) (The Chapman–Kolmogorov equation) For all t, s ≥ 0, x ∈ K and E ∈ B, we have the equation pt (x, dy)ps (y, E). (2.26) pt+s (x, E) = K

E

t +s

t

0

y

x K Fig. 2.12.

Here is an intuitive way of thinking about the above definition of a Markov transition function. The value pt (x, E) expresses the transition probability that a physical particle starting at position x will be found in the set E at time t. The Chapman–Kolmogorov equation (2.26) expresses the idea that a transition from the position x to the set E in time t + s is composed of a transition from x to some position y in time t, followed by a transition from y to the set E in the remaining time s; the latter transition has probability ps (y, E) which depends only on y (see Figure 2.12). Thus a particle “starts afresh”; this property is called the Markov property. The Chapman–Kolmogorov equation (2.26) asserts that pt (x, K) is monotonically increasing as t ↓ 0, so that the limit p+0 (x, K) = lim pt (x, K) t↓0

exists. A transition function pt (x, ·) is said to be normal if it satisfies the condition p+0 (x, K) = 1 for all x ∈ K. The next theorem, due to Dynkin [Dy1, Chapter 4, Section 2], justifies the definition of a transition function, and hence it will be fundamental for our further study of Markov processes:


33

Theorem 2.9. For every Markov process, the function pt (x, ·), defined by formula (2.25), is a Markov transition function. Conversely, every normal Markov transition function corresponds to some Markov process. Here are some important examples of normal transition functions on the line R = (−∞, ∞): Example 2.1 (Uniform motion). If t ≥ 0, x ∈ R and E ∈ B, we let pt (x, E) = χE (x + vt), where v is a constant, and χE (y) = 1 if y ∈ E and = 0 if y ∈ E. This process, starting at x, moves deterministically with constant velocity v. Example 2.2 (Poisson process). If t ≥ 0, x ∈ R and E ∈ B, we let pt (x, E) = e−λt

∞ (λt)n χE (x + n), n! n=0

where λ is a positive constant. This process, starting at x, advances one unit by jumps, and the probability of n jumps during the time 0 and t is equal to e−λt (λt)n /n!. Example 2.3 (Brownian motion). If t > 0, x ∈ R and E ∈ B, we let 1 (y − x)2 pt (x, E) = √ exp − dy, 2t 2πt E and p0 (x, E) = χE (x). This is a mathematical model of one-dimensional Brownian motion. Its character is quite different from that of the Poisson process; the transition function pt (x, E) satisfies the condition pt (x, [x − ε, x + ε]) = 1 − o(t) as t ↓ 0, for all ε > 0 and x ∈ R. This means that the process never stands still, as does the Poisson process. Indeed, this process changes state not by jumps but by continuous motion. A Markov process with this property is called a diffusion process. Example 2.4 (Brownian motion with constant drift). If t > 0, x ∈ R and E ∈ B, we let 1 (y − mt − x)2 √ pt (x, E) = exp − dy, 2t 2πt E and p0 (x, E) = χE (x), where m is a constant.

34

2 Semigroup Theory

This represents Brownian motion with a constant drift of magnitude m superimposed; the process can be represented as {xt + mt}, where {xt } is Brownian motion on R. Example 2.5 (Cauchy process). If t > 0, x ∈ R and E ∈ B, we let t 1 pt (x, E) = dy, π E t2 + (y − x)2 and p0 (x, E) = χE (x). This process can be thought of as the “trace” on the real line of trajectories of two-dimensional Brownian motion, and it moves by jumps (see [Kn, Lemma 2.12]). More precisely, if B1 (t) and B2 (t) are two independent Brownian motions and if T is the first passage time of B1 (t) to x, then B2 (T ) has the Cauchy density 1 |x| , −∞ < y < ∞. π x2 + y 2 Here are two more examples of diffusion processes on the half line R+ = [0, ∞) in which we must take account of the effect of the boundary point 0: Example 2.6 (Reflecting barrier Brownian motion). If t > 0, x ∈ R+ and E ∈ B, we let 1 (y − x)2 (y + x)2 pt (x, E) = √ exp − exp − dy + dy , 2t 2t 2πt E E and p0 (x, E) = χE (x). This represents Brownian motion with a reflecting barrier at x = 0; the process may be represented as {|xt |}, where {xt } is Brownian motion on R. Indeed, since {|xt |} goes from x to y if {xt } goes from x to ±y due to the symmetry of the transition function in Example 2.3 about x = 0, it follows that pt (x, E) = Px {|xt | ∈ E} (y − x)2 (y + x)2 1 exp − exp − = √ dy + dy . 2t 2t 2πt E E Example 2.7 (Sticking barrier Brownian motion). If t > 0, x ∈ R+ and E ∈ B, we let 1 (y − x)2 (y + x)2 exp − exp − pt (x, E) = √ dy − dy 2t 2t 2πt E E 2 x z 1 exp − + 1− √ dz χE (0), 2t 2πt −x


35

and p0 (x, E) = χE (x). This represents Brownian motion with a sticking barrier at x = 0. When a Brownian particle reaches the boundary point 0 for the first time, instead of reflecting it sticks there forever; in this case the state 0 is called a trap. 2.2.3 Path Functions of Markov Processes It is naturally interesting and important to ask the following problem: Problem. Given a Markov transition function pt (x, ·), under which conditions on pt (x, ·) does there exist a Markov process with transition function pt (x, ·) whose paths are almost surely continuous? A Markov process X = (xt , F , Ft , Px ) is said to be right continuous provided that we have, for each x ∈ K, Px {ω ∈ Ω : the mapping t → xt (ω) is a right continuous function from [0, ∞) into K∂ } = 1. Furthermore, we say that X is continuous provided that we have, for each x ∈ K, Px {ω ∈ Ω : the mapping t → xt (ω) is a continuous function from [0, ζ(ω)) into K∂ } = 1, where ζ is the lifetime of the process X . Now we give some useful criteria for path continuity in terms of Markov transition functions (see Dynkin [Dy1, Chapter 6], [Dy2, Chapter 3, Section 2]): Theorem 2.10. Let (K, ρ) be a locally compact, separable metric space and let pt (x, ·) be a normal Markov transition function on K. (i) Assume that the following two conditions are satisfied: (L) For each s > 0 and each compact E ⊂ K, we have the condition lim sup pt (x, E) = 0.

x→∂ 0≤t≤s

(M) For each ε > 0 and each compact E ⊂ K, we have the condition lim sup pt (x, K \ Uε (x)) = 0, t↓0 x∈E

where Uε (x) = {y ∈ K : ρ(y, x) < ε} is an ε-neighborhood of x.

36

2 Semigroup Theory

Then there exists a Markov process X with transition function pt (x, ·) whose paths are right continuous on [0, ∞) and have left-hand limits on [0, ζ) almost surely. (ii) Assume that condition (L) and the following condition (N) (replacing condition (M)) are satisfied: (N) For each ε > 0 and each compact E ⊂ K, we have the condition lim t↓0

1 sup pt (x, K \ Uε (x)) = 0, t x∈E

or equivalently sup pt (x, K \ Uε (x)) = o(t)

as t ↓ 0.

x∈E

Then there exists a Markov process X with transition function pt (x, ·) whose paths are almost surely continuous on [0, ζ). Remark 2.2. It is known (see Dynkin [Dy1, Lemma 6.2]) that if the paths of a Markov process are right continuous, then the transition function pt (x, ·) satisfies the condition lim pt (x, Uε (x)) = 1 for all x ∈ K. t↓0

2.2.4 Strong Markov Processes and Transition Functions A Markov process is called a strong Markov process if the “starting afresh” property holds not only for every fixed moment but also for suitable random times. We shall formulate precisely this “strong” Markov property. Let X = (xt , F , Ft , Px ) be a Markov process. A mapping τ : Ω → [0, ∞] is called a stopping time or Markov time with respect to {Ft } if it satisfies the condition {τ ≤ t} = {ω ∈ Ω : τ (ω) ≤ t} ∈ Ft

for all t ∈ [0, ∞).

Intuitively, this means that the events {τ ≤ t} depend on the process only up to time t, but not on the “future” after time t. It should be noticed that any non-negative constant mapping is a stopping time. If τ is a stopping time with respect to {Ft }, we let Fτ = {A ∈ F : A ∩ {τ ≤ t} ∈ Ft

for all t ∈ [0, ∞)}.

Intuitively, we may think of Fτ as the “past” up to the random time τ . It is easy to verify that Fτ is a σ-algebra. If τ ≡ t0 for some constant t0 ≥ 0, then Fτ reduces to Ft0 . For each t ∈ [0, ∞], we define a mapping Φt : [0, t] × Ω −→ K∂


37

by the formula Φt (s, ω) = xs (ω). A Markov process X = (xt , F , Ft , Px ) is said to be progressively measurable with respect to {Ft } if the mapping Φt is B[0,t] × Ft /B∂ -measurable for each t ∈ [0, ∞], that is, if we have the condition Φ−1 t (E) = {Φt ∈ E} ∈ B[0,t] × Ft

for all E ∈ B∂ .

Here B[0,t] is the σ-algebra of all Borel sets in the interval [0, t] and B∂ is the σ-algebra in K∂ generated by B. It should be noticed that if X is progressively measurable and if τ is a stopping time, then the mapping xτ : ω → xτ (ω) (ω) is Fτ /B∂ -measurable. The next definition expresses the idea of “starting afresh” at random times: Definition 2.2. We say that a progressively measurable Markov process X = (xt , F , Ft , Px ) has the strong Markov property with respect to {Ft } if the following condition is satisfied: For all h ≥ 0, x ∈ K∂ , E ∈ B∂ and all stopping times τ , we have the formula Px {xτ +h ∈ E | Fτ } = ph (xτ , E), or equivalently, Px (A ∩ {xτ +h ∈ E}) =

ph (xτ (ω) (ω), E) dPx (ω) for all A ∈ Fτ . A

We shall state a simple criterion for the strong Markov property in terms of transition functions. Let (K, ρ) be a locally compact, separable metric space. We add a point ∂ to the metric space K as the point at infinity if K is not compact, and as an isolated point if K is compact; so the space K∂ = K ∪ {∂} is compact (see Figure 2.10). Let C(K) be the space of real-valued, bounded continuous functions f (x) on K; the space C(K) is a Banach space with the supremum norm f ∞ = sup |f (x)|. x∈K

We say that a function f ∈ C(K) converges to zero as x → ∂ if, for each ε > 0, there exists a compact subset E of K such that |f (x)| < ε

for all x ∈ K \ E,

and we then write limx→∂ f (x) = 0. We let

C0 (K) = f ∈ C(K) : lim f (x) = 0 . x→∂

The space C0 (K) is a closed subspace of C(K); hence it is a Banach space. Note that C0 (K) may be identified with C(K) if K is compact.

38

2 Semigroup Theory

Now we introduce a useful convention as follows: Any real-valued function f (x) on K is extended to the space K∂ = K ∪ {∂}by setting f (∂) = 0. From this point of view, the space C0 (K) is identified with the subspace of C(K∂ ) which consists of all functions f (x) satisfying the condition f (∂) = 0: C0 (K) = {f ∈ C(K∂ ) : f (∂) = 0} . Furthermore, we can extend a Markov transition function pt (x, ·) on K to a Markov transition function pt (x, ·) on K∂ by the formulas: ⎧ for all x ∈ K and E ∈ B, ⎨ pt (x, E) = pt (x, E) pt (x, {∂}) = 1 − pt (x, K) for all x ∈ K, ⎩ pt (∂, K) = 0, pt (∂, {∂}) = 1. Intuitively this means that a Markovian particle moves in the space K until it “dies” at the time when it reaches the point ∂; hence the point ∂ is called the terminal point. Now we introduce some conditions on the measures pt (x, ·) related to continuity in x ∈ K, for fixed t ≥ 0: Definition 2.3. (i) A Markov transition function pt (x, ·) is called a Feller function if the function Tt f (x) = pt (x, dy)f (y) K

is a continuous function of x ∈ K whenever f is in C(K), that is, if we have the condition f ∈ C(K) =⇒ Tt f ∈ C(K). (ii) We say that pt (x, ·) is a C0 -function if the space C0 (K) is an invariant subspace of C(K) for the operators Tt : f ∈ C0 (K) =⇒ Tt f ∈ C0 (K). Remark 2.3. The Feller property is equivalent to saying that the measures pt (x, ·) depend continuously on x ∈ K in the usual weak topology, for every fixed t ≥ 0. The next theorem gives a useful criterion for the strong Markov property (see [Dy1, Theorem 5.10]): Theorem 2.11. If the transition function pt (x, ·) of a right continuous Markov process X has the C0 -property, then X is a strong Markov process.


39

Furthermore, we state a simple criterion for the strong Markov property in terms of Markov transition functions. To do this, we introduce the following definition: Definition 2.4. A Markov transition function pt (x, ·) on K is said to be uniformly stochastically continuous on K if the following condition is satisfied: For each ε > 0 and each compact E ⊂ K, we have the condition lim sup [1 − pt (x, Uε (x))] = 0, t↓0 x∈E

(2.27)

where Uε (x) = {y ∈ K : ρ(y, x) < ε} is an ε-neighborhood of x. It should be emphasized that every uniformly stochastically continuous transition function is normal and satisfies condition (M) in Theorem 2.10. By combining part (i) of Theorem 2.10 and Theorem 2.11, we obtain the following result (see [Dy1, Theorem 6.3]): Theorem 2.12. Assume that a uniformly stochastically continuous, C0 transition function pt (x, ·) satisfies condition (L). Then it is the transition function of some strong Markov process X whose paths are right continuous and have no discontinuities other than jumps. A continuous strong Markov process is called a diffusion process. The next theorem states a sufficient condition for the existence of a diffusion process with a prescribed Markov transition function: Theorem 2.13. Assume that a uniformly stochastically continuous, C0 transition function pt (x, ·) satisfies conditions (L) and (N). Then it is the transition function of some diffusion process X . This is an immediate consequence of part (ii) of Theorem 2.10 and Theorem 2.12. 2.2.5 Markov Transition Functions and Feller Semigroups The Feller or C0 -property deals with continuity of a Markov transition function pt (x, E) in x, and does not, by itself, have no concern with continuity in t. We give a necessary and sufficient condition on pt (x, E) in order that its associated operators {Tt }t≥0 , defined by the formula pt (x, dy)f (y), f ∈ C0 (K), (2.28) Tt f (x) = K

is strongly continuous in t on the space C0 (K): lim Tt+s f − Tt f ∞ = 0, s↓0

f ∈ C0 (K).

(2.29)

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2 Semigroup Theory

Then we have the following (cf. [Ta2, Theorem 9.2.3]): Theorem 2.14. Let pt (x, ·) be a C0 -transition function on K. Then the associated operators {Tt }t≥0 , defined by formula (2.28) is strongly continuous in t on C0 (K) if and only if pt (x, ·) is uniformly stochastically continuous on K and satisfies condition (L). Proof. (i) The “if” part: Since continuous functions with compact support are dense in C0 (K), it suffices to prove the strong continuity of {Tt } at t = 0: lim Tt f − f ∞ = 0

(2.30)

t↓0

for all such functions f . For any compact subset E of K containing the support supp f of f , we have the inequality Tt f − f ∞ ≤ sup |Tt f (x) − f (x)| + sup |Tt f (x)| x∈E

x∈K\E

≤ sup |Tt f (x) − f (x)| + f ∞ · sup pt (x, supp f ). (2.31) x∈E

x∈K\E

However, condition (L) implies that, for each ε > 0, we can find a compact subset E of K such that, for all sufficiently small t > 0, sup pt (x, supp f ) < ε.

(2.32)

x∈K\E

On the other hand, we have, for each δ > 0, pt (x, dy)(f (y) − f (x)) Tt f (x) − f (x) = Uδ (x) pt (x, dy)(f (y) − f (x)) − f (x)(1 − pt (x, K)), + K\Uδ (x)

and hence sup |Tt f (x) − f (x)|

x∈E

≤

sup |f (y) − f (x)| + 3 f ∞ · sup [1 − pt (x, Uδ (x))] . x∈E

ρ(x,y) 0, sup |Tt f (x) − f (x)| < ε(1 + 3 f ∞ ).

(2.33)

x∈E

Therefore, by carrying inequalities (2.32) and (2.33) into inequality (2.31) we obtain that, for all sufficiently small t > 0, Tt f − f ∞ < ε(1 + 4 f ∞). This proves the desired formula (2.30), that is, the strong continuity (2.29) of {Tt }. (ii) The “only if” part: For any x ∈ K and ε > 0, we define a continuous function fx (y) by the formula (see Figure 2.13) 1 − 1ε ρ(x, y) if ρ(x, y) ≤ ε, fx (y) = (2.34) 0 if ρ(x, y) > ε.

fx

ε

x

ε

Fig. 2.13.

Let E be an arbitrary compact subset of K. Then, for all sufficiently small ε > 0, the functions fx , x ∈ E, are in C0 (K) and satisfy the condition fx − fz ∞ ≤

1 ρ(x, z) ε

for all x, z ∈ E.

(2.35)

However, for any δ > 0, by the compactness of E we can find a finite number of points x1 , x2 , . . ., xn of E such that E=

n

Uδε/4 (xk ),

k=1

and hence min ρ(x, xk ) ≤

1≤k≤n

δε 4

for all x ∈ E.

Thus, by combining this inequality with inequality (2.35) with z := xk we obtain that

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2 Semigroup Theory

min fx − fxk ∞ ≤

1≤k≤n

δ 4

for all x ∈ E.

(2.36)

Now we have, by formula (2.34), 0 ≤ 1 − pt (x, Uε (x)) ≤ 1 − pt (x, dy)fx (y) K∂

= fx (x) − Tt fx (x) ≤ fx − Tt fx ∞ ≤ fx − fxk ∞ + fxk − Tt fxk ∞ + Tt fxk − Tt fx ∞ ≤ 2 fx − fxk ∞ + fxk − Tt fxk ∞

for all x ∈ E.

In view of inequality (2.36), the first term on the last inequality is bounded by δ/2 for the right choice of k. Furthermore, it follows from the strong continuity (2.30) of {Tt } that the second term tends to zero as t ↓ 0 for each k = 1, 2, . . ., n. Consequently, we have, for all sufficiently small t > 0, sup [1 − pt (x, Uε (x))] ≤ δ.

x∈E

This proves the desired condition (2.27), that is, the uniform stochastic continuity of pt (x, ·). Finally, it remains to verify condition (L). Assume, to the contrary, that: For some s > 0 and some compact E ⊂ K, there exist a positive constant ε0 , a sequence {tk }, tk ↓ t (0 ≤ t ≤ s) and a sequence {xk }, xk → ∂, such that (2.37) ptk (xk , E) ≥ ε0 . Now we take a relatively compact subset U of K containing E, and let (see Figure 2.14) ρ(x, K \ U ) f (x) = . ρ(x, E) + ρ(x, K \ U ) Then it follows that the function f (x) is in C0 (K) and satisfies the condition pt (x, dy)f (y) ≥ pt (x, E) ≥ 0. Tt f (x) = K

Therefore, by combining this inequality with inequality (2.37) we obtain that Ttk f (xk ) ≥ ptk (xk , E) ≥ ε0 .

(2.38)

However, we have the inequality Ttk f (xk ) ≤ Ttk f − Tt f ∞ + Tt f (xk ).

(2.39)


43

f

E U Fig. 2.14.

Since the semigroup {Tt } is strongly continuous and since we have the assertion lim Tt f (xk ) = Tt f (∂) = 0, k→∞

we can let k → ∞ in inequality (2.39) to obtain that lim sup Ttk f (xk ) = 0. k→∞

This contradicts inequality (2.38). The proof of Theorem 2.14 is now complete. A family {Tt }t≥0 of bounded linear operators acting on the space C0 (K) is called a Feller semigroup on K if it satisfies the following three conditions: (i) Tt+s = Tt · Ts , t, s ≥ 0 (the semigroup property); T0 = I. (ii) The family {Tt } is strongly continuous in t for all t ≥ 0: lim Tt+s f − Tt f ∞ = 0, s↓0

f ∈ C0 (K).

(iii) The family {Tt } is non-negative and contractive on C0 (K): f ∈ C0 (K), 0 ≤ f (x) ≤ 1 on K =⇒ 0 ≤ Tt f (x) ≤ 1

on K.

Rephrased, Theorem 2.14 gives a characterization of Feller semigroups in terms of Markov transition functions: Theorem 2.15. If pt (x, ·) is a uniformly stochastically continuous C0 transition function on K and satisfies condition (L), then its associated operators {Tt }t≥0 , defined by formula (2.28), form a Feller semigroup on K. Conversely, if {Tt }t≥0 is a Feller semigroup on K, then there exists a uniformly stochastically continuous C0 -transition pt (x, ·) on K, satisfying condition (L), such that formula (2.28) holds. The most important applications of Theorem 2.15 are of course in the second statement.

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2 Semigroup Theory

2.2.6 Generation Theorems of Feller Semigroups In this subsection we prove various generation theorems of Feller semigroups by using the Hille–Yosida theory of semigroups. If {Tt }t≥0 is a Feller semigroup on K, we define its infinitesimal generator A by the formula Tt u − u , u ∈ C0 (K), (2.40) Au = lim t↓0 t provided that the limit (2.40) exists in the space C0 (K). More precisely, the generator A is a linear operator from C0 (K) into itself defined as follows: (1) The domain D(A) of A is the set D(A) = {u ∈ C0 (K) : the limit (2.40) exists} . (2) Au = limt↓0

Tt u−u , t

u ∈ D(A).

The next theorem is a version of the Hille–Yosida theorem adapted to the present context (cf. [Ta2, Theorem 9.3.1 and Corollary 9.3.2]): Theorem 2.16 (Hille–Yosida). (i) Let {Tt }t≥0 be a Feller semigroup on K and let A be its infinitesimal generator. Then we have the following four assertions: (a) The domain D(A) is dense in the space C0 (K). (b) For each α > 0, the equation (αI − A)u = f has a unique solution u in D(A) for any f ∈ C0 (K). Hence, for each α > 0 the Green operator (αI − A)−1 : C0 (K) → C0 (K) can be defined by the formula u = (αI − A)−1 f,

f ∈ C0 (K).

(c) For each α > 0, the operator (αI − A)−1 is non-negative on C0 (K): f ∈ C0 (K), f (x) ≥ 0

on K =⇒ (αI − A)−1 f (x) ≥ 0

on K.

(d) For each α > 0, the operator (αI − A)−1 is bounded on C0 (K) with norm (αI − A)−1 ≤

1 . α

(ii) Conversely, if A is a linear operator from C0 (K) into itself satisfying condition (a) and if there is a non-negative constant α0 such that, for all α > α0 , conditions (b) through (d) are satisfied, then A is the infinitesimal generator of some Feller semigroup {Tt }t≥0 on K. Proof. In view of the Hille–Yosida theory (see [Yo, Chapter IX, Section 7]), it suffices to show that the semigroup {Tt }t≥0 is non-negative if and only if its resolvents (Green operators) {(αI − A)−1 }α>α0 are non-negative.


45

The “only if” part is an immediate consequence of the following expression of (αI − A)−1 in terms of the semigroup {Tt }: ∞ (αI − A)−1 = exp[−αt] Tt dt, α > 0. 0

On the other hand, the “if” part follows from the expression of the semigroup Tt (α) in terms of the Yosida approximation Jα = α(αI − A)−1 : ∞ (αt)n n Jα , Tt (α) = exp[−αt] exp [αtJα ] = exp[−αt] n! n=0

and the definition of the semigroup Tt : Tt = lim Tt (α). α→∞

The proof of Theorem 2.16 is complete. Corollary 2.17. Let K be a compact metric space and let A be the infinitesimal generator of a Feller semigroup on K. Assume that the constant function 1 belongs to the domain D(A) of A and that we have, for some constant c, A1(x) ≤ −c

on K.

(2.41)

Then the operator A = A + cI is the infinitesimal generator of some Feller semigroup on K. Proof. It follows from an application of part (i) of Theorem 2.16 that, for all α > c the operators (αI − A )−1 = ((α − c)I − A)−1 are defined and non-negative on the whole space C(K). However, in view of inequality (2.41) we obtain that α ≤ α − (A1 + c) = (αI − A )1 so that

on K,

α(αI − A )−1 1 ≤ (αI − A )−1 (αI − A )1 = 1

on K.

Hence we have, for all α > c, (αI − A )−1 = (αI − A )−1 1 ∞ ≤

1 . α

Therefore, by applying part (ii) of Theorem 2.16 to the operator A we find that A is the infinitesimal generator of some Feller semigroup on K. The proof of Corollary 2.17 is complete.

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2 Semigroup Theory

Now we write down explicitly the infinitesimal generators of Feller semigroups associated with the transition functions in Examples 2.1 through 2.7 (cf. [DY]). Example 2.8 (Uniform motion). K = R and D(A) = {f ∈ C0 (K) : f ∈ C0 (K)}, Af = vf , f ∈ D(A). Example 2.9 (Poisson process). K = R and D(A) = C0 (K), Af (x) = λ(f (x + 1) − f (x)),

f ∈ D(A).

The operator A is not “local”; the value Af (x) depends on the values f (x) and f (x + 1). This reflects the fact that the Poisson process changes state by jumps. Example 2.10 (Brownian motion). K = R and D(A) = {f ∈ C0 (K) : f ∈ C0 (K), f ∈ C0 (K)}, Af = 12 f , f ∈ D(A). The operator A is “local”, that is, the value Af (x) is determined by the values of f in an arbitrary small neighborhood of x. This reflects the fact that Brownian motion changes state by continuous motion. Example 2.11 (Brownian motion with constant drift). K = R and D(A) = {f ∈ C0 (K) : f ∈ C0 (K), f ∈ C0 (K)}, Af = 12 f + mf , f ∈ D(A). Example 2.12 (Cauchy process). K = R and, the domain D(A) contains C 2 functions on K with compact support, and the infinitesimal generator A is of the form dy 1 +∞ (f (x + y) − f (x)) 2 . Af (x) = π −∞ y The operator A is not “local”, which reflects the fact that the Cauchy process changes state by jumps. Example 2.13 (Reflecting barrier Brownian motion). K = [0, ∞) and D(A) = {f ∈ C0 (K) : f ∈ C0 (K), f ∈ C0 (K), f (0) = 0}, Af = 12 f , f ∈ D(A).


47

Example 2.14 (Sticking barrier Brownian motion). K = [0, ∞) and D(A) = {f ∈ C0 (K) : f ∈ C0 (K), f ∈ C0 (K), f (0) = 0}, Af = 12 f , f ∈ D(A). Finally, here are two more examples where it is difficult to begin with a transition function and the infinitesimal generator is the basic tool of describing the process. Example 2.15 (Sticky barrier Brownian motion). K = [0, ∞) and D(A) = {f ∈ C0 (K) : f ∈ C0 (K), f ∈ C0 (K), f (0) − αf (0) = 0}, Af = 12 f , f ∈ D(A). Here α is a positive constant. This process may be thought of as a “combination” of the reflecting and sticking Brownian motions. The reflecting and sticking cases are obtained by letting α → 0 and α → ∞, respectively. Example 2.16 (Absorbing barrier Brownian motion). K = [0, ∞) where the boundary point 0 is identified with the point at infinity ∂. D(A) = {f ∈ C0 (K) : f ∈ C0 (K), f ∈ C0 (K), f (0) = 0}, Af = 12 f , f ∈ D(A). This represents Brownian motion with an absorbing barrier at x = 0; a Brownian particle “dies” at the first moment when it hits the boundary x = 0. Namely, the point 0 is the terminal point. It is worth pointing out here that a strong Markov process cannot stay at a single position for a positive length of time and then leave that position by continuous motion; it must either jump away or leave instantaneously. We give a simple example of a strong Markov process which changes state not by continuous motion but by jumps when the motion reaches the boundary. Example 2.17. K = [0, ∞). ⎧ ⎨ D(A) = {f ∈ C0 (K) ∩ C 2 (K) : f ∈ C0 (K), f ∈ C0 (K), ∞ f (0) = 2c 0 (f (y) − f (0))dF (y), ⎩ Af = 12 f , f ∈ D(A). Here c is a positive constant and F is a distribution function on the interval (0, ∞). This process may be interpreted as follows. When a Brownian particle reaches the boundary x = 0, it stays there for a positive length of time and then jumps back to a random point, chosen with the function F , in the interior

48

2 Semigroup Theory

(0, ∞). The constant c is the parameter in the “waiting time” distribution at the boundary x = 0. We remark that the boundary condition ∞ (f (y) − f (0)) dF (y) f (0) = 2c 0

depends on the values of f far away from the boundary x = 0, unlike the boundary conditions in Examples 2.13 through 2.16. Although Theorem 2.16 asserts precisely when a linear operator A is the infinitesimal generator of some Feller semigroup, it is usually difficult to verify conditions (b) through (d). So we give useful criteria in terms of the maximum principle (see [BCP], [SU], [Ra], [Ta2, Theorem 9.3.3 and Corollary 9.3.4]): Theorem 2.18 (Hille–Yosida–Ray). Let K be a compact metric space. Then we have the following two assertions: (i) Let B be a linear operator from C(K) = C0 (K) into itself, and assume that: (α) The domain D(B) of B is dense in the space C(K). (β) There exists an open and dense subset K0 of K such that if a function u ∈ D(B) takes a positive maximum at a point x0 of K0 , then we have the inequality Bu(x0 ) ≤ 0. Then the operator B is closable in the space C(K). (ii) Let B be as in part (i), and further assume that: (β ) If a function u ∈ D(B) takes a positive maximum at a point x of K, then we have the inequality Bu(x ) ≤ 0. (γ) For some α0 ≥ 0, the range R(α0 I − B) of α0 I − B is dense in the space C(K). Then the minimal closed extension B of B is the infinitesimal generator of some Feller semigroup on K. Proof. (i) It suffices to show that: {un } ⊂ D(B), un → 0 and Bun → v

in C(K) =⇒ v = 0.

By replacing v by −v if necessary, we assume, to the contrary, that: The function v(x) takes a positive value at some point of K. Then, since K0 is open and dense in K, we can find a point x0 of K0 , a neighborhood U of x0 contained in K0 and a positive constant ε such that we have, for all sufficiently large n, Bun (x) > ε

for all x ∈ U .

(2.42)


49

On the other hand, by condition (α) there exists a function h ∈ D(B) such that h(x0 ) > 1, h(x) < 0 for all x ∈ K \ U . Therefore, since un → 0 in C(K), it follows that the function un (x) = un (x) +

εh(x) 1 + Bh ∞

satisfies the conditions un (x0 ) = un (x0 ) + un (x) = un (x) +

εh(x0 ) > 0, 1 + Bh ∞

εh(x) < 0 for all x ∈ K \ U , 1 + Bh ∞

if n is sufficiently large. This implies that the function un ∈ D(B) takes its positive maximum at a point xn of U ⊂ K0 . Hence we have, by condition (β), Bun (xn ) ≤ 0. However, it follows from inequality (2.42) that Bun (xn ) = Bun (xn ) + ε

Bh(xn ) > Bun (xn ) − ε > 0. 1 + Bh ∞

This is a contradiction. (ii) We apply part (ii) of Theorem 2.16 to the minimal closed extension B of B. The proof is divided into six steps. Step 1: First, we show that u ∈ D(B), (α0 I − B)u ≥ 0

on K =⇒ u ≥ 0 on K.

(2.43)

By condition (γ), we can find a function v ∈ D(B) such that (α0 I − B)v ≥ 1

on K.

Then we have, for any ε > 0, u + εv ∈ D(B), (α0 I − B)(u + εv) ≥ ε

(2.44)

on K.

In view of condition (β ), this implies that the function −(u(x) + εv(x)) does not take any positive maximum on K, so that u(x) + εv(x) ≥ 0 on K.

50

2 Semigroup Theory

Thus, by letting ε ↓ 0 in this inequality we obtain that u(x) ≥ 0

on K.

This proves the desired assertion (2.43). Step 2: It follows from assertion (2.43) that the inverse (α0 I − B)−1 of α0 I − B is defined and non-negative on the range R(α0 I − B). Moreover, it is bounded with norm (α0 I − B)−1 ≤ v ∞ .

(2.45)

Here v(x) is the function which satisfies condition (2.44). Indeed, since g = (α0 I − B)v ≥ 1 on K, it follows that, for all f ∈ C(K), − f ∞ g ≤ f ≤ f ∞ g

on K.

Hence, by the non-negativity of (α0 I − B)−1 we have, for all f ∈ R(α0 I − B), − f ∞v ≤ (α0 I − B)−1 f ≤ f ∞ v

on K.

This proves the desired inequality (2.45). Step 3: Next we show that R(α0 I − B) = C(K).

(2.46)

Let f (x) be an arbitrary element of C(K). By condition (γ), we can find a sequence {un } in D(B) such that fn = (α0 I − B)un → f in C(K). Since the inverse (α0 I − B)−1 is bounded, it follows that un = (α0 I − B)−1 fn converges to some function u ∈ C(K), and hence Bun = α0 un − fn converges to α0 u − f in C(K). Thus we have, by the closedness of B, u ∈ D(B), Bu = α0 u − f, so that (α0 I − B)u = f. This proves the desired assertion (2.46). Step 4: Furthermore, we show that u ∈ D(B), (α0 I − B)u ≥ 0

on K =⇒ u ≥ 0 on K.

(2.47)

Since R(α0 I − B) = C(K), in view of the proof of assertion (2.47) it suffices to show the following: If a function u ∈ D(B) takes a positive maximum at a point x of K, then we have the inequality Bu(x ) ≤ 0. (2.48) Assume, to the contrary, that Bu(x ) > 0.


51

Since there exists a sequence {un } in D(B) such that un → u and Bun → Bu in C(K), we can find a neighborhood U of x and a positive constant ε such that, for all sufficiently large n, Bun (x) > ε

for all x ∈ U .

(2.49)

Furthermore, by condition (α) we can find a function h ∈ D(B) such that h(x ) > 1, h(x) < 0 for all x ∈ K \ U . Then it follows that the function un (x) = un (x) + satisfies the condition

εh(x) 1 + Bh ∞

un (x ) > u(x ) > 0, un (x) < u(x ) for all x ∈ K \ U ,

if n is sufficiently large. This implies that the function un ∈ D(B) takes its positive maximum at a point xn of U . Hence we have, by condition (β ), Bun (xn ) ≤ 0,

xn ∈ U.

However, it follows from inequality (2.49) that Bun (xn ) = Bun (xn ) + ε

Bh(xn ) > Bun (xn ) − ε > 0. 1 + Bh ∞

This is a contradiction. Step 5: In view of Steps 3 and 4, we obtain that the inverse (α0 I − B)−1 of α0 I − B is defined on the whole space C(K), and is bounded with norm (α0 I − B)−1 = (α0 I − B)−1 1 ∞ . Step 6: Finally, we show that: For all α > α0 , the inverse (αI − B)−1 of αI − B is defined on the whole space C(K), and is non-negative and bounded with norm (αI − B)−1 ≤ We let

1 . α

Gα0 = (α0 I − B)−1 .

First, we choose a constant α1 > α0 such that (α1 − α0 ) Gα0 < 1,

(2.50)

52

2 Semigroup Theory

and let α0 < α ≤ α1 . Then, for any f ∈ C(K), the Neumann series u=

∞

I+

! (α0 − α)n Gnα0

Gα0 f

n=1

converges in C(K), and is a solution of the equation u − (α0 − α)Gα0 u = Gα0 f. Hence we have the assertions

u ∈ D(B), (αI − B)u = f.

This proves that R(αI − B) = C(K),

α0 < α ≤ α1 .

(2.51)

Thus, by arguing just as in the proof of Step 1 we obtain that, for any α0 < α ≤ α1 , u ∈ D(B), (αI − B)u ≥ 0 on K =⇒ u ≥ 0 on K. (2.52) By combining assertions (2.51) and (2.52), we find that, for any α0 < α ≤ α1 , the inverse (αI − B)−1 is defined and non-negative on the whole space C(K). We let Gα = (αI − B)−1 , α0 < α ≤ α1 . Then it follows that the operator Gα is bounded with norm Gα ≤

1 . α

(2.53)

Indeed, in view of assertion (2.48) it follows that if a function u ∈ D(B) takes a positive maximum at a point x of K, then we have the inequality Bu(x ) ≤ 0, so that max u(x) = u(x ) ≤ x∈K

1 1 (αI − B)u(x ) ≤ (αI − B)u ∞ . α α

(2.54)

Similarly, if the function u ∈ D(B) takes a negative minimum at a point of K, then (replacing u(x) by −u(x)), we have the inequality − min u(x) = max(−u(x)) ≤ x∈K

x∈K

1 (αI − B)u ∞ . α

(2.55)


53

The desired inequality (2.53) follows from inequalities (2.54) and (2.55). Summing up, we have proved assertion (2.50) for all α0 < α ≤ α1 . Now we assume that assertion (2.50) is proved for all α0 < α ≤ αn−1 , n = 2, 3, . . .. Then, by taking αn = 2αn−1 − or equivalently αn =

α1 , 2

n ≥ 2,

1 2n−2 + α1 , 2

n ≥ 2,

we have, for all αn−1 < α ≤ αn , α − αn−1 αn−1 αn − αn−1 ≤ αn−1 1 = 1 + 22−n < 1.

(α − αn−1 ) Gαn−1 ≤

Hence assertion (2.50) for αn−1 < α ≤ αn is proved just as in the proof of assertion (2.50) for α0 < α ≤ α1 . This proves the desired assertion (2.50). Consequently, by applying part (ii) of Theorem 2.16 to the operator B we obtain that B is the infinitesimal generator of some Feller semigroup on K. The proof of Theorem 2.18 is now complete. Corollary 2.19. Let A be the infinitesimal generator of a Feller semigroup on a compact metric space K and let M be a bounded linear operator on C(K) into itself. Assume that either M or A = A+M satisfies condition (β ). Then the operator A is the infinitesimal generator of some Feller semigroup on K. Proof. We apply part (ii) of Theorem 2.18 to the operator A . First, we remark that A = A + M is a densely defined, closed linear operator from C(K) into itself. Since the semigroup {Tt }t≥0 is non-negative and contractive on C(K), it follows that if a function u ∈ D(A) takes a positive maximum at a point x of K, then we have the inequality Au(x ) = lim t↓0

Tt u(x ) − u(x ) ≤ 0. t

This implies that if M satisfies condition (β ), so does A = A + M . We let Gα0 = (α0 I − A)−1 , α0 > 0. If α0 is so large that Gα0 M ≤ Gα0 · M ≤

M < 1, α0

54

2 Semigroup Theory

then the Neumann series u=

I+

∞

! (Gα0 M )n

Gα0 f

n=1

converges in C(K) for any f ∈ C(K), and is a solution of the equation u − Gα0 M u = Gα0 f. Hence we have the assertions

This proves that

u ∈ D(A) = D(A ), (α0 I − A )u = f.

R(α0 I − A ) = C(K).

Therefore, by applying part (ii) of Theorem 2.18 to the operator A we obtain that A is the infinitesimal generator of some Feller semigroup on K. The proof of Corollary 2.19 is complete.

3 Lp Theory of Pseudo-Differential Operators

In this chapter we present a brief description of the basic concepts and results of the Lp theory of pseudo-differential operators which may be considered as a modern theory of the classical potential theory. In particular, we formulate the Besov space boundedness theorem due to Bourdaud [Bo] (Theorem 3.15) and a useful criterion for hypoellipticity due to H¨ ormander [Ho2] (Theorem 3.16) which play an essential role in the proof of our main results. For detailed studies of pseudo-differential operators, the reader is referred to Chazarain– Piriou [CP], H¨ ormander [Ho3], Kumano-go [Ku] and Taylor [Ty].

3.1 Function Spaces Let Ω be a bounded domain of Euclidean space Rn with smooth boundary Γ = ∂Ω. Its closure Ω = Ω ∪Γ is an n-dimensional, compact smooth manifold with boundary. We may assume the following (see Figures 3.1 and 3.2): (a) The domain Ω is a relatively compact open subset of an n-dimensional, compact smooth manifold M without boundary in which Ω has a smooth boundary Γ . (b) In a neighborhood W of Γ in M a normal coordinate t is chosen so that the points of W are represented as (x , t), x ∈ Γ , −1 < t < 1; t > 0 in Ω, t < 0 in M \ Ω and t = 0 only on Γ . (c) The manifold M is equipped with a strictly positive density μ which, on W , is the product of a strictly positive density ω on Γ and the Lebesgue measure dt on (−1, 1). This manifold M is called the double of Ω. The function spaces we shall treat are the following (cf. [AF], [BL], [Ca], [Fr1], [Tb], [Tr]): (i) The generalized Sobolev spaces H s,p (Ω) and H s,p (M ), consisting of all potentials of order s of Lp functions. When s is integral, these spaces coincide with the usual Sobolev spaces W s,p (Ω) and W s,p (M ), respectively. K. Taira, Boundary Value Problems and Markov Processes, Lecture Notes in Mathematics 1499, DOI 10.1007/978-3-642-01677-6 3, c Springer-Verlag Berlin Heidelberg 2009

55

56


M

Ω = {t >0}

Γ=∂Ω = { t =0}

Fig. 3.1.

W

Γ = {t=0} {0< t < 1} {−1< t< 0} Fig. 3.2.

(ii) The Besov spaces B s,p (Γ ). These are functions spaces defined in terms of the Lp modulus of continuity, and enter naturally in connection with boundary value problems. First, if 1 ≤ p < ∞, we let Lp (Ω) = the space of (equivalence classes of) Lebesgue measurable functions u(x) on Ω such that |u(x)|p is integrable on Ω. The space Lp (Ω) is a Banach space with the norm u p =

1/p |u(x)| dx . p

Ω

For p = ∞, we let L∞ (Ω) = the space of (equivalence classes of) essentially bounded, Lebesgue measurable functions u(x) on Ω. The space L∞ (Ω) is a Banach space with the norm u ∞ = ess supx∈Ω |u(x)|.

3.1 Function Spaces

57

We recall the basic definitions and facts about the Fourier transform. If f ∈ L1 (Rn ), we define its (direct) Fourier transform F f (ξ) by the formula e−ix·ξ f (x)dx, ξ = (ξ1 , ξ2 , . . . , ξn ), F f (ξ) = Rn

√ where i = −1 and x · ξ = x1 ξ1 + x2 ξ2 + . . . + xn ξn . We also denote F f (ξ) by f"(ξ). Similarly, if g ∈ L1 (Rn ), we define its inverse Fourier transform F ∗ g(x) by the formula 1 ∗ eix·ξ g(ξ) dξ. F g(x) = (2π)n Rn We also denote F ∗ g(x) by gˇ(x). We introduce a subspace of L1 (Rn ) which is invariant under the Fourier transform. We define the Schwartz space S(Rn ) = the space of smooth functions ϕ(x) rapidly decreasing at infinity on Rn such that we have, for any non-negative integer j, pj (ϕ) = sup (1 + |x|2 )j/2 |∂ α ϕ(x)| < ∞. x∈Rn |α|≤j

We equip the space S(Rn ) with the topology defined by the countable family {pj } of seminorms. The space S(Rn ) is a Fréchet space. The transforms F and F ∗ map S(Rn ) continuously into itself, and F F ∗ = F ∗ F = I on S(Rn ). Example 3.1. For a > 0, we have the assertion 2

ϕ(x) = e−a|x| ∈ S(Rn ). Furthermore, it is easy to verify the following formulas: π n/2 |x|2 2 e−ix·ξ e−a|x| dx = e− 4a , F ϕ(ξ) = a n R 1 ∗ F (F ϕ)(x) = eix·ξ ϕ(ξ) " dξ = ϕ(x). (2π)n Rn Since the injection of C0∞ (Rn ) into S(Rn ) is continuous, it follows that the dual space S (Rn ) of S(Rn ) consists of those distributions T ∈ D (Rn ) that have continuous extensions to S(Rn ). The elements of S (Rn ) are called tempered distributions on Rn . Roughly speaking, the tempered distributions are those which grow at most polynomially at infinity, since the functions in S(Rn ) die out faster than any power of x at infinity. More precisely, we have the following examples of tempered distributions: (1) The functions in Lp (Rn ), 1 ≤ p ≤ ∞, are tempered distributions.

58


(2) A locally integrable function on Rn is a tempered distribution if it grows at most polynomially at infinity. (3) If u ∈ S (Rn ) and f (x) is a smooth function on Rn all of whose derivatives grow at most polynomially at infinity, then the product f (x)u(x) is a tempered distribution. (4) Any derivative of a tempered distribution is also a tempered distribution. Now we give some concrete and important examples of distributions in the space S (Rn ): Example 3.2. (a) The Dirac measure: δ(x). (b) Riesz potentials: Rα (x) =

Γ ((n − α)/2) 1 , n−α α n/2 2 π Γ (α/2) |x|

0 < α < n.

(c) Newtonian potentials: N (x) =

Γ ((n − 2)/2) 1 , |x|n−2 4π n/2

(d) Bessel potentials: Gα (x) =

1 1 Γ (α/2) (4π)n/2

∞

e−t−

|x|2 4t

n ≥ 3.

t

α−n 2

0

dt , t

α > 0.

It is known (see [AS]) that the function Gα (x) is represented as follows: Gα (x) =

α−n 1 K(n−α)/2 (|x|) |x| 2 , 2(n+α−2)/2 π n/2 Γ (α/2)

where K(n−α)/2 (z) is the modified Bessel function of the third kind (cf. [Wt]). (e) Riesz kernels: Rj (x) =

√ Γ ((n + 1)/2) xj −1 v. p. n+1 , (n+1)/2 |x| π

1 ≤ j ≤ n.

The distribution v. p. (xj /|x|n+1 ) is an extension of v. p. (1/x) to the ndimensional case. The importance of tempered distributions lies in the fact that they have Fourier transforms. The direct and inverse Fourier transforms can be extended to the space S (Rn ) by the following formulas: Fu, ϕ = u, F ϕ , F ∗ v, ϕ = v, F ∗ ϕ ,

u ∈ S (Rn ), ϕ ∈ S(Rn ). v ∈ S (Rn ), ϕ ∈ S(Rn ).

Once again, the transforms F and F ∗ map S (Rn ) continuously into itself, and F F ∗ = F ∗ F = I on S (Rn ). We can calculate explicitly the Fourier transform of the tempered distributions in Example 3.2 as follows:

3.1 Function Spaces

59

Example 3.3. (a) The Dirac measure: (F δ)(ξ) = 1. (b) Riesz potentials: (F Rα )(ξ) =

1 , |ξ|α

0 < α < n.

(c) Newtonian potentials: (F N )(ξ) =

1 , |ξ|2

n ≥ 3.

(d) Bessel potentials: (F Gα )(ξ) =

1 , (1 + |ξ|2 )α/2

α > 0.

(e) Riesz kernels: (F Rj )(ξ) =

ξj , |ξ|

1 ≤ j ≤ n.

If s ∈ R, we define a linear map J s : S (Rn ) −→ S (Rn ) by the formula J s u = F ∗ (1 + |ξ|2 )−s/2 F u ,

u ∈ S (Rn ).

This can be visualized as follows: u ∈ S (Rn ) ⏐ ⏐ F$

Js

−−−−→

S (Rn ) J s u % ⏐ ∗ ⏐F

F u ∈ S (Rn ) −−−−−−−−→ S (Rn ) (1 + |ξ|2 )−s/2 F u (1+|ξ|2 )−s/2

Then it is easy to see that the map J s is an isomorphism of S (Rn ) onto itself and that its inverse is the map J −s . The function J s u is called the Bessel potential of order s of u. (I) Now, if s ∈ R and 1 < p < ∞, we let H s,p (Rn ) = the image of Lp (Rn ) under the mapping J s . We equip H s,p (Rn ) with the norm u s,p = J −s u p for u ∈ H s,p (Rn ). The space H s,p (Rn ) is called the (generalized) Sobolev space of order s. We list some basic topological properties of H s,p (Rn ): (1) The Schwartz space S(Rn ) is dense in each H s,p (Rn ).

60


(2) The space H −s,p (Rn ) is the dual space of H s,p (Rn ), where p = p/(p − 1) is the exponent conjugate to p. (3) If s > t, then we have the inclusions S(Rn ) ⊂ H s,p (Rn ) ⊂ H t,p (Rn ) ⊂ S (Rn ), with continuous injections. (4) If s is a non-negative integer, then the space H s,p (Rn ) is isomorphic to the usual Sobolev space W s,p (Rn ), that is, the space H s,p (Rn ) coincides with the space of functions u ∈ Lp (Rn ) such that Dα u ∈ Lp (Rn ) for |α| ≤ s, and the norm · s,p is equivalent to the norm ⎛ ⎝

|α|≤s

Rn


.

(II) Next, if 1 < p < ∞, we let B 1,p (Rn−1 ) = the space of (equivalence classes of) functions ϕ(x ) ∈ Lp (Rn−1 ) for which the integral |ϕ(x + y ) − 2ϕ(x ) + ϕ(x − y )|p dy dx |y |n−1+p Rn−1 ×Rn−1 is finite. The space B 1,p (Rn−1 ) is a Banach space with respect to the norm |ϕ(x )|p dx |ϕ|1,p = Rn−1

+

Rn−1 ×Rn−1

|ϕ(x + y ) − 2ϕ(x ) + ϕ(x − y )|p dy dx |y |n−1+p

1/p

If p = ∞, we let B 1,∞ (Rn−1 ) = the space of (equivalence classes of) functions ϕ(x ) ∈ L∞ (Rn−1 ) for which the quantity ϕ(· + y ) − 2ϕ(·) + ϕ(· − y ) ∞ sup |y | |y |>0 is finite. The space B 1,∞ (Rn−1 ) is a Banach space with respect to the norm ϕ(· + y ) − 2ϕ(·) + ϕ(· − y ) ∞ . |y | |y |>0

|ϕ|1,∞ = ϕ ∞ + sup If s ∈ R, we let

.

3.1 Function Spaces

B s,p (Rn−1 ) = the image of B 1,p (Rn−1 ) under the mapping J

s−1

61

, where

s−1

is the Bessel potential of order s − 1 on Rn−1 . −s+1 We equip the space B s,p (Rn−1 ) with the norm |ϕ|s,p = J ϕ J

1,p

for

ϕ(x ) ∈ B (R ). The space B (R ) is called the Besov space of order s. We list some basic topological properties of B s,p (Rn−1 ): s,p

n−1

s,p

n−1

(1) The Schwartz space S(Rn−1 ) is dense in each B s,p (Rn−1 ). (2) The space B −s,p (Rn−1 ) is the dual space of B s,p (Rn−1 ), where p = p/(p − 1) is the exponent conjugate to p. (3) If s > t, then we have the inclusions S(Rn−1 ) ⊂ B s,p (Rn−1 ) ⊂ B t,p (Rn−1 ) ⊂ S (Rn−1 ), with continuous injections. (4) If s = m + σ where m is a non-negative integer and 0 < σ < 1, then the Besov space B s,p (Rn−1 ) coincides with the space of functions ϕ(x ) ∈ H m,p (Rn−1 ) such that, for |α| = m, the integral (Slobodecki˘ı seminorm) |Dα ϕ(x ) − Dα ϕ(y )|p dx dy < ∞. |x − y |n−1+pσ Rn−1 ×Rn−1 Furthermore, the norm |ϕ|s,p is equivalent to the norm |Dα ϕ(x )|p dx |α|≤m

+

Rn−1

|α|=m

Rn−1 ×Rn−1

|Dα ϕ(x ) − Dα ϕ(y )|p dx dy |x − y |n−1+pσ

1/p .

Now we define the generalized Sobolev spaces H s,p (Ω), H s,p (M ) and the Besov spaces B s,p (Γ ) for arbitrary values of s. For each s ∈ R, we define H s,p (Ω) = the space of distributions u ∈ D (Ω) such that there exists a function U ∈ H s,p (Rn ) with U |Ω = u, and equip the space H s,p (Ω) with the norm u s,p = inf U s,p , where the infimum is taken over all such U . The space H s,p (Ω) is a Banach space with respect to the norm · s,p . We remark that H 0,p (Ω) = Lp (Ω);

· 0,p = · p .

62


The spaces H s,p (M ) are defined to be locally the spaces H s,p (Rn ), upon using local coordinate systems flattening out M , together with a partition of unity. The spaces B s,p (Γ ) are defined similarly, with H s,p (Rn ) replaced by B s,p (Rn−1 ). The norms of H s,p (M ) and B s,p (Γ ) will be denoted by · s,p and | · |s,p , respectively. We state two important theorems that will be used in the study of boundary value problems in the framework of Sobolev spaces of Lp type (see [AF], [BL], [St1], [Tr]): (I) (The trace theorem) Let 1 < p < ∞. Then the trace map ρ : H s,p (Ω) −→ B s−1/p,p (∂Ω) u −→ u|∂Ω is continuous for all s > 1/p, and is surjective. (II) (The Rellich–Kondrachov theorem) If s > t, then the injections H s,p (M ) −→ H t,p (M ), B s,p (∂Ω) −→ B t,p (∂Ω) are both compact (or completely continuous). Finally, we introduce a space of distributions on Ω which behave locally just like the distributions in H s,p (Rn ): s,p Hloc (Ω) = the space of distributions u ∈ D (Ω) such that

ϕu ∈ H s,p (Rn ) for all ϕ ∈ C0∞ (Ω). s,p We equip the localized Sobolev space Hloc (Ω) with the topology defined by the seminorms u → ϕu s,p as ϕ ranges over C0∞ (Ω). It is easy to verify that s,p s,p (Ω) is a Fréchet space. The localized Besov space Bloc (∂Ω) is defined Hloc s,p n s,p n−1 similarly, with H (R ) replaced by B (R ).

3.2 Fourier Integral Operators In this section, we present a brief description of basic concepts and results of the theory of Fourier integral operators. 3.2.1 Symbol Classes Let Ω be an open subset of Rn . If m ∈ R and 0 ≤ δ < ρ ≤ 1, we let m Sρ,δ (Ω × RN ) = the set of all functions a(x, θ) ∈ C ∞ (Ω × RN ) with the property that, for any compact K ⊂ Ω and

any multi-indices α, β, there exists a positive constant CK,α,β such that we have, for all x ∈ K and θ ∈ RN , α β ∂ ∂ a(x, θ) ≤ CK,α,β (1 + |θ|)m−ρ|α|+δ|β| . θ

x

3.2 Fourier Integral Operators

63

m The elements of Sρ,δ (Ω × RN ) are called symbols of order m. We drop the N m Ω × R and use Sρ,δ when the context is clear. & α of order m with Example 3.4. (1) A polynomial p(x, ξ) = |α|≤m aα (x)ξ ∞ m n coefficients in C (Ω) is in S1,0 (Ω × R ). (2) If m ∈ R, the function

m/2 Ω × Rn (x, ξ) −→ 1 + |ξ|2 m is in S1,0 (Ω × Rn ). (3) A function a(x, θ) ∈ C ∞ (Ω × (RN \ {0}) is said to be positively homogeneous of degree m in θ if it satisfies the condition

for all t > 0 and θ ∈ RN \ {0}.

a(x, tθ) = tm a(x, θ)

If a(x, θ) is positively homogeneous of degree m in θ and if ϕ(θ) is a smooth function such that ϕ(θ) = 0 for |θ| ≤ 1/2 and ϕ(θ) = 1 for |θ| ≥ 1, then the m (Ω × RN ). function ϕ(θ)a(x, θ) is in S1,0 If K is a compact subset of Ω and if j is a non-negative integer, we define m (Ω × RN ) by the formula a seminorm pK,j,m on Sρ,δ m (Ω Sρ,δ

× R ) a −→ pK,j,m (a) = N

sup x∈K, θ∈RN |α|≤j

α β ∂ ∂x a(x, θ) θ . (1 + |θ|)m−ρ|α|+δ|β|

m (Ω × RN ) with the topology defined by the family We equip the space Sρ,δ {pK,j,m } of seminorms where K ranges over all compact subsets of Ω and m (Ω × RN ) is a Fréchet space. j = 0, 1, . . .. The space Sρ,δ We set m S −∞ (Ω × RN ) = Sρ,δ (Ω × RN ). m∈R

The next theorem gives a meaning to a formal sum of symbols of decreasing order: m

Theorem 3.1. Let aj (x, θ) ∈ Sρ,δj (Ω × RN ), mj ↓ −∞, j = 0, 1, . . .. Then m0 there exists a symbol a(x, θ) ∈ Sρ,δ (Ω × RN ), unique modulo S −∞ (Ω × RN ), such that we have, for all positive integer k, a(x, θ) −

k−1

mk aj (x, θ) ∈ Sρ,δ (Ω × RN ).

j=0

If formula (3.1) holds true, we write a(x, θ) ∼

∞ j=0

aj (x, θ).

(3.1)

64


&∞ The formal sum j=0 aj (x, θ) is called an asymptotic expansion of a(x, θ). m A symbol a(x, θ) ∈ S1,0 (Ω × RN ) is said to be classical if there exist smooth functions aj (x, θ), positively homogeneous of degree m − j in θ for |θ| ≥ 1, such that we have, for all positive integer k, a(x, θ) −

k−1

m−k aj (x, θ) ∈ S1,0 (Ω × RN ).

j=0

The homogeneous function a0 (x, θ) of degree m is called the principal part of a(x, θ). We let m (Ω × RN ) = the set of all classical symbols of order m. Scl

For example, the symbols in Example 3.4 are all classical, and they have respectively as principal part the following functions: & (1) pm (x, ξ) = |α|=m aα (x)ξ α . (2) |ξ|m . (3) a(x, θ). m (Ω × RN ) is said to be elliptic of order m if, for A symbol a(x, θ) in Sρ,δ any compact K ⊂ Ω, there exists a positive constant CK such that

|a(x, θ)| ≥ CK (1 + |θ|)m

for all x ∈ K and |θ| ≥

1 CK .

There is a simple criterion in the case of classical symbols. m (Ω × RN ) with principal part a0 (x, θ). Theorem 3.2. Let a(x, θ) be in Scl Then a(x, θ) is elliptic if and only if it satisfies the condition

a0 (x, θ) = 0

for all x ∈ Ω and |θ| = 1.

3.2.2 Phase Functions Let Ω be an open subset of Rn . A function ϕ(x, θ) in C ∞ Ω × RN \ {0} is called a phase function on Ω × (RN \ {0}) if it satisfies the following three conditions: (a) ϕ(x, θ) is real-valued. (b) ϕ(x, θ) is positively homogeneous of degree one in the variable θ. (c) The differential dϕ(x, θ) does not vanish on Ω × RN \ {0} . Example 3.5. Let U be an open subset of Rp and Ω = U × U . The function ϕ(x, y, ξ) = (x − y) · ξ is a phase function on the space Ω × (Rp \ {0}) with n = 2p and N = p.

3.2 Fourier Integral Operators

65

The next lemma will play a fundamental role in defining oscillatory integrals. Lemma 3.3. If ϕ(x, θ) is a phase function on Ω × RN \ {0} , then there exists a first-order differential operator L=

N

∂ ∂ + bk (x, θ) + c(x, θ) ∂θj ∂xk n

aj (x, θ)

j=1

k=1

such that L(eiϕ ) = eiϕ , and that its coefficients aj (x, θ), bk (x, θ), c(x, θ) enjoy the following properties: −1 0 aj (x, θ) ∈ S1,0 ; bk (x, θ), c(x, θ) ∈ S1,0 .

Furthermore, the transpose L of L has coefficients aj (x, θ), bk (x, θ), c (x, θ) in the same symbol classes as aj (x, θ), bk (x, θ), c(x, θ), respectively. For example, if ϕ(x, y, ξ) is a phase function as in Example 3.5 ϕ(x, y, ξ) = (x − y) · ξ,

(x, y) ∈ U × U, ξ ∈ (Rp \ {0}) ,

then the operator L is given by the formula ' p p 1 ξk ∂ 1 − ρ(ξ) ∂ L= √ (x − y ) + j j ∂ξj |ξ|2 ∂xk −1 2 + |x − y|2 j=1 k=1 ( p −ξk ∂ + + ρ(ξ), |ξ|2 ∂yk k=1

where ρ(ξ) is a function in C0∞ (Rp ) such that ρ(ξ) = 1 for |ξ| ≤ 1. 3.2.3 Oscillatory Integrals If Ω is an open subset of Rn , we let ∞ m Ω × RN = Ω × RN . Sρ,δ Sρ,δ m∈R

If ϕ(x, θ) is a phase function on Ω × RN \ {0} , we wish to give a meaning to the integral eiϕ(x,θ) a(x, θ)u(x) dx dθ, u ∈ C0∞ (Ω), (3.2) Iϕ (aw) = Ω×RN

∞ for each symbol a(x, θ) ∈ Sρ,δ Ω × RN .

66


By Lemma 3.3, we can replace eiϕ in formula (3.2) by L(eiϕ ). Then a formal integration by parts gives us that eiϕ(x,θ) L (a(x, θ)w(x, y)) dx dθ. Iϕ (au) = Ω×RN

However, the properties of the coefficients of the transpose L imply that L r−η r maps Sρ,δ continuously into Sρ,δ for all r ∈ R, where η = min(ρ, 1 − δ). By continuing this process, we can reduce the growth of the integrand at infinity until it becomes integrable, and give a meaning to the integral (3.2) for each ∞ (Ω × Rn ). symbol a(x, θ) ∈ Sρ,δ More precisely, we have the following: Theorem 3.4. (i) The linear functional S −∞ Ω × RN a −→ Iϕ (au) ∈ C ∞ extends uniquely to a linear functional on Sρ,δ Ω × RN whose restriction m to each Sρ,δ Ω × RN is continuous. Furthermore, the restriction of the linear m Ω × RN is expressed as the formula functional to Sρ,δ (a) = eiϕ(x,θ) (L )k (a(x, θ)w(x, y)) dx dθ, Ω×RN

where k > (m + N )/η and η = min(ρ, 1 − δ). m Ω × RN , the mapping (ii) For any fixed a(x, θ) ∈ Sρ,δ C0∞ (Ω) u −→ Iϕ (au) = (a) ∈ C

(3.3)

is a distribution of order ≤ k for k > (m + N )/η with η = min(ρ, 1 − δ). ∞ an oscillatory integral, but use the We call the linear functional on Sρ,δ standard notation as in formula (3.2). The distribution (3.3) is called the Fourier integral distribution associated with the phase function ϕ(x, θ) and the amplitude a(x, θ), and will be denoted as follows: eiϕ(x,θ)a(x, θ) dθ. K(x) = RN

If u is a distribution on Ω, then the singular support of u is the smallest closed subset of Ω outside of which u is smooth. The singular support of u is denoted by sing supp u. The next theorem estimates the singular support of a Fourier integral distribution. N \ {0} Theorem 3.5. If ϕ(x, θ) is a phase function on the space Ω × R ∞ Ω × RN , then the distribution and if a(x, θ) is in Sρ,δ K(x) = eiϕ(x,θ) a(x, θ) dθ ∈ D (Ω) RN

satisfies the condition

sing supp K ⊂ x ∈ Ω : dθ ϕ(xθ) = 0 for some θ ∈ RN \ {0} .

3.3 Pseudo-Differential Operators

67

3.2.4 Fourier Integral Operators Let U and V be open subsets of Rp and Rq , respectively. If ϕ(x, y, θ) is a ∞ phase function on U × V × (RN \ {0}) and if a(x, y, θ) ∈ Sρ,δ (U × V × RN ), then there is associated a distribution K ∈ D (U × V ) defined by the formula K(x, y) = eiϕ(x,y,θ)a(x, y, θ) dθ. RN

By applying Theorem 3.5 to our situation, we obtain that sing supp K ⊂ (x, y) ∈ U × V : dθ ϕ(x, y, θ) = 0 for some θ ∈ RN \ {0} . The distribution K ∈ D (U × V ) defines a continuous linear operator A : C0∞ (V ) −→ D (U ) by the formula Av, u = K, u ⊗ v ,

u ∈ C0∞ (U ), v ∈ C0∞ (V ).

The operator A is called the Fourier integral operator associated with the phase function ϕ(x, y, θ) and the amplitude a(x, y, θ), and will be denoted as follows: Av(x) = eiϕ(x,y,θ)a(x, y, θ)v(y) dy dθ, v ∈ C0∞ (V ). V ×RN

The next theorem summarizes some basic properties of the operator A: Theorem 3.6. (i) If dy,θ ϕ(x, y, θ) = 0 on U × V × RN \ {0} , then the operator A maps C0∞ (V ) continuously into C ∞ (U ). (ii) If dx,θ ϕ(x, y, θ) = 0 on U × V × RN \ {0} , then the operator A extends to a continuous linear operator on E (V ) into D (U ). (iii) If dy,θ ϕ(x, y, θ) = 0 and dx,θ ϕ(x, y, θ) = 0 on U × V × RN \ {0} , then we have, for all v ∈ E (V ), sing supp Av ⊂ x ∈ U : dθ ϕ(x, y, θ) = 0 for some y ∈ sing supp v and θ ∈ RN \ {0} .

3.3 Pseudo-Differential Operators Let Ω1 and Ω2 be open subsets of Rn1 and Rn2 , respectively. If K(x1 , x2 ) is a distribution in D (Ω1 × Ω2 ), we can define a continuous linear operator A ∈ L(C0∞ (Ω2 ), D (Ω1 )) by the formula

68


Aψ, ϕ = K, ϕ ⊗ ψ ,

ϕ ∈ C0∞ (Ω1 ), ψ ∈ C0∞ (Ω2 ).

We then write A = Op (K). Since the tensor space C0∞ (Ω1 ) ⊗ C0∞ (Ω2 ) is sequentially dense in C0∞ (Ω1 × Ω2 ), it follows that the mapping D (Ω1 × Ω2 ) K −→ Op (K) ∈ L(C0∞ (Ω2 ), D (Ω1 )) is injective. The next theorem asserts that it is also surjective: Theorem 3.7 (the Schwartz kernel theorem). If A is a continuous linear operator on C0∞ (Ω2 ) into D (Ω1 ), then there exists a unique distribution KA (x1 , x2 ) in D (Ω1 × Ω2 ) such that A = Op (K). The distribution KA is called the kernel of A. Formally we have the formula KA (x1 , x2 ) ψ(x2 ) dx2 , ψ ∈ C0∞ (Ω2 ). Aψ(x1 ) = Ω2

Now we give some important examples of distributions kernels (see Example 3.2): Example 3.6. (a) Riesz potentials: Ω1 = Ω2 = Rn , 0 < α < n. (−Δ)−α/2 u(x) = Rα ∗ u(x) Γ ((n − α)/2) 1 = α n/2 u(y) dy, |x − y|n−α 2 π Γ (α/2) Rn

u ∈ C0∞ (Rn ).

(b) Newtonian potentials: Ω1 = Ω2 = Rn , n ≥ 3. (−Δ)−1 u(x) = N ∗ u(x) Γ ((n − 2)/2) 1 = u(y) dy, |x − y|n−2 4π n/2 n R (c) Bessel potentials: Ω1 = Ω2 = Rn , α > 0. Gα (x − y) u(y) dy, (I − Δ)−α/2 u(x) = Gα ∗ u(x) = Rn

u ∈ C0∞ (Rn ).

u ∈ C0∞ (Rn ).

(d) Riesz operators: Ω1 = Ω2 = Rn , 1 ≤ j ≤ n. Yj u(x) = Rj ∗ u(x) √ Γ ((n + 1)/2) xj − yj = −1 v. p. u(y) dy, |x − y|n+1 π (n+1)/2 n R

u ∈ C0∞ (Rn ).

(e) The Calder´ on–Zygmund integro-differential operator: Ω1 = Ω2 = Rn . n ∂u 1 1/2 (−Δ) u(x) = √ Yj (x) ∂xj −1 j=1 n xj − yj ∂u Γ ((n + 1)/2) v. p. (y) dy, = |x − y|n+1 ∂yj π (n+1)/2 n R j=1 u ∈ C0∞ (Rn ).


69

Let Ω be an open subset of Rn and m ∈ R. A pseudo-differential operator of order m on Ω is a Fourier integral operator of the form Au(x) = ei(x−y)·ξ a(x, y, ξ)u(y) dydξ, u ∈ C0∞ (Ω), (3.4) Ω×Rn

m with some a(x, y, ξ) ∈ Sρ,δ (Ω × Ω × Rn). In other words, a pseudo-differential operator of order m is a Fourier integral operator associated with the phase m function ϕ(x, y, ξ) = (x − y) · ξ and some amplitude a(x, y, ξ) ∈ Sρ,δ (Ω × Ω × n R ). We let

Lm ρ,δ (Ω) = the set of all pseudo-differential operators of order m on Ω. By applying Theorems 3.5 and 3.6 to our situation, we obtain the following three assertions: (1) A pseudo-differential operator A maps C0∞ (Ω) continuously into C ∞ (Ω) and extends to a continuous linear operator A : E (Ω) → D (Ω). (2) The distribution kernel KA (x, y) of a pseudo-differential operator A satisfies the condition sing supp KA ⊂ {(x, x) : x ∈ Ω}, that is, the kernel KA is smooth off the diagonal {(x, x) : x ∈ Ω} in Ω ×Ω. (3) sing supp Au ⊂ sing supp u, u ∈ E (Ω). In other words, Au is smooth whenever u is. This property is referred to as the pseudo-local property. We set

L−∞ (Ω) =

Lm ρ,δ (Ω).

m∈R

The next theorem characterizes the class L−∞ (Ω). Theorem 3.8. The following three conditions are equivalent: (i) A ∈ L−∞ (Ω). (ii) A is written in the form (3.4) with some a ∈ S −∞ (Ω × Ω × Rn ). (iii) A is a regularizer, or equivalently, its distribution kernel KA (x, y) is in the space C ∞ (Ω × Ω). We recall that a continuous linear operator A : C0∞ (Ω) → D (Ω) is said to be properly supported if the following two conditions are satisfied: (a) For any compact subset K of Ω, there exists a compact subset K of Ω such that supp v ⊂ K =⇒ supp Av ⊂ K . (b) For any compact subset K of Ω, there exists a compact subset K ⊃ K of Ω such that supp v ∩ K = ∅ =⇒ supp Av ∩ K = ∅.


70

If A is properly supported, then it maps C0∞ (Ω) continuously into E (Ω) and extends to a continuous linear operator on C ∞ (Ω) into D (Ω). The next theorem states that every pseudo-differential operator can be written as the sum of a properly supported operator and a regularizer. Theorem 3.9. If A ∈ Lm ρ,δ (Ω), then we have the decomposition A = A0 + R, −∞ (Ω). where A0 ∈ Lm ρ,δ (Ω) is properly supported and R ∈ L m If p(x, ξ) ∈ Sρ,δ (Ω×Rn ), then the operator p(x, D), defined by the formula 1 eix·ξ p(x, ξ) u "(ξ) dξ, u ∈ C0∞ (Ω), (3.5) p(x, D)u(x) = (2π)n Rn

is a pseudo-differential operator of order m on Ω, that is, p(x, D) ∈ Lm ρ,δ (Ω). The next theorem asserts that every properly supported pseudo-differential operator can be reduced to the form (3.5). Theorem 3.10. If A ∈ Lm ρ,δ (Ω) is properly supported, then we have the assertion m p(x, ξ) = e−ix·ξ A(eix·ξ ) ∈ Sρ,δ (Ω × Rn ),

and A = p(x, D). m (Ω × Ω × Rn ) is an amplitude for A, then we Furthermore, if a(x, y, ξ) ∈ Sρ,δ have the following asymptotic expansion: 1 ∂ξα Dyα (a(x, y, ξ)) . p(x, ξ) ∼ α! y=x α≥0

The function p(x, ξ) is called the complete symbol of A. We extend the notion of a complete symbol to the whole space Lm ρ,δ (Ω). m If A ∈ Lm (Ω), then we choose a properly supported operator A ∈ L 0 ρ,δ ρ,δ (Ω) −∞ such that A − A0 ∈ L (Ω), and define σ(A) = the equivalence class of the complete symbol of A0 in m (Ω × Rn )/S −∞ (Ω × Rn ). Sρ,δ In view of Theorems 3.10 and 3.11, it follows that σ(A) does not depend on the operator A0 chosen. The equivalence class σ(A) is called the complete symbol of A. It is easy to see that the mapping m n −∞ (Ω × Rn ) Lm ρ,δ (Ω) A −→ σ(A) ∈ Sρ,δ (Ω × R )/S


71

induces an isomorphism −∞ m (Ω) −→ Sρ,δ (Ω × Rn )/S −∞ (Ω × Rn ). Lm ρ,δ (Ω)/L

We shall often identify the complete symbol σ(A) with a representative m in the class Sρ,δ (Ω × Rn ) for notational convenience, and call any member of σ(A) a complete symbol of A. A pseudo-differential operator A ∈ Lm 1,0 (Ω) is said to be classical if its m complete symbol σ(A) has a representative in the class Scl (Ω × Rn ). We let Lm cl (Ω) = the set of all classical pseudo-differential operators of order m on Ω. Then the mapping m n −∞ Lm (Ω × Rn ) cl (Ω) A −→ σ(A) ∈ Scl (Ω × R )/S

induces an isomorphism −∞ m Lm (Ω) −→ Scl (Ω × Rn )/S −∞ (Ω × Rn ). cl (Ω)/L

Also we have the assertion L−∞ (Ω) =

Lm cl (Ω).

m∈R

If A ∈ Lm cl (Ω), then the principal part of σ(A) has a canonical representative σA (x, ξ) ∈ C ∞ (Ω × (Rn \ {0})) which is positively homogeneous of degree m in the variable ξ. The function σA (x, ξ) is called the homogeneous principal symbol of A. The next two theorems assert that the class of pseudo-differential operators forms an algebra closed under the operations of composition of operators and taking the transpose or adjoint of an operator. ∗ Theorem 3.11. If A ∈ Lm ρ,δ (Ω), then its transpose A and its adjoint A are m ∗ both in Lρ,δ (Ω), and the complete symbols σ(A ) and σ(A ) have respectively the following asymptotic expansions:

1 ∂ α Dα (σ(A)(x, −ξ)) , α! ξ x α≥0 1 ∗ σ(A )(x, ξ) ∼ ∂ξα Dxα σ(A)(x, ξ) . α! σ(A )(x, ξ) ∼

α≥0

m Theorem 3.12. If A ∈ Lm ρ ,δ (Ω) and B ∈ Lρ ,δ (Ω) where 0 ≤ δ < ρ ≤ 1 and if one of them is properly supported, then the composition AB is in


72

+m Lm (Ω) with ρ = min(ρ , ρ ), δ = max(δ , δ ), and we have the following ρ,δ asymptotic expansion:

σ(AB)(x, ξ) ∼

1 ∂ α (σ(A)(x, ξ)) · Dxα (σ(B)(x, ξ)) . α! ξ

α≥0

A pseudo-differential operator A ∈ Lm ρ,δ (Ω) is said to be elliptic of order m if its complete symbol σ(A) is elliptic of order m. In view of Theorem 3.2, it follows that a classical pseudo-differential operator A ∈ Lm cl (Ω) is elliptic if and only if its homogeneous principal symbol σA (x, ξ) does not vanish on the space Ω × (Rn \ {0}). The next theorem states that elliptic operators are the “invertible” elements in the algebra of pseudo-differential operators. Theorem 3.13. An operator A ∈ Lm ρ,δ (Ω) is elliptic if and only if there exists a properly supported operator B ∈ L−m ρ,δ (Ω) such that: AB ≡ I mod L−∞ (Ω), BA ≡ I mod L−∞ (Ω). Such an operator B is called a parametrix for A. In other words, a parametrix for A is a two-sided inverse of A modulo L−∞ (Ω). We observe that a parametrix is unique modulo L−∞ (Ω). The next theorem proves the invariance of pseudo-differential operators under change of coordinates. Theorem 3.14. Let Ω1 and Ω2 be two open subsets of Rn and let χ : Ω1 → Ω2 be a C ∞ diffeomorphism. If A ∈ Lm ρ,δ (Ω1 ), where 1 − ρ ≤ δ < ρ ≤ 1, then the mapping Aχ : C0∞ (Ω2 ) −→ C ∞ (Ω2 ) v −→ A(v ◦ χ) ◦ χ−1 is in Lm ρ,δ (Ω2 ), and we have the asymptotic expansion σ(Aχ )(y, η) ∼

1 ∂ξα σ(A) (x,t χ (x) · η) · Dzα eir(x,z,η) α! z=x

(3.6)

α≥0

with

r(x, z, η) = χ(z) − χ(x) − χ (x) · (z − x), η .

Here x = χ−1 (y),χ (x) is the derivative of χ at x and t χ (x) is its transpose. Remark 3.1. Formula (3.6) shows that σ(Aχ )(y, η) ≡ σ(A) x,t χ (x) · η Note that the mapping

m−(ρ−δ)

mod Sρ,δ

.


73

Ω2 × Rn (y, η) −→ x,t χ (x) · η ∈ Ω1 × Rn is just a transition map of the cotangent bundle T ∗ (Rn ). This implies that n the principal symbol σm (A) of A ∈ Lm ρ,δ (R ) can be invariantly defined on ∗ n T (R ) when 1 − ρ ≤ δ < ρ ≤ 1. The situation may be represented by the following diagram: A

C0∞ (Ω1 ) −−−−→ C ∞ (Ω1 ) ⏐ % ⏐χ∗ ⏐ χ∗ ⏐ $ C0∞ (Ω2 ) −−−−→ C ∞ (Ω2 ) Aχ

Here χ∗ v = v ◦ χ is the pull-back of v by χ and χ∗ u = u ◦ χ−1 is the pushforward of u by χ, respectively. A differential operator of order m with smooth coefficients on Ω is cons,p s,p s−m,p s−m,p (Ω) (resp. Bloc (Ω)) into Hloc (Ω) (resp. Bloc (Ω)) for all tinuous on Hloc s ∈ R. This result extends to pseudo-differential operators: Theorem 3.15 (the Besov space boundedness theorem). Every properly supported operator A ∈ Lm 1,δ (Ω), 0 ≤ δ < 1, extends to a continuous linear operator s,p s−m,p (Ω) −→ Hloc (Ω) A : Hloc for all s ∈ R and 1 < p < ∞, and also it extends to a continuous linear operator s,p s−m,p A : Bloc (Ω) −→ Bloc (Ω) for all s ∈ R and 1 ≤ p ≤ ∞. For a proof of Theorem 3.15, the reader might refer to Bourdaud [Bo, Theorem 1] (see also [Ta5, Appendix A]). Now we define the concept of a pseudo-differential operator on a manifold, and transfer all the machinery of pseudo-differential operators to manifolds. Let M be an n-dimensional compact smooth manifold without boundary. Theorem 3.14 leads us to the following: Definition 3.1. Let 1 − ρ ≤ δ < ρ ≤ 1. A continuous linear operator A : C ∞ (M ) → C ∞ (M ) is called a pseudo-differential operator of order m ∈ R if it satisfies the following two conditions: (i) The distribution kernel KA (x, y) of A is smooth off the diagonal {(x, x) : x ∈ M } in M × M . (ii) For any chart (U, χ) on M , the mapping Aχ : C0∞ (χ(U )) −→ C ∞ (χ(U )) u −→ A(u ◦ χ) ◦ χ−1 belongs to the class Lm ρ,δ (χ(U )).


74

We let Lm ρ,δ (M ) = the set of all pseudo-differential operators of order m on M , and set

L−∞ (M ) =

Lm ρ,δ (M ).

m∈R

Some results about pseudo-differential operators on Rn stated above are also true for pseudo-differential operators on M . In fact, pseudo-differential operators on M are defined to be locally pseudo-differential operators on Rn . For example, we have the following five assertions: (1) A pseudo-differential operator A extends to a continuous linear operator A : D (M ) → D (M ). (2) sing supp Au ⊂ sing supp u, u ∈ D (M ). (3) A continuous linear operator A : C ∞ (M ) → D (M ) is a —em regularizer if and only if it is in the class L−∞ (M ). (4) The class Lm ρ,δ (M ) is stable under the operations of composition of operators and taking the transpose or adjoint of an operator. (5) (The Besov space boundedness theorem) A pseudo-differential operator A ∈ Lm 1,δ (M ), 0 ≤ δ < 1, extends to a continuous linear operator A : H s,p (M ) −→ H s−m,p (M ) for all s ∈ R and 1 < p < ∞, and also it extends to a continuous linear operator A : B s,p (M ) −→ B s−m,p (M ) for all s ∈ R and 1 ≤ p ≤ ∞. A pseudo-differential operator A ∈ Lm 1,0 (M ) is said to be classical if, for any chart (U, χ) on M , the mapping Aχ : C0∞ (χ(U )) → C ∞ (χ(U )) belongs to the class Lm cl (χ(U )). We let Lm cl (M ) = the set of all classical pseudo-differential operators of order m on M . We observe that

L−∞ (M ) =

Lm cl (M ).

m∈R

If (U, χ) is a chart on M , there is associated a hoLet A ∈ mogeneous principal symbol σAχ ∈ C ∞ (χ(U ) × (Rn \ {0})). In view of Remark 3.1, by smoothly patching together the functions σAχ we can obtain a smooth function σA (x, ξ) on T ∗ (M ) \ {0} = {(x, ξ) ∈ T ∗ (M ) : ξ = 0}, which is positively homogeneous of degree m in the variable ξ. The function σA (x, ξ) is called the homogeneous principal symbol of A. Lm cl (M ).


75

A classical pseudo-differential operator A ∈ Lm cl (M ) is said to be elliptic of order m if its homogeneous principal symbol σA (x, ξ) does not vanish on the bundle T ∗ (M ) \ {0} of non-zero cotangent vectors. Then we have the following assertion: (6) An operator A ∈ Lm cl (M ) is elliptic if and only if there exists a parametrix B ∈ L−m (M ) for A: cl AB ≡ I mod L−∞ (M ), BA ≡ I mod L−∞ (M ). Let Ω be an open subset of Rn . A properly supported pseudo-differential operator A on Ω is said to be hypoelliptic if it satisfies the condition sing supp u = sing supp Au,

u ∈ D (Ω).

For example, Theorem 3.13 asserts that elliptic operators are hypoelliptic. We remark that this notion may be transferred to manifolds. The following criterion for hypoellipticity is due to H¨ ormander (cf. [Ho2, Theorem 4.2]): Theorem 3.16. Let A = p(x, D) ∈ Lm ρ,δ (Ω) be properly supported. Assume that, for any compact K ⊂ Ω and any multi-indices α, β, there exist positive constants CK,α,β , CK and a real number μ such that we have, for all x ∈ K and |ξ| ≥ CK , α β Dξ Dx p(x, ξ) ≤ CK,α,β |p(x, ξ)| (1 + |ξ|)−ρ|α|+δ|β| ,

(3.7a)

|p(x, ξ)|−1 ≤ CK (1 + |ξ|) .

(3.7b)

μ

Then there exists a parametrix B ∈ Lμρ,δ (Ω) for A.

4 Lp Approach to Elliptic Boundary Value Problems

In this chapter we study elliptic boundary value problems in the framework of Lp -spaces, by using the Lp theory of pseudo-differential operators. For more thorough treatments of this subject, the reader might refer to Hörmander [Ho1], Seeley [Se2], Taylor [Ty, Chapter XI] and also Taira [Ta2, Chapter 8] (L2 -version).

4.1 The Dirichlet Problem In this section we shall consider the Dirichlet problem in the framework of Sobolev spaces of Lp type. This is a generalization of the classical potential approach to the Dirichlet problem. Let Ω be a bounded domain of Euclidean space Rn with smooth boundary Γ = ∂Ω. Its closure Ω = Ω ∪Γ is an n-dimensional, compact smooth manifold with boundary. We may assume that Ω is the closure of a relatively compact open subset Ω of an n-dimensional, compact smooth manifold M without boundary in which Ω has a smooth boundary Γ . This manifold M is the double of Ω (see Figure 4.1). We let n n ∂2 ∂ aij (x) + bi (x) + c(x) A= ∂x ∂x ∂x i j i i,j=1 i=1 be a second-order, elliptic differential operator with real coefficients such that: (1) aij ∈ C ∞ (M ) and aij (x) = aji (x) for all x ∈ M , 1 ≤ i, j ≤ n, and there exists a positive constant a0 such that n


on T ∗ (M ).

i,j=1

Here T ∗ (M ) is the cotangent bundle of the double M . K. Taira, Boundary Value Problems and Markov Processes, Lecture Notes in Mathematics 1499, DOI 10.1007/978-3-642-01677-6 4, c Springer-Verlag Berlin Heidelberg 2009

77

78


M

Ω

Γ= ∂ Ω

Fig. 4.1.

(2) bi ∈ C ∞ (M ) for all 1 ≤ i ≤ n. (3) c ∈ C ∞ (M ) and c(x) ≤ 0 in M . Furthermore, for simplicity, we assume that: The function c(x) does not vanish identically on the double M .

(4.1)

The next theorem states the existence of a volume potential for A, which plays the same role for A as the Newtonian potential plays for the Laplacian (cf. [Se1, Theorem 1] and [Ta2, Theorem 8.2.1]): Theorem 4.1. (i) The operator A : C ∞ (M ) → C ∞ (M ) is bijective, and its inverse Q is a classical elliptic pseudo-differential operator of order −2 on M . (ii) The operators A and Q extend respectively to isomorphisms A : H s,p (M ) −→ H s−2,p (M ), Q : H s−2,p (M ) −→ H s,p (M ) for all s ∈ R, which are still inverses of each other. Next we construct a surface potential for A, which is a generalization of the classical Poisson kernel for the Laplacian. We let Kv = Q(v ⊗ δ)|Γ , v ∈ C ∞ (Γ ). Here v(x ) ⊗ δ(t) is a distribution on M defined by the formula v ⊗ δ, ϕ · μ = v, ϕ(·, 0) · ω ,

ϕ(x , t) ∈ C ∞ (M ),

where μ is a strictly positive density on M and ω is a strictly positive density on Γ = {t = 0}, respectively. Then we have the following (cf. [Ta2, Theorem 8.2.2]):

4.1 The Dirichlet Problem

79

Theorem 4.2. (i) The operator K is a classical elliptic pseudo-differential operator of order −1 on Γ . (ii) The operator K : C ∞ (Γ ) → C ∞ (Γ ) is bijective, and its inverse L is a classical elliptic pseudo-differential operator of first order on Γ . Furthermore, the operators K and L extend respectively to isomorphisms K : B σ,p (Γ ) −→ B σ+1,p (Γ ), L : B σ+1,p (Γ ) −→ B σ,p (Γ ) for all σ ∈ R, which are still inverses of each other. Now we let P ϕ = Q(Lϕ ⊗ δ)|Ω ,

ϕ ∈ C ∞ (Γ ).

Then the operator P maps C ∞ (Γ ) continuously into C ∞ (Ω), and extends to a continuous linear operator P : B s−1/p,p (Γ ) −→ H s,p (Ω) for all s ∈ R. Furthermore, we have, for all ϕ ∈ B s−1/p,p (Γ ), AP ϕ = AQ(Lϕ ⊗ δ)|Ω = (Lϕ ⊗ δ)|Ω = 0 in Ω, P ϕ|Γ = KLϕ = ϕ on Γ . The operator P is called the Poisson operator. We let N (A, s, p) = {u ∈ H s,p (Ω) : Au = 0 in Ω} ,

s ∈ R.

Since the injection H s,p (Ω) → D (Ω) is continuous, it follows that the null space N (A, s, p) is a closed subspace of H s,p (Ω); hence it is a Banach space. Then we have the following fundamental result (cf. [Se2, Theorems 5 and 6]): Theorem 4.3. The Poisson operator P maps the Besov space B s−1/p,p (Γ ) isomorphically onto the null space N (A, s, p) for all s ∈ R. By combining Theorems 4.1 and 4.3, we can obtain the following existence and uniqueness theorem for the Dirichlet problem (cf. [ADN]): Theorem 4.4. Let s ≥ 2. The Dirichlet problem Au = f in Ω, u=ϕ on Γ

(4.2)

has a unique solution u(x) in H s,p (Ω) for any f ∈ H s−2,p (Ω) and any ϕ ∈ B s−1/p,p (Γ ). Furthermore, we can prove the following existence and uniqueness theorem for the Neumann problem (cf. [ADN]):


80

Theorem 4.5. Let s ≥ 2. The Neumann problem Au = f in Ω, ∂u on Γ ∂n = ϕ

(4.3)

has a unique solution u(x) in H s,p (Ω) for any f ∈ H s−2,p (Ω) and any ϕ ∈ B s−1−1/p,p (Γ ). Here n is the unit interior normal to the boundary Γ . By Theorem 4.5, we can introduce a linear operator GN : H s−2,p (Ω) −→ H s,p (Ω) as follows: For any f ∈ H s−2,p (Ω), the function GN f ∈ H s,p (Ω) is the unique solution of the problem Au = f in Ω, ∂u on Γ . ∂n = 0 The operator GN is called the Green operator for the Neumann problem.

4.2 Formulation of a Boundary Value Problem If u ∈ H 2,p (Ω) = W 2,p (Ω), we can define its traces γ0 u and γ1 u respectively by the formulas γ0 u = u|Γ, ∂u γ1 u = ∂n , Γ and let γu = {γ0 u, γ1 u} . Then we have the following (cf. [St1]): Theorem 4.6 (the trace theorem). The trace map γ : H 2,p (Ω) −→ B 2−1/p,p (Γ ) ⊕ B 1−1/p,p (Γ ) is continuous and surjective for all 1 < p < ∞. We define a boundary condition ∂u Bu := a(x ) + b(x )u = a(x )γ1 u + b(x )γ0 u, ∂n

u ∈ H 2,p (Ω),

Γ

where a(x ) and b(x ) are real-valued, smooth functions on Γ . Then we have the following: Proposition 4.7. The mapping B : H 2,p (Ω) −→ B 1−1/p,p (Γ ) is continuous for all 1 < p < ∞. Now we can formulate our boundary value problem for (A, B) as follows: Given functions f ∈ Lp (Ω) and ϕ ∈ B 2−1/p,p (Γ ), find a function u ∈ H 2,p (Ω) such that Au = f in Ω, (4.4) Bu = ϕ on Γ .

4.3 Reduction to the Boundary

81

4.3 Reduction to the Boundary In this section, by using the operators P and GN we shall show that problem (4.4) can be reduced to the study of a pseudo-differential operator on the boundary. First, we remark that every function u(x) in H 2,p (Ω) can be written in the following form: u(x) = v(x) + w(x), (4.5) where

v = GN (Au) ∈ H 2,p (Ω), w = u − v ∈ N (A, 2, p) = z ∈ H 2,p (Ω) : Az = 0 in Ω .

Since the operator GN : Lp (Ω) → H 2,p (Ω) is continuous, it follows that the decomposition (4.5) is continuous; more precisely, we have the inequalities v 2,p ≤ C Au p ≤ C u 2,p ; w 2,p ≤ u 2,p + v 2,p ≤ C u 2,p . Here the letter C denotes a generic positive constant. Now we assume that u ∈ H 2,p (Ω) is a solution of the boundary value problem (4.4) Au = f in Ω, Bu = ϕ on Γ . Then, by virtue of the decomposition (4.5) of u(x) this is equivalent to saying that w ∈ H 2,p (Ω) is a solution of the boundary value problem Aw = 0 in Ω, (4.6) Bw = ϕ − Bv on Γ . Here v(x) = GN f ∈ H 2,p (Ω) and w(x) = u(x) − v(x). However, Theorem 4.3 asserts that the spaces N (A, 2, p) and B 2−1/p,p (Γ ) are isomorphic in such a way that: γ0

N (A, 2, p) −→ B 2−1/p,p (Γ ). N (A, 2, p) ←− B 2−1/p,p (Γ ). P

Therefore, we find that w ∈ H 2,p (Ω) is a solution of problem (4.6) if and only if ψ(x ) ∈ B 2−1/p,p (Γ ) is a solution of the equation BP ψ = ϕ − Bv Here ψ = γ0 w, or equivalently, w = P ψ. Summing up, we obtain the following:

on Γ .

(4.7)

82


Proposition 4.8. For functions f ∈ Lp (Ω) and ϕ ∈ B 2−1/p,p (Γ ), there exists a solution u ∈ H 2,p (Ω) of problem (4.4) if and only if there exists a solution ψ ∈ B 2−1/p,p (Γ ) of equation (4.7). Furthermore, the solutions u(x) and ψ(x ) are related as follows: u = GN f + P ψ. We remark that equation (4.7) is a generalization of the classical Fredholm integral equation. We let T : C ∞ (Γ ) −→ C ∞ (Γ ) ϕ −→ BP ϕ. Then we have, by condition (4.2), T = a(x )Π + b(x ), where

∂ (P ϕ) , Πϕ = γ1 P ϕ = ∂n Γ

ϕ ∈ C ∞ (Γ ).

It is known (cf. [Ho1], [Se2]) that the operator Π is a classical pseudodifferential operator of first order on Γ ; hence the operator T is a classical pseudo-differential operator of first order on the boundary Γ . Consequently, Proposition 4.8 asserts that problem (4.4) can be reduced to the study of the first-order pseudo-differential operator T on the boundary Γ . We shall formulate this fact more precisely in terms of functional analysis. First, we remark that the operator T : C ∞ (Γ ) → C ∞ (Γ ) extends to a continuous linear operator T : B s,p (Γ ) −→ B s−1,p (Γ ),

s ∈ R.

Then we have the formula T ϕ = BP ϕ,

ϕ ∈ B 2−1/p,p (Γ ),

since the Poisson operator P : B 2−1/p,p (Γ ) −→ N (A, 2, p) and the boundary operator B : H 2,p (Ω) −→ B 1−1/p,p (Γ ) are both continuous. We associate with problem (4.4) a linear operator A : H 2,p (Ω) −→ Lp (Ω) ⊕ B 2−1/p,p (Γ ) as follows.


83

(a) The domain D(A) of A is the space D(A) = u ∈ H 2,p (Ω) = W 2,p (Ω) : Bu ∈ B 2−1/p,p (Γ ) . (b) Au = {Au, Bu}, u ∈ D(A). Note that the space B 2−1/p,p (Γ ) is a right boundary space associated with the Dirichlet condition: a(x ) ≡ 0 and b(x ) ≡ 1 on Γ . Since the operators A : H 2,p (Ω) → Lp (Ω) and B : H 2,p (Ω) → B 1−1/p,p (Γ ) are both continuous, it follows that A is a closed operator. Furthermore, the operator A is densely defined, since the domain D(A) contains the space C ∞ (Ω). Similarly, we associate with equation (4.7) a linear operator T : B 2−1/p,p (Γ ) −→ B 2−1/p,p (Γ ) as follows. (α) The domain D(T ) of T is the space D(T ) = ϕ ∈ B 2−1/p,p (Γ ) : T ϕ ∈ B 2−1/p,p (Γ ) . (β) T ϕ = T ϕ, ϕ ∈ D(T ). Then the operator T is a densely defined, closed operator, since the operator T : B 2−1/p,p (Γ ) → B 1−1/p,p (Γ ) is continuous and since the domain D(T ) contains the space C ∞ (Γ ). The next theorem states that A has regularity property if and only if T has. Theorem 4.9. The following two conditions are equivalent: u ∈ Lp (Ω), Au ∈ Lp (Ω) and Bu ∈ B 2−1/p,p (Γ ) =⇒ u ∈ H 2,p (Ω). (4.8) ϕ ∈ B −1/p,p (Γ ) and T ϕ ∈ B 2−1/p,p (Γ ) =⇒ ϕ ∈ B 2−1/p,p (Γ ).

(4.9)

Proof. (i) First, we show that assertion (4.8) implies assertion (4.9). To do this, assume that ϕ ∈ B −1/p,p (Γ )

and T ϕ ∈ B 2−1/p,p (Γ ).

Then, by letting u = P ϕ we obtain that u ∈ Lp (Ω), Au = 0

and Bu = T ϕ ∈ B 2−1/p,p (Γ ).

Hence it follows from condition (4.8) that u ∈ H 2,p (Ω),


84

so that, by Theorem 4.6, ϕ = γ0 u ∈ B 2−1/p,p (Γ ). (ii) Conversely, we show that assertion (4.9) implies estimate (4.8). To do this, assume that u ∈ Lp (Ω), Au ∈ Lp (Ω)

and Bu ∈ B 2−1/p,p (Γ ).

Then the distribution u(x) can be decomposed as follows: u(x) = v(x) + w(x), where

v = GN (Au) ∈ H 2,p (Ω), w = u − v ∈ N (A, 0, p) = {z ∈ Lp (Ω) : Az = 0 in Ω} .

Moreover, Theorem 4.3 asserts that the distribution w(x) can be written in the form w = P ϕ, ϕ = γ0 w ∈ B −1/p,p (Γ ). Hence we have, by Theorem 4.6, T ϕ = BP ϕ = Bw = Bu − Bv = Bu − b(x )γ0 v ∈ B 2−1/p,p (Γ ), since γ1 v = 0. Thus it follows from condition (4.9) that ϕ ∈ B 2−1/p,p (Γ ), so that again, by Theorem 4.3, w = P ϕ ∈ H 2,p (Ω). This proves that

u = v + w ∈ H 2,p (Ω).

The proof of Theorem 4.9 is complete. The next theorem states that a priori estimates for A are entirely equivalent to corresponding a priori estimates for T . Theorem 4.10. The following two estimates are equivalent: u 2,p ≤ C Au p + |Bu|2−1/p,p + u p , u ∈ D(A). |ϕ|2−1/p,p ≤ C |T ϕ|2−1/p,p + |ϕ|−1/p,p , ϕ ∈ D(T ).

(4.10) (4.11)

Here and in the following the letter C denotes a generic positive constant.


85

Proof. (i) First, we show that estimate (4.10) implies estimate (4.11). By taking u = P ϕ with ϕ ∈ D(T ) in estimate (4.10), we obtain that (4.12) P ϕ 2,p ≤ C |T ϕ|2−1/p,p + P ϕ p . However, Theorem 4.3 asserts that the Poisson operator P maps the Besov space B s−1/p,p (Γ ) isomorphically onto the null space N (A, s, p) for all s ∈ R. Thus the desired estimate (4.11) follows from estimate (4.12). (ii) Conversely, we show that estimate (4.11) implies estimate (4.10). To do this, we express a function u ∈ D(A) in the form u(x) = v(x) + w(x), where

v = GN (Au) ∈ H 2,p (Ω), w = u − v ∈ N (A, 2, p) = z ∈ H 2,p (Ω) : Az = 0 in Ω .

Then we have, by Theorem 4.5 with s := 2, v 2,p = GN (Au) 2,p ≤ C Au p .

(4.13)

Furthermore, by applying estimate (4.11) to the distribution γ0 w we obtain that |γ0 w|2−1/p,p ≤ C |T (γ0 w)|2−1/p,p + |γ0 w|−1/p,p = C |Bw|2−1/p,p + |γ0 w|−1/p,p ≤ C |Bu|2−1/p,p + |Bv|2−1/p,p + |γ0 w|−1/p,p . In view of Theorem 4.3, this proves that w 2,p ≤ C |Bu|2−1/p,p + |Bv|2−1/p,p + w p ≤ C |Bu|2−1/p,p + |Bv|2−1/p,p + u p + v p ≤ C |Bu|2−1/p,p + |Bv|2−1/p,p + u p + v 2,p .

(4.14)

However, since γ1 v = 0, it follows from an application of Theorem 4.6 that |Bv|2−1/p,p = |b(x )γ0 v|2−1/p,p ≤ C v 2,p .

(4.15)

Thus, by carrying estimates (4.13) and (4.15) into estimate (4.14) we obtain that w 2,p ≤ C Au p + |Bu|2−1/p,p + u p . (4.16) Therefore, the desired estimate (4.10) follows from estimates (4.13) and (4.16), since u(x) = v(x) + w(x). The proof of Theorem 4.10 is complete.

5 Proof of Theorem 1.1

This chapter is devoted to the proof of Theorem 1.1. The idea of our proof is stated as follows. First, we reduce the study of the boundary value problem ' (A − λ)u = f in D, (1.1) ∂u Lu = μ(x ) ∂n + γ(x )u = ϕ on ∂D to that of a first-order pseudo-differential operator T (λ) = LP (λ) on the boundary ∂D, just as in Section 4.3. Then we prove that conditions (A) and (B) are sufficient for the validity of the a priori estimate (1.2) u 2,p ≤ C(λ) f p + |ϕ|2−1/p,p + u p . More precisely, we construct a parametrix S(λ) for T (λ) in the H¨ ormander class L01,1/2 (∂D) (Lemma 5.2), and apply the Besov-space boundedness theorem (Theorem 3.15) to S(λ) to obtain the desired estimate (1.2) (Lemma 5.1).

5.1 Boundary Value Problem with Spectral Parameter Let D be a bounded domain of Euclidean space RN with smooth boundary ∂D. Its closure D = D ∪ ∂D is an N -dimensional, compact smooth manifold with boundary. We may assume that D is the closure of a relatively compact " without open subset D of an N -dimensional, compact smooth manifold D " is the boundary in which D has a smooth boundary ∂D. This manifold D double of D (see Figure 5.1). We let A=

N i,j=1

∂2 ∂ + bi (x) + c(x) ∂xi ∂xj i=1 ∂xi N

aij (x)


87

88


D

D

∂D

Fig. 5.1.

be a second-order, elliptic differential operator with real coefficients such that: " and aij (x) = aji (x) for all x ∈ D, " 1 ≤ i, j ≤ N , and there (1) aij ∈ C ∞ (D) exists a positive constant a0 such that N


" on T ∗ (D),

i,j=1

" is the cotangent bundle of the double D. " where T ∗ (D) i ∞ " (2) b ∈ C (D) for all 1 ≤ i ≤ N . " and c(x) ≤ 0 in D. (3) c ∈ C ∞ (D) In this chapter we consider the elliptic boundary value problem with spectral parameter (A − λ)u = f in D, (1.1) ∂u + γ(x )u = ϕ on ∂D. Lu = μ(x ) ∂n Here we recall that: (4) λ is a complex parameter. (5) μ(x ) and γ(x ) are real-valued, smooth functions on the boundary ∂D. (6) n = (n1 , n2 , . . . , nN ) is the unit interior normal to the boundary ∂D (see Figure 1.1). The purpose of this chapter is to prove Theorem 1.1. More precisely, we prove the a priori estimate u 2,p ≤ C(λ) (A − λ)u p + |Lu|2−1/p,p + u p , (1.2) provided that the following two conditions (A) and (B) are satisfied: (A) μ(x ) ≥ 0 on ∂D. (B) γ(x ) < 0 on M := {x ∈ ∂D : μ(x ) = 0}.

5.2 Proof of Estimate (1.2)

89

5.2 Proof of Estimate (1.2) The proof of Estimate (1.2) is divided into three steps. Step I: It suffices to show that estimate (1.2) holds true for some λ0 > 0, since we have, for all λ ∈ C, (A − λ0 )u = (A − λ)u + (λ − λ0 )u. We take a positive constant λ0 so large that the function c(x) − λ0 satisfies the condition " of D. c(x) − λ0 < 0 on the double D (5.1) This condition (5.1) implies that condition (4.1) is satisfied for the operator A − λ0 . Therefore, by applying Theorems 4.4 and 4.3 to the operator A − λ0 we can obtain the following two fundamental results: (a) The Dirichlet problem

(A − λ0 )w = 0 w=ϕ

in D, on ∂D

has a unique solution w ∈ H t,p (D) for any function ϕ ∈ B t−1/p,p (∂D) with t ∈ R. (b) The Poisson operator P (λ0 ) : B t−1/p,p (∂D) −→ H t,p (D), defined by w = P (λ0 )ϕ, is an isomorphism of the space B t−1/p,p (∂D) onto the null space N (A − λ0 , t, p) = {u ∈ H t,p (D) : (A − λ0 )u = 0 in D} for all t ∈ R; and its inverse is the trace operator γ0 on the boundary ∂D. We let T (λ0 ) : C ∞ (∂D) −→ C ∞ (∂D) ϕ −→ LP (λ0 )ϕ. Then we have the formula T (λ0 ) = μ(x )Π(λ0 ) + γ(x ), where

∂ Π(λ0 )ϕ = (P (λ0 )ϕ) . ∂n ∂D

It is known that the operator Π(λ0 ) is a classical pseudo-differential operator of first order on the boundary ∂D and that its complete symbol is given by the following formula (cf. [Ta2, Section 10.2]): √ √ p1 (x , ξ ) + −1 q1 (x , ξ ) + p0 (x , ξ ) + −1 q0 (x , ξ ) + terms of order ≤ −1 depending on λ0 ,

90


where p1 (x , ξ ) < 0 on the bundle T ∗ (∂D) \ {0} of non-zero cotangent vectors.

(5.2)

For example, if A is the usual Laplacian Δ=

∂2 ∂2 ∂2 + 2 + ...+ 2 , 2 ∂x1 ∂x2 ∂xN

then we have the formula p1 (x , ξ ) = minus the length |ξ | of ξ with respect to the Riemannian metric of ∂D induced by the natural metric of RN . Therefore, we obtain that the operator T (λ0 ) = μ(x )Π(λ0 ) + γ(x ) is a classical pseudo-differential operator of first order on the boundary ∂D and further that its complete symbol t(x , ξ ; λ0 ) is given by the following formula: √ t(x , ξ ; λ0 ) = μ(x ) p1 (x , ξ ) + −1 q1 (x , ξ ) √ + [γ(x ) + μ(x )p0 (x , ξ )] + −1 μ(x )q0 (x , ξ ) + terms of order ≤ −1 depending on λ0 .

(5.3)

Then, by arguing just as in Section 4.3 we can prove that the question of the validity of a priori estimates and the question of regularity for solutions of problem (1.1) for λ = λ0 are reduced to the corresponding questions for the operator T (λ0 ) (cf. Theorems 4.9 and 4.10). Step II: Therefore, in order to prove estimate (1.2) for λ = λ0 it suffices to show the following: Lemma 5.1. Assume that conditions (A) and (B) are satisfied: (A) μ(x ) ≥ 0 on ∂D. (B) γ(x ) < 0 on M = {x ∈ ∂D : μ(x ) = 0}. Then we have, for all s ∈ R, ϕ ∈ D (∂D), T (λ0 )ϕ ∈ B s,p (∂D) =⇒ ϕ ∈ B s,p (∂D).

(5.4)

Furthermore, for any t < s, there exists a positive constant Cs,t such that |ϕ|s,p ≤ Cs,t (|T (λ0 )ϕ|s,p + |ϕ|t,p ) .

(5.5)

Proof. (a) The proof of Lemma 5.1 is based on the following lemma (cf. [Ka, Theorem 3.1]):


91

Lemma 5.2. Assume that conditions (A) and (B) are satisfied. Then, for each point x of ∂D, we can find a neighborhood U (x ) of x such that: For any compact K ⊂ U (x ) and any multi-indices α, β, there exist positive constants CK,α,β and CK such that we have, for all x ∈ K and all |ξ | ≥ CK , α β −|α|+(1/2)|β| , (5.6a) Dξ Dx t(x , ξ ; λ0 ) ≤ CK,α,β |t(x , ξ ; λ0 )| (1 + |ξ|) |t(x , ξ ; λ0 )|

−1

≤ CK .

(5.6b)

Granting Lemma 5.2 for the moment, we shall prove Lemma 5.1. m (b) First, we cover ∂D by a finite number of local charts {(Uj , χj )}j=1 in each of which inequalities (5.6a) and (5.6b) hold true. Since the operator T (λ0 ) satisfies conditions (3.7a) and (3.7b) of Theorem 3.16 with μ := 0, ρ := 1 and δ = 1/2, it follows from an application of the same theorem that there exists a parametrix S(λ0 ) in the class L01,1/2 (Uj ) for T (λ0 ): ' T (λ0 )S(λ0 ) ≡ I mod L−∞ (Uj ), S(λ0 )T (λ0 ) ≡ I

mod L−∞ (Uj ).

m

m

Let {ϕj }j=1 be a partition of unity subordinate to the covering {Uj }j=1 , and choose a function ψj (x ) ∈ C0∞ (Uj ) such that ψj (x ) = 1 on supp ϕj , so that ϕj (x )ψj (x ) = ϕj (x ). Now we may assume that ϕ ∈ B t,p (∂D) for some t < s and that T (λ0 )ϕ ∈ s,p B (∂D). We remark that the operator T (λ0 ) can be written in the following form: m m T (λ0 ) = ϕj T (λ0 )ψj + ϕj T (λ0 )(1 − ψj ). j=1

j=1

However, by applying Theorems 3.12 and 3.8 to our situation we obtain that the second terms ϕj T (λ0 )(1 − ψj ) are in L−∞ (∂D). Indeed, it suffices to note that ϕj (x ) (1 − ψj (x )) = ϕj (x ) − ϕj (x ) = 0. Hence we are reduced to the study of the first terms ϕj T (λ0 )ψj . This implies that we have only to prove the following local version of assertions (5.4) and (5.5): ψj ϕ ∈ B t,p (Uj ), T (λ0 )ψj ϕ ∈ B s,p (Uj ) =⇒ ψj ϕ ∈ B s,p (Uj ). |T (λ0 )ψj ϕ|2s,p + |ψj ϕ|2t,p . |ψj ϕ|s,p ≤ Cs,t

(5.7) (5.8)

However, by applying the Besov-space boundedness theorem (Theorem 3.15) σ,p to our situation we obtain that the parametrix S(λ0 ) maps Bloc (Uj ) continuously into itself for all σ ∈ R. This proves the desired assertions (5.7) and (5.8), since we have the assertion ψj ϕ ≡ S(λ0 ) (T (λ0 )ψj ϕ)

mod C −∞ (Uj ).

92


Lemma 5.1 is proved, apart from the proof of Lemma 5.2. Step III: Proof of Lemma 5.2 The proof of Lemma 5.2 is divided into five steps. Step III-1: First, we verify condition (5.6b). By assertions (5.3) and (5.2), we can find positive constants c0 and c1 such that we have, for all sufficiently large |ξ |, |t(x , ξ ; λ0 )| ≥ μ(x ) |p1 (x , ξ ) + p0 (x , ξ )| − γ(x ) c μ(x )|ξ | − 12 γ(x ) if 0 ≤ μ(x ) ≤ c1 , ≥ c00 if c1 ≤ μ(x ) ≤ 1, 2 μ(x )|ξ | − γ(x ) since γ(x ) < 0 on M = {x ∈ ∂D : μ(x ) = 0}. Hence we have, for all sufficiently large |ξ |, |t(x , ξ ; λ0 )| ≥ C (μ(x )|ξ | + 1) .

(5.9)

Here in the following the letter C denotes a generic positive constant. Inequality (5.9) implies the desired condition (5.6b): |t(x , ξ ; λ0 )| ≥ C.

(5.10)

Step III-2: Next we verify condition (5.6a) for |α| = 1 and |β| = 0. Since we have, for all sufficiently large |ξ |, α Dξ t(x , ξ ; λ0 ) ≤ C μ(x ) + |ξ |−1 , it follows from inequality (5.9) that α D t(x , ξ ; λ0 ) ≤ C(1 + |ξ |)−1 (μ(x )|ξ | + 1) ξ ≤ C(1 + |ξ |)−1 |t(x , ξ ; λ0 )| . This inequality proves the desired condition (5.6a) for |α| = 1 and |β| = 0. Step III-3: We verify condition (5.6a) for |β| = 1 and |α| = 0. To do this, we need the following elementary lemma on non-negative functions. Lemma 5.3. Let f (x) be a non-negative, C 2 function on R such that we have, for some positive constant c, sup |f (x)| ≤ c.

(5.11)

x∈R

Then we have the inequality |f (x)| ≤

) ) 2c f (x)

on R.

(5.12)


93

Proof. In view of Taylor’s formula, it follows that 0 ≤ f (y) = f (x) + f (x)(y − x) +

f (ξ) (y − x)2 , 2

where ξ is between x and y. Thus, by letting z = x−y we obtain from estimate (5.11) that f (ξ) 2 z 2 c ≤ f (x) + f (x)z + z 2 for all z ∈ R. 2

0 ≤ f (x) + f (x)z +

This implies the desired inequality (5.12) if we take the discriminant of the quadratic equation. Step III-4: Since we have, for all sufficiently large |ξ |, β β Dx t(x , ξ ; λ0 ) ≤ C Dx μ(x ) · |ξ | + μ(x )|ξ | + 1 , it follows from an application of Lemma 5.3 and inequalities (5.9) and (5.10) that *) + β μ(x ) |ξ | + 1 + (μ(x )|ξ | + 1) Dx t(x , ξ ; λ0 ) ≤ C + * 1/2 ≤ C |ξ |1/2 (μ(x )|ξ | + 1) + (μ(x )|ξ | + 1) −1/2 +1 ≤ C |t(x , ξ ; λ0 )| |ξ |1/2 |t(x , ξ ; λ0 )| ≤ C |t(x , ξ ; λ0 )| (1 + |ξ |)

1/2

.

This inequality proves the desired condition (5.6a) for |β| = 1 and |α| = 0. Step III-5: Similarly, we can verify condition (5.6a) for the general case: |α| + |β| = k, k ∈ N. Now the proof of Lemma 5.1 and hence that of Theorem 1.1 is complete.

6 A Priori Estimates

This Chapter 6 and the next Chapter 7 are devoted to the proof of Theorem 1.2. In this chapter we study the operator Ap , and prove a priori estimates for the operator Ap − λI (Theorem 6.3) which will play a fundamental role in the next chapter. In the proof we make good use of Agmon’s method (Proposition 6.4). This is a technique of treating a spectral parameter λ as a second-order, elliptic differential operator of an extra variable and relating the old problem to a new problem with the additional variable. Recall that the operator Ap is a unbounded linear operator from Lp (D) into itself defined by the following formulas: (a) The domain of definition D(Ap ) of Ap is the space

∂u + γ(x )u = 0 . D(Ap ) = u ∈ H 2,p (D) = W 2,p (D) : Lu = μ(x ) ∂n (1.3) (b) Ap u = Au, u ∈ D(Ap ). We remark that the operator Ap is densely defined, since the domain D(Ap ) contains the space C0∞ (D). First, we have the following: Lemma 6.1. Assume that conditions (A) and (B) are satisfied: (A) μ(x ) ≥ 0 on ∂D. (B) γ(x ) < 0 on M = {x ∈ ∂D : μ(x ) = 0}. Then we have the a priori estimate u 2,p ≤ C ( Au p + u p ) ,

u ∈ D(Ap ).

(6.1)

Proof. The a priori estimate (6.1) follows immediately from estimate (1.2) of Theorem 1.1 with ϕ := 0. K. Taira, Boundary Value Problems and Markov Processes, Lecture Notes in Mathematics 1499, DOI 10.1007/978-3-642-01677-6 6, c Springer-Verlag Berlin Heidelberg 2009

95

96


Corollary 6.2. The operator Ap is a closed operator. Proof. Let {uj } be an arbitrary sequence in the domain D(Ap ) such that: uj −→ u in Lp (D), Auj −→ v in Lp (D). Then, by applying estimate (6.1) to the sequence {uj } we find that {uj } is a Cauchy sequence in the space W 2,p (D). This proves that u ∈ W 2,p (D), and that uj −→ u

in W 2,p (D).

Hence we have the formula Au = lim Auj = v j→∞

in Lp (D),

and also, by Proposition 4.7 with B := L, Lu = lim Luj = 0 j→∞

in B 1−1/p,p (∂D).

Summing up, we have proved that u ∈ D(Ap ) and Ap u = v. The proof of Corollary 6.2 is complete. The next theorem is an essential step in the proof of Theorem 1.2 (cf. the proof of Theorem 7.1 in Chapter 7): Theorem 6.3. Assume that conditions (A) and (B) are satisfied. Then, for every −π < θ < π, there exists a positive constant R(θ) depending on θ such that if λ = r2 eiθ and |λ| = r2 ≥ R(θ), we have, for all u ∈ W 2,p (D) satisfying Lu = 0 on ∂D (i. e., u ∈ D(Ap )), |u|2,p + |λ|1/2 · |u|1,p + |λ| · u p ≤ C(θ) (A − λ)u p ,

(6.2)

with a positive constant C(θ) depending on θ. Here | · |j,p , j = 1, 2, is the seminorm on the space W 2,p (D) defined by the formula ⎛ |u|j,p = ⎝


.

D |α|=j

Proof. The proof of Theorem 6.3 is divided into two steps. Step I: We shall make use of a method essentially due to Agmon (cf. [Ag], [Fu], [LM], [Ta1]). We introduce an auxiliary variable y of the unit circle S = R/2πZ,


97

and replace the complex parameter λ by the second-order differential operator −eiθ

∂2 , ∂y 2

−π < θ < π.

Namely, we replace the operator A − λ by the operator ∂2 , Λ(θ) = A + eiθ 2 , ∂y

−π < θ < π,

and consider instead of the problem with spectral parameter (A − λ)u = f in D, ∂u Lu = μ(x ) ∂n + γ(x )u = 0 on ∂D the following boundary value problem: ' , u = A + eiθ ∂ 22 u Λ(θ), , = f, ∂y L, u=

∂, u μ(x ) ∂n

in D × S, on ∂D × S.

+ γ(x ), u=0

(1.1)

(6.3)

, We remark that the operator Λ(θ) is elliptic for −π < θ < π. Then we have the following result, analogous to Lemma 6.1: Proposition 6.4. Assume that conditions (A) and (B) are satisfied. Then we have, for all u , ∈ W 2,p (D × S) satisfying L, u = 0 on ∂D × S, , , u + , , u 2,p ≤ C(θ) Λ(θ), u p , (6.4) p

, with a positive constant C(θ) depending on θ. Proof. We reduce the study of problem (6.3) to that of a pseudo-differential operator on the boundary, just as in problem (1.1). We can prove that Theorems 4.3 and 4.4 remain valid for the operator , Λ(θ) = A + eiθ ∂ 2 /∂y 2 , −π < θ < π: (, a) The Dirichlet problem

, w Λ(θ) ,=0 w ,=ϕ ,

in D × S, on ∂D × S

has a unique solution w , ∈ H t,p (D×S) for any function ϕ , ∈ B t−1/p,p (∂D× S) with t ∈ R. (,b) The Poisson operator P,(θ) : B t−1/p,p (∂D × S) −→ H t,p (D × S), defined by w , = P,(θ)ϕ, , is an isomorphism of the space B t−1/p,p (∂D × , , u = S) onto the null space N (Λ(θ), t, p) = {, u ∈ H t,p (D × S) : Λ(θ), 0 in D × S} for all t ∈ R; and its inverse is the trace operator on the boundary ∂D × S.

98


We let T,(θ) : C ∞ (∂D × S) −→ C ∞ (∂D × S) ϕ , −→ LP, (θ)ϕ. , Then the operator T,(θ) can be decomposed as follows: , T,(θ) = μ(x )Π(θ) + γ(x ), where

∂ , , ϕ P (θ)ϕ , Π(θ) ,= , ∂n ∂D×S

(6.5)

ϕ , ∈ C ∞ (∂D × S).

, It is known that the operator Π(θ) is a classical pseudo-differential operator of first order on the boundary ∂D × S and that its complete symbol is given by the following formula (cf. [Ta2, Section 10.2]): √ p,1 (x , ξ , y, η; θ) + −1 q,1 (x , ξ , y, η; θ) √ + p,0 (x , ξ , y, η; θ) + −1 q,0 (x , ξ , y, η; θ) + terms of order ≤ −1, where p,1 (x , ξ , y, η; θ) < 0 on the bundle T ∗ (∂D × S) \ {0} of non-zero cotangent vectors, for −π < θ < π. (6.6) For example, if A is the usual Laplacian Δ=

∂2 ∂2 ∂2 + + . . . + , ∂x21 ∂x22 ∂x2N

then we have the formula p,1 (x , ξ , y, η; θ) ⎡ * +1/2 ⎤1/2 2 2 2 2 4 2 2 | + cos θ · η + sin θ · η + |ξ | + cos θ · η |ξ ⎥ ⎢ = −⎣ ⎦ . 2 , Therefore, we obtain that the operator T,(θ) = μ(x )Π(θ) + γ(x ) is a classical pseudo-differential operator of first order on the boundary ∂D × S and that its complete symbol is given by the following formula: √ μ(x ) p,1 (x , ξ , y, η; θ) + −1 q,1 (x , ξ , y, η; θ) √ + [γ(x ) + μ(x ), p0 (x , ξ , y, η; θ)] + −1 μ(x ), q0 (x , ξ , y, η; θ) + terms of order ≤ −1.

(6.7)


99

Then, by virtue of assertions (6.7) and (6.6) we can verify that the operator , T (θ) satisfies conditions (3.7a) and (3.7b) of Theorem 3.16 with μ := 0, ρ := 1 and δ := 1/2, just as in the proof of Lemma 5.2. Hence we obtain the following result, analogous to Lemma 5.1: Lemma 6.5. Assume that conditions (A) and (B) are satisfied. Then we have, for all s ∈ R, ϕ , ∈ D (∂D × S), T,(θ)ϕ , ∈ B s,p (∂D × S) =⇒ ϕ , ∈ B s,p (∂D × S). ,s,t such that Furthermore, for any t < s, there exists a positive constant C ,s,t |T,(θ)ϕ| , s,p + |ϕ| (6.8) |ϕ| , s,p ≤ C , t,p . The desired estimate (6.4) follows from estimate (6.8) with s := 2 − 1/p and t := −1/p, just as in the proof of Theorem 4.10. The proof of Proposition 6.4 is complete. Step II: Now let u(x) be an arbitrary function in the domain D(Ap ): u ∈ W 2,p (D) and Lu = 0 on ∂D. We choose a function ζ(y) in C ∞ (S) such that ⎧ ⎨ 0 ≤ ζ(y) ≤ 3 1 on 4 S, supp ζ ⊂ π3 , 5π 3 , ⎩ ζ(y) = 1 for π2 ≤ y ≤

3π 2 ,

and let v,η (x, y) = u(x) ⊗ ζ(y)eiηy ,

x ∈ D, y ∈ S, η ≥ 0.

Then we have the assertions v,η ∈ W 2,p (D × S), 2 iθ ∂ , Λ(θ), vη = A + e v,η ∂y 2 = (A − η 2 eiθ )u ⊗ ζ(y)eiηy + 2(iη)eiθ u ⊗ ζ (y)eiηy + eiθ u ⊗ ζ (y)eiηy , and also

L, vη (x , y) = Lu(x ) ⊗ ζ(y)eiηy = 0 on ∂D × S.

Thus, by applying inequality (6.4) to the functions v,η (x, y) = u(x) ⊗ ζ(y)eiηy ,

x ∈ D, y ∈ S, η ≥ 0,

we obtain that

, iηy u ⊗ ζeiηy ≤ C(θ) u ⊗ ζeiηy . , Λ(θ)(u ⊗ ζe ) + 2,p p p

(6.9)

100


We can estimate each term of inequality (6.9) as follows: 1/p p p u ⊗ ζeiηy = |u(x)| |ζ(y)| dxdy = ζ p · u p. (6.10) p D×S , Λ(θ)(u ⊗ ζeiηy ) ≤ (A − η 2 eiθ )u ⊗ ζeiηy p p +2η u ⊗ ζ eiηy p + u ⊗ ζ eiηy p ≤ ζ p · (A − η 2 eiθ )up u ⊗ ζeiηy p 2,p

+ (2η ζ p + ζ p ) u p . α Dx,y (u(x) ⊗ ζ(y)eiηy )p dxdy = |α|≤2

≥

D×S

|α|≤2

3π/2

π/2

D

=

k+|β|≤2

≥π

(6.11)

D

|β|=2

D

α Dx,y (u(x) ⊗ eiηy )p dxdy

3π/2

k β η D u(x)p dxdy

π/2

β D u(x)p dx + η p Dβ u(x)p dx p |u(x)| dx

|β|=1

+η 2p D = π |u|p2,p + η p |u|p1,p + η 2p u pp .

D

(6.12)

Therefore, by carrying these three inequalities (6.10), (6.11) and (6.12) into , (θ) independent inequality (6.9) we obtain that, with a positive constant C of η, , (θ) (A − η 2 eiθ )u + η u p . |u|2,p + η |u|1,p + η 2 u p ≤ C p If η is so large that

, (θ), η ≥ 2C

then we can eliminate the last term on the right-hand side to obtain that , (θ) (A − η 2 eiθ )u . |u|2,p + η |u|1,p + η 2 u p ≤ 2C p This proves the desired inequality (6.2) if we take λ := η 2 eiθ , , (θ)2 , R(θ) := 4C , (θ). C(θ) := 2C The proof of Theorem 6.3 is now complete.


In this chapter we prove Theorem 1.2 (Theorems 7.1 and 7.9). Once again we make use of Agmon’s method in the proof of Theorems 7.1 and 7.9. In particular, Agmon’s method plays an important role in the proof of the surjectivity of the operator Ap − λI (Proposition 7.2).

7.1 Proof of Theorem 1.2, Part (i) First, we prove part (i) of Theorem 1.2: Theorem 7.1. Assume that conditions (A) and (B) are satisfied: (A) μ(x ) ≥ 0 on ∂D. (B) γ(x ) < 0 on M = {x ∈ ∂D : μ(x ) = 0}. Then, for every 0 < ε < π/2, there exists a positive constant rp (ε) such that the resolvent set of Ap contains the set Σp (ε) = {λ = r2 eiθ : r ≥ rp (ε), −π + ε ≤ θ ≤ π − ε}, and that the resolvent (Ap − λI)−1 satisfies the estimate (Ap − λI)−1 ≤ cp (ε) |λ|

for all λ ∈ Σp (ε),

(1.4)

where cp (ε) is a positive constant depending on ε. Proof. The proof of Theorem 7.1 is divided into three steps. Step I: By estimate (6.2), it follows that if λ = r2 eiθ , −π < θ < π and if |λ| = r2 ≥ R(θ), then we have, for all u ∈ D(Ap ), |u|2,p + |λ|1/2 · |u|1,p + |λ| · u p ≤ C(θ) (Ap − λI)u p . However, we find from the proof of Theorem 6.3 that the constants R(θ) and C(θ) depend continuously on θ ∈ (−π, π), so that they may be chosen K. Taira, Boundary Value Problems and Markov Processes, Lecture Notes in Mathematics 1499, DOI 10.1007/978-3-642-01677-6 7, c Springer-Verlag Berlin Heidelberg 2009

101

102


uniformly in θ ∈ [−π + ε, π − ε], for every ε > 0. This proves the existence of the constants rp (ε) and cp (ε), that is, we have, for all λ = r2 eiθ satisfying r ≥ rp (ε) and −π + ε ≤ θ ≤ π − ε, |u|2,p + |λ|1/2 · |u|1,p + |λ| · u p ≤ cp (ε) (Ap − λI)u p .

(7.1)

By estimate (7.1), it follows that the operator Ap − λI is injective and its range R(Ap − λI) is closed in Lp (D), for all λ ∈ Σp (ε). Step II: We show that the operator Ap − λI is surjective for all λ ∈ Σp (ε), that is, (7.2) R(Ap − λI) = Lp (D) for all λ ∈ Σp (ε). To do this, it suffices to show that the operator Ap −λI is a Fredholm operator with (7.3) ind (Ap − λI) = 0 for all λ ∈ Σp (ε), since Ap − λI is injective for all λ ∈ Σp (ε). Here we recall that a densely defined, closed linear operator T with domain D(T ) from a Banach space X into itself is called a Fredholm operator if it satisfies the following three conditions: (i) The null space N (T ) = {x ∈ D(T ) : T x = 0} of T has finite dimension, that is, dim N (T ) < ∞. (ii) The range R(T ) = {T x : x ∈ D(T )} of T is closed in X. (iii) The range R(T ) has finite codimension in X, that is, codim R(T ) = dim X/R(T ) < ∞. In this case the index ind T of T is defined by the formula ind T = dim N (T ) − codim R(T ). Step II-1: We reduce the study of the operator Ap − λI (λ ∈ Σp (ε)) to that of a pseudo-differential operator on the boundary, just as in the proof of Theorem 1.1. Let T (λ) be a classical pseudo-differential operator of first order on the boundary ∂D defined as follows: T (λ) = LP (λ) = μ(x )Π(λ) + γ(x ),

λ ∈ Σp (ε),

(7.4)

where Π(λ) : C ∞ (∂D) −→ C ∞ (∂D) ∂ ϕ −→ (P (λ)ϕ) . ∂n ∂D Since the operator T (λ) : C ∞ (∂D) → C ∞ (∂D) extends to a continuous linear operator T (λ) : B t,p (∂D) → B t−1,p (∂D) for all t ∈ R, we can introduce a densely defined, closed linear operator Tp (λ) : B 2−1/p,p (∂D) −→ B 2−1/p,p (∂D) as follows.

7.1 Proof of Theorem 1.2, Part (i)

103

(α) The domain D (Tp (λ)) of Tp (λ) is the space D (Tp (λ)) = ϕ ∈ B 2−1/p,p (∂D) : T (λ)ϕ ∈ B 2−1/p,p (∂D) . (β) Tp (λ)ϕ = T (λ)ϕ, ϕ ∈ D (Tp (λ)). Then we can obtain the following three results (cf. [Ta2, Section 8.3]): (I) The null space N (Ap − λI) of Ap − λI has finite dimension if and only if the null space N (Tp (λ)) of Tp (λ) has finite dimension, and we have the formula dim N (Ap − λI) = dim N (Tp (λ)) . (II) The range R(Ap − λI) of Ap − λI is closed if and only if the range R (Tp (λ)) of Tp (λ) is closed; and R(Ap − λI) has finite codimension if and only if R (Tp (λ)) has finite codimension, and we have the formula codim R(Ap − λI) = codim R (Tp (λ)) . (III) The operator Ap − λI is a Fredholm operator if and only if the operator Tp (λ) is a Fredholm operator, and we have the formula ind (Ap − λI) = ind Tp (λ). Therefore, the desired assertion (7.3) is reduced to the following assertion: ind Tp (λ) = 0

for all λ ∈ Σp (ε).

(7.5)

Step II-2: To prove assertion (7.5), we shall make use of Agmon’s method just as in Chapter 6. Let T,(θ) be the classical pseudo-differential operator of first order on the boundary ∂D × S introduced in Chapter 6 (see formula (6.5)): , T,(θ) = LP,(θ) = μ(x )Π(θ) + γ(x ),

−π < θ < θ,

where , Π(θ) : C ∞ (∂D × S) −→ C ∞ (∂D × S) ∂ , ϕ˜ −→ P (θ)ϕ˜ . ∂n ∂D×S We define a densely defined, closed linear operator T,p (θ) : B 2−1/p,p (∂D × S) −→ B 2−1/p,p (∂D × S) as follows.

104


(˜ α) The domain D T,p (θ) of T,p (θ) is the space D T,p (θ) = ϕ˜ ∈ B 2−1/p,p (∂D × S) : T,(θ)ϕ˜ ∈ B 2−1/p,p (∂D × S) . ˜ T,p (θ)ϕ˜ = T,(θ)ϕ, ˜ ϕ˜ ∈ D T,p (θ) . (β) Then the most fundamental relationship between the operators T,p (θ) and Tp (λ) is stated as follows: Proposition 7.2. If ind T,p (θ) is finite, then there exists a finite subset K of Z such that the operator Tp (λ ) is bijective for all λ = 2 eiθ satisfying ∈ Z\ K. Granting Proposition 7.2 for the moment, we shall prove Theorem 7.1. Step III: End of Proof of Theorem 7.1 Step III-1: We show that if conditions (A) and (B) are satisfied, then we have the assertion ind T,p (θ) = dim N (T,p (θ)) − codim R(T,p (θ)) < ∞.

(7.6)

To this end, we need a useful criterion for Fredholm operators (cf. [Ta2, Theorem 3.7.6]): Lemma 7.3 (Peetre). Let X, Y , Z be Banach spaces such that X ⊂ Z is a compact injection, and let T be a closed linear operator with D(T ) from X into Y with domain D(T ). Then the following two conditions are equivalent: (i) The null space N (T ) of T has finite dimension and the range R(T ) of T is closed in Y . (ii) There is a positive constant C such that x X ≤ C ( T x Y + x Z )

for all x ∈ D(T ).

(7.7)

Proof. (i) =⇒ (ii): Since the null space N (T ) has finite dimension, we can find a closed topological complement X0 in X: X = N (T ) ⊕ X0 .

(7.8)

This gives that D(T ) = N (T ) ⊕ (D(T ) ∩ X0 ) . Namely, every element x of D(T ) can be written in the form x = x0 + x1 ,

x0 ∈ D(T ) ∩ X0 , x1 ∈ N (T ).

Moreover, since the range R(T ) is closed in Y , it follows from an application of the closed graph theorem [Yo, Chapter II, Section 6, Theorem 1] that there exists a positive constant C such that x0 X ≤ C T x0 Y = T x Y .

(7.9)


105

Here and in the following the letter C denotes a generic positive constant independent of x. On the other hand, it should be noticed that all norms on a finite dimensional linear space are equivalent. This gives that x1 X ≤ C x1 Z .

(7.10)

Moreover, since the injection X → Z is compact and hence is continuous, we obtain that x1 Z ≤ x Z + x0 Z ≤ x Z + C x0 X . (7.11) Thus we have, by inequalities (7.10) and (7.11), x1 X ≤ C ( x Z + x0 X ).

(7.12)

Therefore, by combining inequalities (7.9) and (7.12) we obtain the desired inequality x X ≤ x0 X + x1 X ≤ C ( T x Y + x Z )

for all x ∈ D(T ),

(7.7)

since T x0 = T x. (ii) =⇒ (i): By inequality (7.7), it follows that x X ≤ C x Z

for all x ∈ N (T ).

(7.13)

However, the null space N (T ) is closed in X, and so it is a Banach space. Since the injection X → Z is compact, we obtain from inequality (7.13) that the closed unit ball {x ∈ N (T ) : x X ≤ 1} of the Banach space N (T ) is compact. Hence it follows from an application of [Yo, Chapter III, Section 2, Corollary 2] that dim N (T ) < ∞. Let X0 be a closed topological complement of N (T ) as in decomposition (7.8). To prove the closedness of R(T ), it suffices to show that x X ≤ C T x Y

for all x ∈ D(T ) ∩ X0 .

Assume, to the contrary, that: For every n ∈ N, there is an element xn of D(T ) ∩ X0 such that xn X > n T xn Y . If we let

xn =

xn , xn X

then we have the assertions xn ∈ D(T ) ∩ X0 , 1 T xn Y < . n

xn X = 1,

(7.14a) (7.14b)

106


Since the injection X → Z is compact, by passing to a subsequence we may assume that the sequence {xn } is a Cauchy sequence in Z. Then, by combining inequalities (7.14b) and (7.7) we find that the sequence {xn } is a Cauchy sequence in X, and hence it converges to some element x of X0 ⊂ X. Thus, by assertions (7.14a) and (7.14b) it follows that x X = lim xn X = 1, n

and further that

x ∈ D(T ),

T x = 0,

since the operator T is closed. Summing up, we have proved that x ∈ N (T ), x X = 1. However, this is a contradiction. Indeed, we then have the assertion x ∈ N (T ) ∩ X0 = {0}. The proof of Lemma 7.3 is complete. By using estimate (6.8) with s := 2 − 1/p, we obtain that ,t |T,(θ)ϕ| |ϕ| ˜ 2−1/p,p ≤ C ˜ 2−1/p,p + |ϕ| ˜ t,p , ϕ˜ ∈ D(T,p (θ)),

(7.15)

where t < 2 − 1/p. However, it follows from an application of the Rellich– Kondrachov theorem that the injection B 2−1/p,p (∂D × S) → B t,p (∂D × S) is compact for t < 2 − 1/p. Thus, by applying Peetre’s lemma (Lemma 7.3) with X = Y := B 2−1/p,p (∂D × S), Z := B t,p (∂D × S), T := T,p (θ), we obtain that the range R T,p (θ) is closed in B 2−1/p,p (∂D × S) and that dim N T,p (θ) < ∞. (7.16) On the other hand, by formula (6.7) we find that the complete symbol of the adjoint T,(θ)∗ is given by the following formula (cf. Theorem 3.11): √ μ(x ) p˜1 (x , ξ , y, η; θ) − −1 q˜1 (x , ξ , y, η; θ) n−1 p0 (x , ξ , y, η; θ) − ∂xj μ(x ) · ∂ξj q˜1 (x , ξ , y, η; θ) + γ(x ) + μ(x )˜

√ − −1 μ(x )˜ q0 (x , ξ , y, η; θ) +

j=1

n−1 j=1

+ terms of order ≤ −1.

∂xj μ(x ) · ∂ξj p˜1 (x , ξ , y, η; θ)


107

However, it follows from an application of Lemma 5.3 that ∂xj μ(x ) = 0 on M = {x ∈ ∂D : μ(x ) = 0}. Thus we can easily verify that the pseudo-differential operator T,(θ)∗ satisfies conditions (3.7a) and (3.7b) of Theorem 3.16 with μ := 0, ρ := 1 and δ := 1/2. This implies that estimate (7.15) holds true for the adjoint operator T,p (θ)∗ of T,p (θ): ˜ −2+1/p,p ≤ C ˜ τ,p , ψ˜ ∈ D T,p (θ)∗ , ˜ −2+1/p,p + |ψ| ,τ |T,(θ)∗ ψ| |ψ| where τ < −2 + 1/p and p = p/(p − 1) is the exponent conjugate to p. Hence we have, by the closed range theorem ([Yo, Chapter VII, Section 5, Theorem]) and Peetre’s lemma (Lemma 7.3), (7.17) codim R T,p (θ) = dim N T,p (θ)∗ < ∞,

since the injection B −2+1/p,p (∂D × S) → B τ,p (∂D × S) is compact for τ < −2 + 1/p. Therefore, the desired assertion (7.6) follows by combining assertions (7.16) and (7.17). Step III-2: By assertion (7.6), we can apply Proposition 7.2 to obtain that the operator Tp (2 eiθ ) : B 2−1/p,p (∂D) → B 2−1/p,p (∂D) is bijective if ∈ Z \ K for some finite subset K of Z. In particular, we have the assertion ind Tp (λ0 ) = 0

for all λ0 = 2 eiθ with ∈ Z \ K.

(7.18)

However, in view of formulas (7.4) and (5.3) it follows that, for any given λ, λ0 ∈ Σp (ε), we can find a classical pseudo-differential operator K(λ, λ0 ) of order −1 on the boundary ∂D such that T (λ) = T (λ0 ) + K(λ, λ0 ). Furthermore, it follows from an application of the Rellich–Kondrachov theorem that the operator K(λ, λ0 ) : B 2−1/p,p (∂D) −→ B 2−1/p,p (∂D) is compact. Hence we have the assertion ind Tp (λ) = ind Tp (λ0 ) for all λ, λ0 ∈ Σp (ε).

(7.19)

Therefore, the desired assertion (7.5) (and hence assertion (7.3)) follows by combining assertions (7.18) and (7.19). Step III-3: Summing up, we have proved that the operator Ap − λI is bijective for all λ ∈ Σp (ε) and that its inverse (Ap − λI)−1 satisfies estimate (1.4). Theorem 7.1 is proved, apart from the proof of Proposition 7.2. The proof of Proposition 7.2 will be given in the next subsection, due to its length.

108


7.1.1 Proof of Proposition 7.2 The proof of Proposition 7.2 is divided into three steps. Step 1: First, we study the null spaces N T,p (θ) and N (Tp (λ )) when λ = 2 eiθ with ∈ Z: N T,p (θ) = ϕ˜ ∈ B 2−1/p,p (∂D × S) : T,(θ)ϕ˜ = 0 , N (Tp (λ )) = ϕ ∈ B 2−1/p,p (∂D) : T (λ )ϕ = 0 . Since the pseudo-differential operators T,(θ) and T (λ ) are both hypoelliptic, it follows that N T,p (θ) = ϕ˜ ∈ C ∞ (∂D × S) : T,(θ)ϕ˜ = 0 , N (Tp (λ )) = {ϕ ∈ C ∞ (∂D) : T (λ )ϕ = 0} . Therefore, we can apply [Ta2, Proposition 8.4.6] to obtain the following most , important relationship between the null spaces N Tp (θ) and N (Tp (λ )) when λ = 2 eiθ with ∈ Z: Lemma 7.4. The following two conditions are equivalent: (1) dim N T,p (θ) < ∞. (2) There exists a finite subset I of Z such that dim N Tp (2 eiθ ) < ∞ dim N Tp (2 eiθ ) = 0

if ∈ I, if ∈ I.

Moreover, in this case we have the formulas 5 N T,p (θ) = N Tp (2 eiθ ) ⊗ eiy , ∈I

dim N T,p (θ) = dim N Tp (2 eiθ ) . ∈I

Step 2: Secondly, we study the ranges R T,p (θ) and R (Tp (λ )) when λ = 2 eiθ with ∈ Z. To do this, we consider the adjoint operators T,p (θ)∗ and Tp (λ )∗ of T,p (θ) and Tp (λ ), respectively. The next lemma allows us to give a characterization of the adjoint operators T,p (θ)∗ and Tp (λ )∗ in terms of pseudo-differential operators (cf. [Ta2, Lemma 8.4.8]): Lemma 7.5. Let M be a compact smooth manifold without boundary. If T is a classical pseudo-differential operator of order m on M , we define a densely defined, closed linear operator


T : B s,p (M ) −→ B s−m+1,p (M )

109

(s ∈ R)

as follows. (a) The domain D(T ) of T is the space D(T ) = ϕ ∈ B s,p (M ) : T ϕ ∈ B s−m+1,p (M ) . (b) T ϕ = T ϕ, ϕ ∈ D(T ). Then the adjoint operator T ∗ of T is characterized as follows: (c) The domain D(T ∗ ) of T ∗ is contained in the space ψ ∈ B −s+m−1,p (M ) : T ∗ ψ ∈ B −s,p (M ) , where p = p/(p − 1) and T ∗ ∈ Lm cl (M ) is the adjoint of T . (d) T ∗ ψ = T ∗ ψ, ψ ∈ D(T ∗ ).

Proof. Let ψ be an arbitrary element of D(T ∗ ) ⊂ B −s+m−1,p (M ), and let {ψj } be a sequence in C ∞ (M ) such that ψj → ψ in B −s+m−1,p (M ). Then we have, for all ϕ ∈ C ∞ (M ) ⊂ D(T ), (T ∗ ψ, ϕ) = (ψ, T ϕ) = (ψ, T ϕ) = lim (ψj , T ϕ) j→∞

= lim (T ∗ ψj , ϕ) j→∞

= (T ∗ ψ, ϕ) . This proves that

T ∗ ψ = T ∗ ψ ∈ B −s,p (M ).

The proof of Lemma 7.5 is complete. We remark that the pseudo-differential operators T (λ)∗ and T,(θ)∗ also satisfy conditions (3.7a) and (3.7b) of Theorem 3.16 with μ := 0, ρ := 1 and δ := 1/2; hence they are hypoelliptic. Therefore, by applying Lemma 7.5 to the operators T,(θ) and T (λ ) we obtain the following: Lemma 7.6. The null spaces N T,p (θ)∗ and N (Tp (λ )∗ ) are characterized respectively as follows: N T,p (θ)∗ = ψ˜ ∈ C ∞ (∂D × S) : T,(θ)∗ ψ˜ = 0 . N (Tp (λ )∗ ) = {ψ ∈ C ∞ (∂D) : T (λ )∗ ψ = 0} .

110


By using Lemma 7.6, we find that Lemma 7.4 remains valid for the adjoint operators T,p (θ)∗ and Tp (λ )∗ (cf. [Ta2, Lemma 8.4.10]): Lemma 7.7. The following two conditions are equivalent: (1) dim N T,p (θ)∗ < ∞. (2) There exists a finite subset J of Z such that ' dim N Tp (2 eiθ )∗ < ∞ if ∈ J, dim N Tp (2 eiθ )∗ = 0 if ∈ J. Moreover, in this case we have the formula dim N T,p (θ)∗ = dim N Tp (2 eiθ )∗ . ∈J

Hence, by combining Lemma 7.7 and the closed range theorem ([Yo, Chapter VII, Section 5, we obtain the most important relation Theorem]) ship between codim R T,p (θ) and codim R (Tp (λ )) when λ = 2 eiθ , ∈ Z (cf. [Ta2, Proposition 8.4.11]): Lemma 7.8. The following two conditions are equivalent: (1) codim R T,p (θ) < ∞. (2) There exists a finite subset J of Z such that codim R Tp (2 eiθ ) < ∞ codim N Tp (2 eiθ ) = 0

if ∈ J, if ∈ J.

Moreover, in this case we have the formula codim R T,p (θ) = codim R Tp (2 eiθ )∗ . ∈J

Step 3: Proposition 7.2 is an immediate consequence of Lemmas 7.4 and 7.8, with K := I ∪ J. The proof of Proposition 7.2 is now complete. Summing up, we have proved Theorem 7.1 and hence part (i) of Theorem 1.2.

7.2 Proof of Theorem 1.2, Part (ii) Part (ii) of Theorem 1.2 may be proved as follows. Theorem 7.1 asserts that, for sufficiently large με > 0, the operator Ap − με I satisfies condition (2.1) (see Figure 7.1). Thus, by applying Theorem 2.2 (and Remark 2.1) to the operator Ap − με I we obtain part (ii) of Theorem 1.2:

7.2 Proof of Theorem 1.2, Part (ii)

111

Σω

Fig. 7.1.

Theorem 7.9. Assume that conditions (A) and (B) are satisfied. Then the operator Ap generates a semigroup Uz on Lp (D) which is analytic in the sector Δε = {z = t + is : z = 0, | arg z| < π/2 − ε} for any 0 < ε < π/2, and enjoys the following three properties: p z (a) The operators Ap Uz and dU dz are bounded operators on L (D) for each z ∈ Δε , and satisfy the relation

dUz = Ap Uz dz

for all z ∈ Δε .

0 (ε), M 1 (ε) and με (b) For each 0 < ε < π/2, there exist positive constants M such that 0 (ε)eμε ·Re z Uz ≤ M for all z ∈ Δε , M1 (ε) με ·Re z e for all z ∈ Δε . Ap Uz ≤ |z| (c) For each u0 ∈ Lp (D), we have, as z → 0, z ∈ Δε , Uz u0 −→ u0

in Lp (D).

The proof of Theorem 1.2 is now complete.

8 Proof of Theorem 1.3, Part (i)

This Chapter 8 and the next Chapter 9 are devoted to the proof of Theorem 1.3 and Theorem 1.4. In this chapter we prove part (i) of Theorem 1.3. In the proof we make use of Sobolev’s imbedding theorems (Theorems 8.1 and 8.2) and a λ-dependent localization argument due to Masuda [Ma] (cf. Lemma 8.4) in order to adjust estimate c (ε) p −1 (Ap − λI) ≤ |λ|

for all λ ∈ Σp (ε)

(1.4)

to obtain the desired estimate (A − λI)−1 ≤

c(ε) |λ|

for all λ ∈ Σ(ε).

(1.6)

Here we recall that

2,p 2,p ∂u D(Ap ) = u ∈ H (D) = W (D) : Lu = μ(x ) + γ(x )u = 0 . (1.3) ∂n (1.5) D(A) = u ∈ C0 (D \ M ) : Au ∈ C0 (D \ M ), Lu = 0 .

8.1 The Space C0 (D \ M ) First, we consider a one-point compactification K∂ = K ∪ {∂} of the space K = D \ M. We say that two points x and y of D are equivalent modulo M if x = y or x, y ∈ M ; we then write x ∼ y. It is easy to verify that this relation ∼ enjoys the so-called equivalence laws. We denote by D/M the totality of equivalence classes modulo M . On the set D/M we define the quotient topology induced by the projection K. Taira, Boundary Value Problems and Markov Processes, Lecture Notes in Mathematics 1499, DOI 10.1007/978-3-642-01677-6 8, c Springer-Verlag Berlin Heidelberg 2009

113

114


q : D −→ D/M. Namely, a subset O of D/M is defined to be open if and only if the inverse image q −1 (O) of O is open in D. It is easy to see that the topological space D/M is a one-point compactification of the space D \ M and that the point at infinity ∂ corresponds to the set M (see Figure 8.1): K∂ := D/M, ∂ := M.

∂D

D

q

D/M

∂

M Fig. 8.1.

Furthermore, we obtain the following two assertions: ˜ ◦ q is (i) If u ˜ is a continuous function defined on K∂ , then the function u continuous on D and constant on M . (ii) Conversely, if u is a continuous function defined on D and constant on M , then it defines a continuous function u ˜ on K∂ . In other words, we have the following isomorphism: C(K∂ ) ∼ = u ∈ C(D) : u(x) is constant on M .

(8.1)

Now we introduce a closed subspace of C(K∂ ) as in Subsection 2.2.1: C0 (K) = {u ∈ C(K∂ ) : u(∂) = 0} . Then we have, by assertion (8.1), C0 (K) ∼ = C0 (D \ M ) = u ∈ C(D) : u(x) = 0 on M .

(8.2)

8.2 Sobolev’s Imbedding Theorems It is the imbedding characteristics of Sobolev spaces of Lp type that render these spaces so useful in the study of partial differential equations. We need the following imbedding properties of Sobolev spaces:

8.3 Proof of Part (i) of Theorem 1.3

115

Theorem 8.1 (Sobolev). Let D be a bounded domain in the Euclidean space RN with boundary ∂D of class C 2 . Then we have the following two assertions: (i) If 1 ≤ p < N , we have the continuous injection W 2,p (D) ⊂ W 1,q (D)

for

1 p

−

1 N

≤

1 q

≤ p1 .

(ii) If N/2 < p < ∞, p = N , we have the continuous injection W 2,p (D) ⊂ C ν (D)

for 0 < ν ≤ 2 −

N p.

Theorem 8.2 (Gagliardo–Nirenberg). Let D be a bounded domain in RN with boundary of class C 2 , and 1 ≤ p, r ≤ ∞. Then we have the following assertions: (i) If p = N and if 1 1 = +θ q N

1 2 − p N

+ (1 − θ)

1 r

for

1 2

≤ θ ≤ 1,

then we have, for all u ∈ W 2,p (D) ∩ Lr (D), u 1,q ≤ C1 u θ2,p u 1−θ , r with a positive constant C1 = C1 (D, p, r, θ). (ii) If N/2 < p < ∞, p = N and if N N 0≤ν 0, there exists a positive constant rp (ε) such that if λ = r2 eiθ with r ≥ rp (ε) and −π + ε ≤ θ ≤ π − ε, we have, for all u ∈ D(Ap ), |λ|1/2 |u|C 1 (D) + |λ| · |u|C(D) ≤ Cp (ε)|λ|N/2p (A − λ)u p ,

(8.4)

with a positive constant Cp (ε). Here

2,p 2,p ∂u + γ(x )u = 0 . D(Ap ) = u ∈ H (D) = W (D) : Lu = μ(x ) ∂n Proof. First, by applying Theorem 8.2 with p := r > N , θ := N/p and ν := 0 we obtain from the Gagliardo–Nirenberg inequality (8.3) that . |u|C(D) ≤ C|u|1,p u 1−N/p p N/p

(8.5)

Here and in the following the letter C denotes a generic positive constant depending on p and ε, but independent of u and λ. Combining inequality (7.1) with inequality (8.5), we find that N/p 1−N/p |u|C(D) ≤ C |λ|−1/2 (A − λ)u p |λ|−1 (A − λ)u p = C|λ|−1+N/2p (A − λ)u p , so that |λ| · |u|C(D) ≤ C|λ|N/2p (A − λ)u p

for all u ∈ D(Ap ).

(8.6)

Similarly, by applying inequality (8.5) to the functions Di u ∈ W 1,p (D), 1 ≤ i ≤ n, we obtain that |Di u|C(D) ≤ C|Di u|1,p Di u 1−N/p p N/p

N/p

1−N/p

≤ C|u|2,p |u|1,p

1−N/p ≤ C ( (A − λ)u p )N/p |λ|−1/2 (A − λ)u p = C|λ|−1/2+N/2p (A − λ)u p . This proves that |λ|1/2 |u|C 1 (D) ≤ C|λ|N/2p (A − λ)u p

for all u ∈ D(Ap ).

(8.7)

Therefore, the desired inequality (8.4) follows by combining inequalities (8.6) and (8.7).


117

The next lemma proves estimate (1.6): Lemma 8.4. Assume that conditions (A) and (B) are satisfied. Then, for every ε > 0, there exists a positive constant r(ε) such that if λ = r2 eiθ with r ≥ r(ε) and −π + ε ≤ θ ≤ π − ε, we have, for all u ∈ D(A), |λ|1/2 |u|C 1 (D) + |λ| · |u|C(D) ≤ c(ε)|(A − λ)u|C(D) ,

(8.8)

with a positive constant c(ε). Here D(A) = u ∈ C0 (D \ M ) : Au ∈ C0 (D \ M ), Lu = 0 . Proof. We shall make use of a λ-dependent localization argument due to Masuda [Ma] in order to adjust the term (A − λ)u p in inequality (8.4) to obtain inequality (8.8). First, we remark that A ⊂ Ap

for all 1 < p < ∞.

Indeed, since we have, for any u ∈ D(A), u ∈ C(D) ⊂ Lp (D), Au ∈ C(D) ⊂ Lp (D)

and Lu = 0,

it follows from an application of Theorem 4.9 and Lemma 5.1 that u ∈ W 2,p (D). (1) Let x0 be an arbitrary point of the closure D = D ∪ ∂D. If x0 is a boundary point and if χ is a smooth coordinate transformation such that χ maps B(x0 , η0 ) ∩ D into B(0, δ) ∩ RN + and flattens a part of the boundary ∂D into the plane xN = 0 (see Figure 8.2), then we let G0 = B(x0 , η0 ) ∩ D, G = B(x0 , η) ∩ D, 0 < η < η0 , G = B(x0 , η/2) ∩ D, 0 < η < η0 . Here B(x, η) denotes the open ball of radius η about x. Similarly, if x0 is an interior point and if χ is a smooth coordinate transformation such that χ maps B(x0 , η0 ) into B(0, δ), then we let (see Figure 8.3) G0 = B(x0 , η0 ), G = B(x0 , η), 0 < η < η0 , G = B(x0 , η/2), 0 < η < η0 . (2) Now we take a function θ(t) in C0∞ (R) such that θ(t) equals one near the origin, and define ϕ(x) = θ(|x |2 ) θ(xN ),

x = (x , xN ).

118


xN

∂D

D

B(x 0,η 0)

B (0;±)

n

χ

x = (x1; : : : ; x N−1)

Fig. 8.2.

G0 = B(x0 ; ´0 ) ∂D

D G G

x0

Fig. 8.3.

Here we may assume that the function ϕ(x) is chosen so that supp ϕ ⊂ B(0, 1), ϕ(x) = 1 on B(0, 1/2). We introduce a localizing function x − x0 xN − t |x − x0 |2 ϕ0 (x, η) ≡ ϕ θ =θ , η η2 η We remark that

x0 = (x0 , t) ∈ D.

supp ϕ0 ⊂ B(x0 , η), ϕ0 (x, η) = 1 on B(x0 , η/2).

Then we have the following: Claim 8.5. If u ∈ D(A), then it follows that ϕ0 (x, η)u ∈ D(Ap ) for all 1 < p < ∞.


119

Proof. (i) First, we recall that u ∈ W 2,p (D)

for all 1 < p < ∞.

Hence we have the assertion ϕ0 (x, η)u ∈ W 2,p (D). (ii) Secondly, it is easy to verify (see Figure 8.4) that the function ϕ0 (x, η)u, x ∈ D, satisfies the boundary condition L(ϕ0 (x, η)u) = 0

D

on ∂D.

@D

B(x0 ; ´)

B(x0; ´)

x0 x0

Fig. 8.4.

Indeed, this is obvious if we have the condition supp (ϕ0 (x, η)u) ⊂ B(x0 , η),

x0 ∈ D.

Moreover, if we have the condition supp (ϕ0 (x, η)u) ⊂ B(x0 , η) ∩ D,

x0 ∈ ∂D,

then it follows that

|x − x0 |2 1 ∂ (ϕ0 (x, η)) = θ (0) · θ = 0, ∂xN η η2 xN =0

since θ (0) = 0. This proves that ∂ (ϕ0 (x, η)) = 0 ∂n

on ∂D.

120


Therefore, we have the assertion ∂ (ϕ0 (x, η)u) + γ(x )ϕ0 (x, η)u ∂n ∂ (ϕ0 (x, η)) u = ϕ0 (x, η)(Lu) + μ(x ) ∂n = 0 on ∂D,

L(ϕ0 (x, η)u) = μ(x )

since Lu = 0 on ∂D. Summing up, we have proved that ϕ0 (x, η)u ∈ D(Ap ) for all 1 < p < ∞. The proof of Claim 8.5 is complete. (3) Now we take a positive number p such that N < p < ∞. Then, by Claim 8.5 we can apply inequality (8.4) to the function ϕ0 (x, η)u, u ∈ D(A), to obtain that |λ|1/2 |u|C 1 (G ) + |λ| · |u|C(G ) ≤ |λ|1/2 |ϕ0 (x, η)u|C 1 (G ) + |λ| · |ϕ0 (x, η)u|C(G ) = |λ|1/2 |ϕ0 (x, η)u|C 1 (D) + |λ| · |ϕ0 (x, η)u|C(D) ≤ C|λ|N/2p (A − λ)(ϕ0 (x, η)u) Lp (D) = C|λ|N/2p (A − λ)(ϕ0 (x, η)u) Lp (G ) , since we have the assertions

0 < η < η0 ,

(8.9)

ϕ0 (x, η) = 1 on G , supp (ϕ0 (x, η)u) ⊂ G .

However, we have the formula (A − λ)(ϕ0 (x, η)u) = ϕ0 (x, η) ((A − λ)u) + [A, ϕ0 (x, η)]u,

(8.10)

where [A, ϕ0 (x, η)] is the commutator of A and ϕ0 (x, η) defined by the formula [A, ϕ0 (x, η)]u = A(ϕ0 (x, η)u) − ϕ0 (x, η)Au =2

N

aij (x)

i,j=1

∂ϕ0 ∂u ∂xi ∂xj

⎞ N 2 ϕ ∂ ∂ϕ 0 0 ⎠ u. (8.11) +⎝ aij (x) + bi (x) ∂x ∂x ∂x i j i i,j=1 i=1 ⎛

N


121

Here we need the following elementary inequality: Claim 8.6. We have, for all v ∈ C j (G ), j = 0, 1, 2, v W j,p (G ) ≤ |G |1/p v C j (G ) , where |G | denotes the measure of G . Proof. It suffices to note that we have, for all w ∈ C(G ), |w(x)|p dx ≤ |G | |w|pC(G ) . G

This proves Claim 8.6. Since we have (see Figure 8.3), for some positive constant c, |G | ≤ |B(x0 , η)| ≤ cη N , it follows from an application of Claim 8.6 that ϕ0 (x, η)((A − λ)u) Lp (G ) ≤ c1/p η N/p |(A − λ)u|C(G ) ,

0 < η < η0 . (8.12)

Furthermore, we remark that

|Dα ϕ0 (x, η)| = O η −|α|

as η ↓ 0.

Hence it follows from an application of Claim 8.6 that ∂ϕ0 ∂u ∂xi ∂xj p ≤ Cη |u|1,p,G ≤ Cη −1+N/p |u|C 1 (G ) , L (G ) 2 ∂ ϕ0 ∂xi ∂xj u p ≤ ηC2 |u|Lp (G ) ≤ Cη −2+N/p |u|C(G ) , L (G ) ∂ϕ0 ∂xi u p ≤ Cη |u|Lp (G ) ≤ Cη −1+N/p |u|C(G ) , L (G )

0 < η < η0 , (8.13) 0 < η < η0 , (8.14) 0 < η < η0 . (8.15)

By using inequalities (8.13), (8.14) and (8.15), we obtain from formula (8.11) that [A, ϕ0 (x, η)]u Lp (G ) ≤ C η −1+N/p |u|C 1 (G ) + η −2+N/p |u|C(G ) + η −1+N/p |u|C(G ) (8.16) ≤ C η −1+N/p |u|C 1 (D) + η −2+N/p |u|C(D) , 0 < η < η0 . In view of formula (8.10), it follows from inequalities (8.12) and (8.16) that (A − λ)(ϕ0 (x, η)u) Lp (G ) ≤ ϕ0 (x, η)((A − λ)u) Lp (G ) + [A, ϕ0 (x, η)]u Lp (G ) ≤ Cη N/p |(A − λ)u|C(G ) + η −1 |u|C 1 (D) + η −2 |u|C(D) , 0 < η < η0 .

(8.17)

122


Therefore, by combining inequalities (8.9) and (8.17) we obtain that |λ|1/2 |u|C 1 (G ) + |λ| · |u|C(G ) ≤ C|λ|N/2p (A − λ)(ϕ0 (x, η)u) Lp (G ) ≤ C|λ|N/2p η N/p |(A − λ)u|C(G ) + η −1 |u|C 1 (G ) + η −2 |u|C(G ) ≤ C|λ|N/2p η N/p |(A − λ)u|C(D) + η −1 |u|C 1 (D) + η −2 |u|C(D) , 0 < η < η0 .

(8.18)

We remark (see Figure 8.5) that the closure D = D ∪ ∂D can be covered by a finite number of sets of the forms B(x0 , η/2) ∩ D,

x0 ∈ ∂D,

and B(x0 , η/2),

D

x0 ∈ D.

@D

B(x0 ; ´ =2)

B(x0 ; ´ =2)

x0 x0

Fig. 8.5.

Hence, by taking the supremum of inequality (8.18) over x ∈ D we find that |λ|1/2 |u|C 1 (D) + |λ| · |u|C(D) ≤ C|λ|N/2p η N/p |(A − λ)u|C(D) + η −1 |u|C 1 (D) + η −2 |u|C(D) , 0 < η < η0 .

(8.19)

(4) We now choose the localization parameter η. We let η=

η0 K, |λ|1/2


123

where K is a positive constant (to be chosen later) satisfying the condition 0 0, considered from C(D) into itself, is non-negative and continuous (bounded) with norm 0 0 Gα = Gα 1 = max G0α 1(x). ∞

x∈D

Proof. Let f (x) be an arbitrary function in C θ (D) such that f (x) ≥ 0 on D. Then, by applying the weak maximum principle (see Theorem A.1) with A := A − α to the function −G0α f we obtain from formula (9.3) that G0α f ≥ 0

on D.

This proves the non-negativity of G0α . Since G0α is non-negative, we have, for all f ∈ C θ (D), −G0α f ∞ ≤ G0α f ≤ G0α f ∞ This implies the continuity of G0α with norm G0α = G0α 1 ∞ . The proof of Lemma 9.2 is complete.

on D.

128

9 Proof of Theorem 1.3, Part (ii)

Similarly, we have the following result: Lemma 9.3. The operator Hα , α > 0, considered from C(∂D) into C(D), is non-negative and continuous (bounded) with norm Hα = Hα 1 ∞ = max Hα 1(x). x∈D

More precisely, we have the following fundamental results: Theorem 9.4. (i) (a) The operator G0α , α > 0, can be uniquely extended to a non-negative, bounded linear operator on C(D) into itself, denoted again by G0α , with norm 0 0 G = G 1 ≤ 1 . (9.5) α α ∞ α (b) For any f ∈ C(D), we have the assertion G0α f = 0

on ∂D.

(c) For all α, β > 0, the resolvent equation holds: G0α f − G0β f + (α − β)G0α G0β f = 0,

f ∈ C(D).

(9.6)

(d) For any f ∈ C(D), we have the assertion lim αG0α f (x) = f (x)

α→+∞

for all x ∈ D.

(9.7)

Furthermore, if f (x ) = 0 on ∂D, then this convergence is uniform in x ∈ D, that is, we have the assertion lim αG0α f = f

α→+∞

in C(D).

(9.8)

(e) The operator G0α maps C k+θ (D) into C k+2+θ (D) for any non-negative integer k. (ii) (a ) The operator Hα , α > 0, can be uniquely extended to a nonnegative, bounded linear operator on C(∂D) into C(D), denoted again by Hα , with norm Hα = 1. (b ) For any ϕ ∈ C(∂D), we have the assertion Hα ϕ = ϕ

on ∂D.

(c ) For all α, β > 0, we have the equation Hα ϕ − Hβ ϕ + (α − β)G0α Hβ ϕ = 0,

ϕ ∈ C(∂D).

(9.9)

(d ) The operator Hα maps C k+2+θ (∂D) into C k+2+θ (D) for any non-negative integer k.

9.1 General Existence Theorem for Feller Semigroups

129

Proof. (i) (a) By making use of Friedrichs’ mollifiers, we find that the H¨ older space C θ (D) is dense in C(D) and further that non-negative functions can be approximated by non-negative smooth functions. Hence, by Lemma 9.2 it follows that the operator G0α : C θ (D) → C 2+θ (D) can be uniquely extended to a non-negative, bounded linear operator G0α : C(D) → C(D) with norm G0α = G0α 1 ∞ . Furthermore, since the function G0α 1 satisfies the conditions (A − α)G0α 1 = −1 in D, on ∂D, G0α 1 = 0 by applying Theorem A.2 with A := A − α we obtain that G0α = G0α 1 ∞ ≤

1 . α

(b) This assertion follows from formula (9.3), since the space C θ (D) is dense in C(D) and since the operator G0α : C(D) → C(D) is bounded. (c) We find from the uniqueness theorem for problem (9.2) (Theorem 9.1) that equation (9.6) holds true for all f ∈ C θ (D). Indeed, it suffices to note that the function v = G0α f − G0β f + (α − β)G0α G0β f ∈ C 2+θ (D) satisfies the conditions

(α − A)v = 0 v=0

in D, on ∂D,

so that v = 0 in D. Therefore, we obtain that the resolvent equation (9.6) holds true for all f ∈ C(D), since the space C θ (D) is dense in C(D) and since the operators G0α and G0β are bounded. (d) First, let f (x) be an arbitrary function in C θ (D) satisfying the boundary condition f |∂D = 0. Then it follows from an application of the uniqueness theorem for problem (9.2) (Theorem 9.1) that we have, for all α, β, f − αG0α f = G0α ((β − A)f ) − βG0α f. Indeed, the both sides satisfy the same equation (α − A)u = −Af in D and have the same boundary value 0 on ∂D. Thus we have, by estimate (9.5), f − αG0α f ∞ ≤ so that

1 β (β − A)f ∞ + f ∞ , α α

lim f − αG0α f ∞ = 0.

α→+∞

(9.10)

130


Now let f (x) be an arbitrary function in C(D) satisfying the boundary condition f |∂D = 0. By means of mollifiers, we can find a sequence {fj } in C θ (D) such that fj −→ f in C(D) as j → ∞, fj = 0 on ∂D. Then we have, by estimate (9.5) and assertion (9.10) with f := fj , f − αG0α f ∞ ≤ f − fj ∞ + fj − αG0α fj ∞ + αG0α (fj − f ) ∞ ≤ 2 f − fj ∞ + fj − αG0α fj ∞ , and hence

lim sup f − αG0α f ∞ ≤ 2 f − fj ∞ . α→+∞

This proves the desired assertion (9.8), since f − fj ∞ → 0 as j → ∞. To prove assertion (9.7), let f (x) be an arbitrary function in C(D) and let x be an arbitrary point of D. If we take a function ψ(y) in C(D) such that ⎧ ⎨ 0 ≤ ψ(y) ≤ 1 on D, ψ(y) = 0 in a neighborhood of x, ⎩ ψ(y) = 1 near the set ∂D, then it follows from the non-negativity of G0α and estimate (9.5) that 0 ≤ αG0α ψ(x) + αG0α (1 − ψ)(x) = αG0α 1(x) ≤ 1.

(9.11)

However, by applying assertion (9.8) to the function 1 − ψ(y) we have the assertion lim αG0α (1 − ψ)(x) = (1 − ψ)(x) = 1

α→+∞

for all x ∈ D.

In view of inequalities (9.11), this implies that lim αG0α ψ(x) = 0 for all x ∈ D.

α→+∞

Thus, since we have the inequalities − f ∞ ψ ≤ f ψ ≤ f ∞ ψ

on D,

it follows that, for x ∈ D, |αG0α (f ψ)(x)| ≤ f ∞ · αG0α ψ(x) −→ 0 as α → +∞. Therefore, by applying assertion (9.8) to the function (1−ψ(y))f (y) we obtain that f (x) = ((1 − ψ)f ) (x) = lim αG0α ((1 − ψ)f ) (x) α→+∞

=

lim αG0α f (x) α→+∞

for all x ∈ D.


131

This proves the desired assertion (9.7). (ii) (a ) Since the space C 2+θ (∂D) is dense in C(∂D), by Lemma 9.3 it follows that the operator Hα : C 2+θ (∂D) → C 2+θ (D) can be uniquely extended to a non-negative, bounded linear operator Hα : C(∂D) → C(D). Furthermore, by applying Theorem A.2 with A := A−α we have the assertion Hα = Hα 1 ∞ = 1. (b ) This assertion follows from formula (9.4), since the space C 2+θ (∂D) is dense in C(∂D) and since the operator Hα : C(∂D) → C(D) is bounded. (c ) We find from the uniqueness theorem for problem (9.2) that equation (9.9) holds true for all ϕ ∈ C 2+θ (∂D). Indeed, it suffices to note that the function w = Hα ϕ − Hβ ϕ + (α − β)G0α Hβ ϕ ∈ C 2+θ (D) satisfies the conditions

(α − A)w = 0 w=0

in D, on ∂D,

so that w=0

in D.

Therefore, we obtain that the desired equation (9.9) holds true for all ϕ ∈ C(∂D), since the space C 2+θ (∂D) is dense in C(∂D) and since the operators G0α and Hα are bounded. The proof of Theorem 9.4 is now complete. Summing up, we have the following diagrams for the operators G0α and Hα : G0

α C(D) −−−− → % ⏐ ⏐

C(D) % ⏐ ⏐

G0

α C θ (D) −−−− → C 2+θ (D)

C(∂D) % ⏐ ⏐

H

α −−−− →

C(D) % ⏐ ⏐

H

α → C 2+θ (D) C 2+θ (∂D) −−−−

Now we consider the following boundary value problem in the framework of the spaces of continuous functions. (α − A)u = f in D, (9.12) Lu = 0 on ∂D. To do this, we introduce three operators associated with problem (9.12).

132


(I) First, we introduce a linear operator A : C(D) −→ C(D) as follows. (a) The domain D(A) of A is the space C 2 (D). &N &N 2 u ∂u (b) Au = i,j=1 aij (x) ∂x∂i ∂x + i=1 bi (x) ∂x + c(x)u, u ∈ D(A). j i Then we have the following: Lemma 9.5. The operator A has its minimal closed extension A in the space C(D). Proof. We apply part (i) of Theorem 2.18 to the operator A. Assume that a function u ∈ C 2 (D) takes a positive maximum at an interior point x0 of D: u(x0 ) = max u(x) > 0. x∈D

Then it follows that ∂u (x0 ) = 0, 1 ≤ i ≤ N, ∂xi N ∂2u aij (x0 ) (x0 ) ≤ 0, ∂xi ∂xj i,j=1 since the matrix (aij (x)) is positive definite. Hence we have the assertion Au(x0 ) =

N i,j=1

aij (x0 )

∂ 2u (x0 ) + c(x0 )u(x0 ) ≤ 0. ∂xi ∂xj

This implies that the operator A satisfies condition (β) of Theorem 2.18 with K0 := D and K := D. Therefore, Lemma 9.5 follows from an application of the same theorem. The proof of Lemma 9.5 is complete. Remark 9.1. Since the injection: C(D) → D (D) is continuous, we have the formula Au =

N i,j=1

∂2u ∂u + bi (x) + c(x)u, ∂xi ∂xj ∂x i i=1 N

aij (x)

u ∈ C(D),

where the right-hand side is taken in the sense of distributions. The operators A and A can be visualized as follows: A

C(D) −−−−→ C(D) % % ⏐ ⏐ ⏐ ⏐ A

C 2 (D) −−−−→ C(D)


133

The extended operators G0α : C(D) → C(D) and Hα : C(∂D) → C(D), α > 0, still satisfy formulas (9.3) and (9.4) respectively in the following sense: Lemma 9.6. (i) For any f ∈ C(D), we have the assertions 0 Gα f ∈ D(A), (αI − A)G0α f = f in D. (ii) For any ϕ ∈ C(∂D), we have the assertions Hα ϕ ∈ D(A), (αI − A)Hα ϕ = 0 in D. Here D(A) is the domain of the closed extension A. Proof. (i) By making use of Friedrichs’ mollifiers, we can choose a sequence {fj } in C θ (D) such that fj → f in C(D) as j → ∞. Then it follows from the boundedness of G0α that G0α fj −→ G0α f

in C(D),

and further that (α − A)G0α fj = fj −→ f Hence we have the assertions 0 Gα f ∈ D(A), (αI − A)G0α f = f

in C(D).

in D.

since the operator A : C(D) → C(D) is closed. (ii) Similarly, part (ii) is proved, since the space C 2+θ (∂D) is dense in C(∂D) and since the operator Hα : C(∂D) → C(D) is bounded. The proof of Lemma 9.6 is complete. Corollary 9.7. Every function u in D(A) can be written in the following form: (9.13) u = G0α (αI − A)u + Hα (u|∂D ) for all α > 0. Proof. We let

w = u − G0α (αI − A)u − Hα (u|∂D ).

Then it follows from Lemma 9.6 that the function w is in D(A) and satisfies the conditions (αI − A)w = 0 in D, w=0 on ∂D. However, in light of Remark 9.1, by applying Lemma 5.1 with λ0 := α and Theorem 4.9 with A := A − α to the Dirichlet case (μ(x ) ≡ 0 and γ(x ) ≡ −1 on ∂D) we obtain that w ∈ C ∞ (D).

134


Therefore, it follows from an application of Theorem 9.1 with A := A − α that w = 0. This proves the desired formula (9.13). The proof of Corollary 9.7 is complete. (II) Secondly, we introduce a linear operator LG0α : C(D) −→ C(∂D) as follows.

0 0 older space C θ (D) with 0 < θ < 1. (a) The domain D LG α of LG α 0is the H¨ 0 0 (b) LGα f = L Gα f , f ∈ D LGα . Then we have the following: Lemma 9.8. The operator LG0α , α > 0, can be uniquely extended to a nonnegative, bounded linear operator LG0α : C(D) → C(∂D). Proof. Let f (x) be an arbitrary function in D(LG0α ) = C θ (D) such that f (x) ≥ 0 on D. Then we have the assertions ⎧ 0 ⎨ Gα f ∈ C 2+θ (D), G0 f ≥ 0 on D, ⎩ α on ∂D, G0α f |∂D = 0 and hence ∂ (G0 f ) + γ(x )G0α f ∂n α ∂ = μ(x ) (G0α f ) ≥ 0 on ∂D. ∂n

LG0α f = μ(x )

This proves that the operator LG0α is non-negative. By the non-negativity of LG0α , we have, for all f ∈ D(LG0α ), −LG0α f ∞ ≤ LG0α f ≤ LG0α f ∞

on ∂D.

This implies the boundedness of LG0α with norm LG0α = LG0α 1 ∞ . Recall that the space C θ (D) is dense in C(D) and that non-negative functions can be approximated by non-negative smooth functions. Hence we find that the operator LG0α can be uniquely extended to a non-negative, bounded linear operator LG0α : C(D) → C(∂D). The proof of Lemma 9.8 is complete.


135

The operators LG0α and LG0α can be visualized as follows: LG0

C(D) −−−−α→ % ⏐ ⏐

C(∂D) % ⏐ ⏐

LG0

C θ (D) −−−−α→ C 1+θ (∂D) The next lemma states a fundamental relationship between the operators LG0α and LG0β for α, β > 0: Lemma 9.9. For any f ∈ C(D), we have the formula LG0α f − LG0β f + (α − β)LG0α G0β f = 0

for all α, β > 0.

(9.14)

Proof. Choose a sequence {fj } in C θ (D) such that fj → f in C(D) as j → ∞, just as in Lemma 9.6. Then, by using the resolvent equation (9.6) with f := fj we have the formula LG0α fj − LG0β fj + (α − β)LG0α G0β fj = 0. Hence, the desired formula (9.14) follows by letting j → ∞, since the operators LG0α , LG0β and G0β are all bounded. The proof of Lemma 9.9 is complete. (III) Finally, we introduce a linear operator LHα : C(∂D) −→ C(∂D) as follows. (a) The domain D (LHα ) of LHα is the space C 2+θ (∂D). (b) LHα ψ = L (Hα ψ), ψ ∈ D (LHα ). Then we have the following: Lemma 9.10. The operator LHα , α > 0, has its minimal closed extension LHα in the space C(∂D). Proof. We apply part (i) of Theorem 2.18 to the operator LHα . To do this, it suffices to show that the operator LHα satisfies condition (β ) with K := ∂D (or condition (β) with K := K0 = ∂D) of the same theorem. Assume that a function ψ in D(LHα ) = C 2+θ (∂D) takes its positive maximum at some point x of ∂D. Since the function Hα ψ is in C 2+θ (D) and satisfies (A − α)Hα ψ = 0 in D, Hα ψ = ψ on ∂D, by applying the weak maximum principle (Theorem A.1) with A := A − α to the function Hα ψ we find that the function Hα ψ takes its positive maximum

136


at the boundary point x ∈ ∂D. Thus we can apply the boundary point lemma (Lemma A.3) with A := A − α to obtain that ∂ (Hα ψ)(x ) < 0. ∂n Hence we have the inequality LHα ψ(x ) =

N −1

αij (x )

i,j=1

∂2ψ ∂ (x ) + μ(x ) (Hα ψ)(x ) ∂xi ∂xj ∂n

+γ(x )ψ(x ) ≤ 0. This verifies condition (β ) of Theorem 2.18. Therefore, Lemma 9.10 follows from an application of the same theorem. The proof of Lemma 9.10 is complete. Remark 9.2. The operator LHα enjoys the following property: If a function ψ in the domain D LHα takes its positive maximum at some point x of ∂D, then we have the inequality LHα ψ(x ) ≤ 0.

(9.15)

The operators LHα and LHα can be visualized as follows: C(∂D) % ⏐ ⏐

LH

−−−−α→

C(∂D) % ⏐ ⏐

LH

C 2+θ (∂D) −−−−α→ C 1+θ (∂D) The next lemma states a fundamental relationship between the operators LHα and LHβ for α, β > 0: Lemma 9.11. The domain D LHα of LHα does not depend on α > 0; so we denote by D the common domain. Then we have the formula LHα ψ − LHβ ψ + (α − β)LG0α Hβ ψ = 0

for all α, β > 0 and ψ ∈ D. (9.16)

Proof. Let ψ(x ) be an arbitrary function in D(LHβ ), and choose a sequence {ψj } in D(LHβ ) = C 2+θ (∂D) such that, as j → ∞, ψj −→ ψ in C(∂D), LHβ ψj −→ LHβ ψ in C(∂D). Then it follows from the boundedness of Hβ and LG0α that


137

LG0α (Hβ ψj ) = LG0α (Hβ ψj ) −→ LG0α (Hβ ψ) in C(∂D). Therefore, by using formula (9.9) with ϕ := ψj we obtain that, as j → ∞, LHα ψj = LHβ ψj − (α − β)LG0α (Hβ ψj ) −→ LHβ ψ − (α − β)LG0α (Hβ ψ) in C(∂D). This implies that

ψ ∈ D(LHα ), LHα ψ = LHβ ψ − (α − β)LG0α (Hβ ψ),

since the operator LHα : C(∂D) → C(∂D) is closed. Conversely, by interchanging α and β we have the assertion D(LHα ) ⊂ D(LHβ ), and so D(LHα ) = D(LHβ ). The proof of Lemma 9.11 is complete. Now we can prove a general existence theorem for Feller semigroups on ∂D in terms of boundary value problem (9.12). The next theorem asserts that the operator LHα is the infinitesimal generator of some Feller semigroup on ∂D if and only if problem (9.12) is solvable for sufficiently many functions ϕ in the space C(∂D): Theorem 9.12. (i) If the operator LHα , α > 0, is the infinitesimal generator of a Feller semigroup on ∂D, then, for each positive constant λ, the boundary value problem (α − A)u = 0 in D, (9.17) (λ − L)u = ϕ on ∂D has a solution u ∈ C 2+θ (D) for any ϕ in some dense subset of C(∂D). (ii) Conversely, if, for some non-negative constant λ, problem (9.17) has a solution u ∈ C 2+θ (D) for any ϕ in some dense subset of C(∂D), then the operator LHα is the infinitesimal generator of some Feller semigroup on ∂D. Proof. (i) If the operator LHα generates a Feller semigroup on ∂D, by applying part (i) of Theorem 2.18 with K := ∂D to the operator LHα we obtain that R λI − LHα = C(∂D) for each λ > 0. This implies that the range R (λI − LHα ) is a dense subset of C(∂D) for each λ > 0. However, if ϕ ∈ C(∂D) is in the range R (λI − LHα ) and if ϕ = (λI − LHα ) ψ with ψ ∈ C 2+θ (∂D), then the function u = Hα ψ ∈ C 2+θ (D) is a solution of problem (9.17). This proves part (i) of Theorem 9.12.

138


(ii) We apply part (ii) of Theorem 2.18 with K := ∂D to the operator LHα . To do this, it suffices to show that the operator LHα satisfies condition (γ) of the same theorem, since it satisfies condition (β ), as is shown in the proof of Lemma 9.10. By the uniqueness theorem for problem (9.2), it follows that any function u ∈ C 2+θ (D) which satisfies the homogeneous equation (α − A)u = 0

in D

can be written in the form: u = Hα (u|∂D ) ,

u|∂D ∈ C 2+θ (∂D) = D (LHα ) .

Thus we find that if there exists a solution u ∈ C 2+θ (D) of problem (9.17) for a function ϕ ∈ C(∂D), then we have the assertion (λI − LHα ) (u|∂D ) = ϕ, and so ϕ ∈ R (λI − LHα ) . Hence, if, for some non-negative constant λ, problem (9.12) has a solution u ∈ C 2+θ (D) for any ϕ in some dense subset of C(∂D), then the range R (λI − LHα ) is dense in C(∂D). This verifies condition (γ) (with α0 := λ) of Theorem 2.18. Therefore, part (ii) of Theorem 9.12 follows from an application of the same theorem. Now the proof of Theorem 9.12 is complete. Remark 9.3. Intuitively, Theorem 9.12 asserts that we can “piece together” a Markov process (Feller semigroup) on the boundary ∂D with A-diffusion in the interior D to construct a Markov process (Feller semigroup) on the closure D = D ∪ ∂D. The situation may be represented schematically by Figure 9.1. We conclude this section by giving a precise meaning to the boundary conditions Lu for functions u in D(A). We let D(L) = u ∈ D(A) : u|∂D ∈ D , where D is the common domain of the operators LHα for all α > 0. We remark that the space D(L) contains C 2+θ (D), since C 2+θ (∂D) = D (LHα ) ⊂ D. Corollary 9.7 asserts that every function u in D(L) ⊂ D(A) can be written in the form u = G0α (αI − A)u + Hα (u|∂D ) for all α > 0. (9.13) Then we define the boundary condition Lu by the formula Lu = LG0α (αI − A)u + LHα (u|∂D ) , u ∈ D(L). The next lemma justifies definition (9.18) of Lu for each u ∈ D(L):

(9.18)


139

@D

D

Fig. 9.1.

Lemma 9.13. The right-hand side of formula (9.18) depends only on u, not on the choice of expression (9.13). Proof. Assume that

αI − A u + Hα (u|∂D ) = G0β βI − A u + Hβ (u|∂D ) ,

u = G0α

where α, β > 0. Then it follows from formula (9.14) with f := αI − A u and formula (9.18) with ψ := u|∂D that LG0α αI − A u + LHα (u|∂D ) = LG0β αI − A u − (α − β)LG0α G0β αI − A u +LHβ (u|∂D ) − (α − β)LG0α Hβ (u|∂D ) = LG0β ((βI − A)u) + LHβ (u|∂D ) +(α − β) LG0β u − LG0α G0β αI − A u − LG0α Hβ (u|∂D ) . (9.19) However, the last term of formula (9.19) vanishes. Indeed, it follows from formula (9.13) with α := β and formula (9.14) with f := u that LG0β u − LG0α G0β αI − A u − LG0α Hβ (u|∂D ) = LG0β u − LG0α G0β βI − A u + Hβ (u|∂D ) + (α − β)G0β u = LG0β u − LG0α u − (α − β)LG0α G0β u = 0. Therefore, we obtain from formula (9.19) that LG0α αI − A u + LHα (u|∂D ) = LG0β βI − A u + LHβ (u|∂D ) . The proof of Lemma 9.13 is complete.

140


9.2 Feller Semigroups with Reflecting Barrier Now we consider the Neumann boundary condition ∂u LN u ≡ . ∂n ∂D We recall that the boundary condition LN is supposed to correspond to the reflection phenomenon. The next theorem (formula (9.21)) asserts that we can “piece together” a Markov process on ∂D with A-diffusion in D to construct a Markov process on D = D ∪ ∂D with reflecting barrier (cf. [BCP, Théorème XIX]): Theorem 9.14. We define a linear operator AN : C(D) −→ C(D) as follows. (a) The domain D(AN ) of AN is the space D(AN ) = u ∈ D(A) : u|∂D ∈ DN , LN u = 0 ,

(9.20)

where DN is the common domain of the operators LN Hα for all α > 0. (b) AN u = Au, u ∈ D(AN ). Then the operator AN is the infinitesimal generator of some Feller semi−1 , α > 0, is given by group on D, and the Green operator GN α = (αI − AN ) the following formula: −1 0 GN LN G0α f , f ∈ C(D). (9.21) α f = Gα f − Hα LN Hα Proof. We apply part (ii) of Theorem 2.16 to the operator AN defined by formula (9.20). The proof is divided into eight steps. Step 1: First, we prove that: The operator LN Hα is the generator of some Feller semigroup on ∂D, for any sufficiently large α > 0. We introduce a linear operator TN (α) : B 2−1/p,p (∂D) −→ B 2−1/p,p (∂D) as follows. (a) The domain D(TN (α)) of TN (α) is the space D(TN (α)) = ϕ ∈ B 2−1/p,p (∂D) : LN Hα ϕ ∈ B 2−1/p,p (∂D) .

9.2 Feller Semigroups with Reflecting Barrier

141

(b) TN (α)ϕ = LN Hα ϕ, ϕ ∈ D(TN (α)). Here it should be emphasized that the harmonic operator Hα is essentially the same as the Poisson operator P (α) introduced in Chapter 5. Then, by arguing just as in the proof of Theorem 7.1 with μ(x ) := 1 and γ(x ) := 0 on ∂D we obtain that The operator TN (α) is bijective for any sufficiently large α > 0. (9.22) Furthermore, it maps the space C ∞ (∂D) onto itself. Since we have the assertion LN Hα = TN (α)

on C ∞ (∂D),

it follows from assertion (9.22) that the operator LN Hα also maps C ∞ (∂D) onto itself, for any sufficiently large α > 0. This implies that the range R(LN Hα ) is a dense subset of C(∂D). Hence, by applying part (ii) of Theorem 9.12 we obtain that the operator LN Hα generates a Feller semigroup on ∂D, for any sufficiently large α > 0. Step 2: Next we prove that: The operator LN Hβ generates a Feller semigroup on ∂D, for any β > 0. We take a positive constant α so large that the operator LN Hα generates a Feller semigroup on ∂D. We apply Corollary 2.19 with K := ∂D to the operator LN Hβ for β > 0. By formula (9.16), it follows that the operator LN Hβ can be written as LN Hβ = LN Hα + Nαβ , where Nαβ = (α − β)LN G0α Hβ is a bounded linear operator on C(∂D) into itself. Furthermore, assertion (9.16) implies that the operator LN Hβ satisfies condition (β ) of Theorem 2.18. Therefore, it follows from an application of Corollary 2.19 that the operator LN Hβ also generates a Feller semigroup on ∂D. Step 3: Now we prove that: The equation L N Hα ψ = ϕ has a unique solution ψ in D(LN Hα ) for any ϕ ∈ C(∂D); hence the inverse LN Hα

−1

of LN Hα can be defined on the whole space C(∂D).

Furthermore, the operator −LN Hα on the space C(∂D).

−1

is non-negative and bounded (9.23)

142


Since the function Hα 1(x) takes its positive maximum 1 only on the boundary ∂D, we can apply the boundary point lemma (Lemma A.3) to obtain that ∂ (Hα 1) < 0 on ∂D. ∂n

(9.24)

Hence the Neumann boundary condition implies that L N Hα 1 = and so

∂ (Hα 1) < 0 on ∂D, ∂n

(−LN Hα 1)(x ) = − max LN Hα 1(x ) > 0. α = min x ∈∂D

x ∈∂D

Furthermore, by using Corollary 2.17 with K := ∂D, A := LN Hα and c := α we obtain that the operator LN Hα + α I is the infinitesimal generator of some Feller semigroup on ∂D. Therefore, since α > 0, it follows from an application of part (i) of Theorem 2.16 with A := LN Hα + α I that the equation −LN Hα ψ = α I − (LN Hα + α I) ψ = ϕ has a unique solution ψ ∈ D(LN Hα ) for any ϕ ∈ C(∂D), and further that the −1 −1 is non-negative and bounded operator −LN Hα = α I − (LN Hα + α I) on the space C(∂D) with norm −1 1 −1 . ≤ −LN Hα = α I − (LN Hα + α I) α Step 4: By assertion (9.23), we can define the right-hand side of formula (9.21) for all α > 0. We prove that: −1 GN , α = (αI − AN )

α > 0.

(9.25)

In view of Lemmas 9.6 and 9.13 with L := LN , it follows that we have, for all f ∈ C(D), ⎧ −1 N 0 0f ⎪ G f = G f − H L H L G ∈ D(A), ⎪ α N α N α α α ⎪ ⎨ −1 N LN G0α f ∈ D LN Hα = DN , Gα f |∂D = −LN Hα ⎪ ⎪ ⎪ ⎩ LN (GN f ) = LN G0 f − LN Hα LN Hα −1 LN G0 f = 0, α α α and (αI − A)(GN α f ) = f. Hence we have proved that, for all f ∈ C(D), N Gα f ∈ D (AN ) , (αI − AN ) GN α f = f.


143

This proves that (αI − AN ) GN α = I

on C(D).

Therefore, in order to prove formula (9.25) it suffices to show the injectivity of the operator αI − AN for α > 0. Assume that: u ∈ D (AN ) and (αI − AN ) u = 0. Then, by Corollary 9.7 it follows that the function u can be written as u = Hα (u|∂D ) , u|∂D ∈ DN = D LN Hα . Thus we have the assertion LN Hα (u|∂D ) = LN u = 0. In view of assertion (9.23), this implies that u|∂D = 0, so that u = Hα (u|∂D ) = 0 in D. This proves the injectivity of αI − AN for α > 0. Step 5: The non-negativity of GN α , α > 0, follows immediately from for−1 0 mula (9.21), since the operators Gα , Hα , −LN Hα and LN G0α are all nonnegative. Step 6: We prove that the operator GN α is bounded on the space C(D) with norm N Gα ≤ 1 for all α > 0. (9.26) α To do this, it suffices to show that, for all α > 0, GN α1≤

1 α

on D.

(9.27)

since GN α is non-negative on C(D). First, it follows from the uniqueness property of solutions of problem (9.2) that (9.28) αG0α 1 + Hα 1 = 1 + G0α (c(x)) on D. Indeed, the both sides satisfy the same equation (α − A)u = α in D and have the same boundary value 1 on ∂D. By applying the boundary operator LN to the both hand sides of equality (9.28), we obtain that −LN Hα 1 = −LN 1 − LN G0α (c(x)) + αLN G0α 1 ∂ 0 =− Gα (c(x)) + αLN G0α 1 ∂n ≥ αLN G0α 1

∂D

on ∂D,

144


since G0α (c(x)) = 0 on ∂D and G0α (c(x)) ≤ 0 on D. Hence we have, by the −1 non-negativity of −LN Hα , −LN Hα

−1

1 LN G0α 1 ≤ α

on ∂D.

(9.29)

By using formula (9.21) with f := 1, inequality (9.29) and equality (9.28), we obtain that −1 0 GN LN G0α 1 α 1 = Gα 1 + Hα −LN Hα ≤ G0α 1 +

1 Hα 1 α

1 1 + G0 (c(x)) α α α 1 on D, ≤ α =

since the operators Hα and G0α are non-negative and since G0α (c(x)) ≤ 0 on D. Therefore, we have proved the desired assertion (9.27) for all α > 0. Step 7: Finally, we prove that: The domain D (AN ) is dense in the space C(D).

(9.30)

Step 7-1: Before the proof, we need some lemmas on the behavior of G0α , −1 Hα and −LN Hα as α → +∞: Lemma 9.15. For all f ∈ C(D), we have the assertion 3 4 lim αG0α f + Hα (f |∂D ) = f in C(D). α→+∞

(9.31)

Proof. Choose a positive constant β and let g := f − Hβ (f |∂D ). Then, by using formula (9.9) with ϕ := f |∂D we obtain that 3 4 αG0α g − g = αG0α f + Hα (f |∂D ) − f − βG0α Hβ (f |∂D ).

(9.32)

However, we have, by estimate (9.5), lim G0α Hβ (f |∂D ) = 0 in C(D),

α→+∞

and, by assertion (9.8), lim αG0α g = g

α→+∞

in C(D),

since g|∂D = 0. Therefore, the desired formula (9.31) follows by letting α → +∞ in formula (9.32). The proof of Lemma 9.15 is complete.


145

Lemma 9.16. The function ∂ (Hα 1) ∂n ∂D diverges to −∞ uniformly and monotonically as α → +∞. Proof. First, formula (9.9) with ϕ := 1 gives that Hα 1 = Hβ 1 − (α − β)G0α Hβ 1. Thus, in view of the non-negativity of G0α and Hα it follows that α ≥ β =⇒ Hα 1 ≤ Hβ 1

on D.

Since Hα 1|∂D = Hβ 1|∂D = 1, this implies that the functions ∂ (Hα 1) ∂n ∂D are monotonically non-increasing in α. Furthermore, by using formula (9.7) with f := Hβ 1 we find that the function β Hα 1(x) = Hβ 1(x) − 1 − αG0α Hβ 1(x) α converges to zero monotonically as α → +∞, for each interior point x of D. Now, for any given positive constant K we can construct a function u ∈ C 2 (D) such that u = 1 on ∂D, ∂u ≤ −K on ∂D. ∂n

(9.33a) (9.33b)

Indeed, it follows from an application of Theorem 9.1 that, for any integer m > 0, the function u = (Hα0 1)m belongs to C 2+θ (D) and satisfies condition (9.33a). Furthermore, we have the assertion ∂u ∂ (H = m 1) α0 ∂n ∂D ∂n ∂D ∂ (Hα0 1) (x ) . ≤ m max x ∈∂D ∂n In view of inequality (9.24) with α := α0 , this implies that the function m u = (Hα0 u) satisfies condition (9.33b) for m sufficiently large.

146


@D U U

U @D

@D

Fig. 9.2.

We take a function u ∈ C 2 (D) which satisfies conditions (9.33a) and (9.33b), and choose a neighborhood U of ∂D, relative to D, with smooth boundary ∂U such that (see Figure 9.2) u≥

1 2

on U .

(9.34)

Recall that the function Hα 1 converges to zero in the interior D monotonically as α → +∞. Since u = Hα 1 = 1 on the boundary ∂D, by using Dini’s theorem we can find a positive constant α (depending on u and hence on K) such that Hα 1 ≤ u on ∂U \ ∂D, α > 2 Au ∞.

(9.35a) (9.35b)

It follows from inequalities (9.34) and (9.35b) that (A − α)(Hα 1 − u) = αu − Au α ≥ − Au ∞ 2 > 0 in U . Thus, by applying the weak maximum principle (Theorem A.1) with A := A − α to the function Hα 1 − u we obtain that the function Hα 1 − u may take its positive maximum only on the boundary ∂U . However, conditions (9.33a) and (9.35a) imply that Hα 1 − u ≤ 0 on ∂U = (∂U \ ∂D) ∪ ∂D. Therefore, we have the assertion


Hα 1 ≤ u

147

on U = U ∪ ∂U ,

and hence

∂u ∂ (Hα 1) ≤ ≤ −K on ∂D, ∂n ∂n since u|∂D = Hα 1|∂D = 1. The proof of Lemma 9.16 is complete. In the following we shall use the notation |ϕ(x )| ϕ ∞ = max x ∈∂D

for a function ϕ(x ) defined on the boundary ∂D. −1 Now we can study the behavior of the operator norm − LN Hα as α → +∞: Corollary 9.17. limα→+∞ − LN Hα

−1

= 0.

Proof. By Lemma 9.16, it follows that the function LN Hα 1(x ) =

∂ (Hα 1) (x ), ∂n

x ∈ ∂D,

diverges to −∞ monotonically as α → +∞. By Dini’s theorem, this convergence is uniform in x ∈ ∂D. Hence the function 1 LN Hα 1(x ) converges to zero uniformly in x ∈ ∂D as α → +∞. This implies that −1 −1 −LN Hα = −LN Hα 1 ∞ 1 ≤ −→ 0 as α → +∞. LN Hα 1 ∞ Indeed, it suffices to note that −LN Hα 1(x ) 1 (−LN Hα 1(x )) ≤ 1= |LN Hα 1(x )| LN Hα 1 ∞

for all x ∈ ∂D.

The proof of Corollary 9.17 is complete. Step 7-2: Proof of Assertion (9.30) In view of formula (9.25) and inequality (9.26), it suffices to prove that lim αGN α f − f ∞ = 0,

α→+∞

since the space C ∞ (D) is dense in C(D).

f ∈ C ∞ (D),

(9.36)

148


First, we remark that N −1 0 0 αGα f − f = L f − αH L H G f − f αG α N α N α α ∞ ∞ 0 ≤ αGα f + Hα (f |∂D ) − f ∞ −1 + −αHα LN Hα LN G0α f − Hα (f |∂D ) ∞ ≤ αG0α f + Hα (f |∂D ) − f ∞ −1 + −αLN Hα LN G0α f − f |∂D . ∞

Thus, in view of assertion (9.31) it suffices to show that * + −1 LN G0α f − f |∂D = 0 in C(∂D). lim −αLN Hα α→+∞

(9.37)

We take a constant β such that 0 < β < α, and write f = G0β g + Hβ ϕ, where (cf. formula (9.13)):

g = (β − A)f ∈ C θ (D), ϕ = f |∂D ∈ C 2+θ (∂D).

Then, by using equations (9.6) (with f := g) and (9.9) we obtain that G0α f = G0α G0β g + G0α Hβ ϕ 1 0 Gβ g − G0α g + Hβ ϕ − Hα ϕ . = α−β Hence we have the assertion −1 LN G0α f − f |∂D −αLN Hα ∞ α α −1 0 0 L −L ϕ − ϕ = H G g − L G g + L H ϕ + N α N β N α N β α − β α−β ∞ α −1 0 ≤ −LN Hα · LN Gβ g + LN Hβ ϕ ∞ α−β α β −1 ϕ ∞ . + −LN Hα · LN G0α · g ∞ + α−β α−β By Corollary 9.17, it follows that the first term on the last inequality converges to zero as α → +∞: α −1 −LN Hα · LN G0β g + LN Hβ ϕ∞ −→ 0 as α → +∞. α−β For the second term, by using formula (9.6) with f := 1 and the non-negativity of G0β and LN G0α we find that

9.3 Proof of Theorem 1.4

149

LN G0α = LN G0α 1 ∞ = LN G0β 1 − (α − β)LN G0α G0β 1 ∞ ≤ LN G0β 1 ∞ . Hence the second term also converges to zero as α → +∞: α −1 −LN Hα · LN G0α · g ∞ −→ 0 as α → +∞. α−β It is clear that the third term converges to zero as α → +∞: β ϕ ∞ −→ 0 as α → +∞. α−β Therefore, we have proved assertion (9.37) and hence the desired assertion (9.36). The proof of assertion (9.30) is complete. Step 8: Summing up, we have proved that the operator AN , defined by formula (9.20), satisfies conditions (a) through (d) in Theorem 2.16. Hence it follows from an application of the same theorem that the operator AN is the infinitesimal generator of some Feller semigroup on D. The proof of Theorem 9.14 is now complete.

9.3 Proof of Theorem 1.4 We apply part (ii) of Theorem 2.16 to the operator A defined by formula (1.5). First, we simplify the boundary condition Lu = μ(x )

∂u + γ(x )u = 0 ∂n

on ∂D.

Assume that the following conditions (A) and (B ) are satisfied: (A) μ(x ) ≥ 0 on ∂D. (B ) γ(x ) ≤ 0 on ∂D and γ(x ) < 0 on M = {x ∈ ∂D : μ(x ) = 0}. Then it follows that

μ(x ) − γ(x ) > 0

on ∂D.

Thus we find that the boundary condition μ(x )

∂u + γ(x )u = 0 on ∂D ∂n

is equivalent to the boundary condition γ(x ) μ(x ) ∂u + u = 0 on ∂D. μ(x ) − γ(x ) ∂n μ(x ) − γ(x )

150


However, if we let μ ˜ (x ) =

μ(x ) , μ(x ) − γ(x )

γ˜ (x ) =

γ(x ) , μ(x ) − γ(x )

then we have the assertions μ ˜(x )

∂u + γ˜(x )u = 0 on ∂D ∂n

and 0≤μ ˜(x ) ≤ 1 on ∂D, ˜(x ) − 1 on ∂D. γ˜ (x ) = μ Namely, we may assume that the boundary condition L is of the form Lu = μ(x ) with

∂u + (μ(x ) − 1)u = 0 ∂n

0 ≤ μ(x ) ≤ 1

on ∂D,

on ∂D.

Next we express the boundary condition L in terms of the Dirichlet and Neumann conditions. It follows from an application of Lemmas 9.8 and 9.10 that LG0α = μ(x ) LN G0α , and that

LHα = μ(x ) LN Hα + μ(x ) − 1.

Hence, in view of definition (9.18) we obtain that Lu = LG0α (αI − A)u + LHα (u|∂D ) = μ(x )LN G0α (αI − A)u + μ(x )LN Hα (u|∂D ) + (μ(x ) − 1)(u|∂D ) = μ(x ) LN G0α (αI − A)u + LN Hα (u|∂D ) + (μ(x ) − 1)(u|∂D ) = μ(x )LN u + (μ(x ) − 1)(u|∂D ),

u ∈ D(L).

This proves the desired formula L = μ(x )LN + μ(x ) − 1. Therefore, the next theorem proves Theorem 1.4: Theorem 9.18. We define a linear operator A : C0 (D \ M ) −→ C0 (D \ M ) as follows (cf. formula (9.20)).

(9.38)


151

(a) The domain D(A) of A is the space ' D(A) =

u ∈ C0 (D \ M ) : Au ∈ C0 (D \ M ), (

Lu = μ(x )LN u + (μ(x ) − 1) (u|∂D ) = 0 .

(9.39)

(b) Au = Au, u ∈ D(A). Assume that the following condition (A ) is satisfied: (A ) 0 ≤ μ(x ) ≤ 1 on ∂D. Then the operator A is the infinitesimal generator of some Feller semigroup {Tt }t≥0 on D \ M , and the Green operator Gα = (αI − A)−1 , α > 0, is given by the following formula: −1 Gα f = GN , f ∈ C0 (D \ M ). (9.40) LGN α f − Hα LHα αf Here GN α is the Green operator for the Neumann condition LN given by formula (9.21). Proof. We apply part (ii) of Theorem 2.16 to the operator A. The proof is divided into six steps. Step 1: First, we prove that: If condition (A ) is satisfied, then the operator LHα is the generator of some Feller semigroup on ∂D, for any sufficiently large α > 0. We introduce a linear operator T (α) : B 2−1/p,p (∂D) −→ B 2−1/p,p (∂D) as follows. (a) The domain D(T (α)) of T (α) is the space D(T (α)) = ϕ ∈ B 2−1/p,p (∂D) : LHα ϕ ∈ B 2−1/p,p (∂D) . (b) T (α)ϕ = LHα ϕ, ϕ ∈ D(T (α)). Then, by arguing just as in the proof of Theorem 7.1 with γ(x ) := μ(x )−1 on ∂D we obtain that The operator T (α) is bijective for any sufficiently large α > 0. (9.41) Furthermore, it maps the space C ∞ (∂D) onto itself. Since we have the assertion LHα = T (α)

on C ∞ (∂D),

152


it follows from assertion (9.41) that the operator LHα also maps C ∞ (∂D) onto itself, for any sufficiently large α > 0. This implies that the range R(LHα ) is a dense subset of C(∂D). Hence, by applying part (ii) of Theorem 9.12 we obtain that the operator LHα generates a Feller semigroup on ∂D, for any sufficiently large α > 0. Step 2: Next we prove that: The operator LHβ generates a Feller semigroup on ∂D, for any β > 0. We take a positive constant α so large that the operator LHα generates a Feller semigroup on ∂D. We apply Corollary 2.19 with K := ∂D to the operator LHβ for β > 0. By formula (9.16), it follows that the operator LHβ can be written as LHβ = LHα + Mαβ , where Mαβ = (α − β)LG0α Hβ is a bounded linear operator on C(∂D) into itself. Furthermore, assertion (9.15) implies that the operator LHβ satisfies condition (β ) of Theorem 2.18. Therefore, it follows from an application of Corollary 2.19 that the operator LHβ also generates a Feller semigroup on ∂D. Step 3: Now we prove that: If condition (A ) is satisfied, then the equation LHα ψ = ϕ has a unique solution ψ in D LHα for any ϕ ∈ C(∂D); hence the inverse LHα

−1

of LHα can be defined on the whole space C(∂D).

Furthermore, the operator −LHα on the space C(∂D).

−1

is non-negative and bounded (9.42)

Since we have, by inequality (9.24), LHα 1 = μ(x )

∂ (Hα 1) + μ(x ) − 1 < 0 on ∂D, ∂n

it follows that (−LHα 1(x )) = − max LHα 1(x ) > 0. kα = min x ∈∂D

x ∈∂D

In view of Lemma 9.16, we find that the constants kα are increasing in α > 0: α ≥ β > 0 =⇒ kα ≥ kβ . Furthermore, by using Corollary 2.17 with K := ∂D, A := LHα and c := kα we obtain that the operator LHα + kα I is the infinitesimal generator of some


153

Feller semigroup on ∂D. Therefore, since kα > 0, it follows from an application of part (i) of Theorem 2.16 with A := LHα + kα I that the equation −LHα ψ = kα I − (LHα + kα I) ψ = ϕ has a unique solution ψ ∈ D LHα for any ϕ ∈ C(∂D), and further that the −1 −1 = kα I − (LHα + kα I) is non-negative and bounded operator −LHα on the space C(∂D) with norm −1 1 −1 . −LHα = kα I − (LHα + kα I) ≤ kα Step 4: By assertion (9.42), we can define the operator Gα by formula (9.40) for all α > 0. We prove that: Gα = (αI − A)−1 ,

α > 0.

(9.43)

By Lemma 9.6 and Theorem 9.14, it follows that we have, for all f ∈ C0 (D \ M ), Gα f ∈ D(A), and A(Gα f ) = αGα f − f. Furthermore, we obtain that the function Gα f satisfies the boundary condition −1 LGN = 0 on ∂D. (9.44) L(Gα f ) = LGN α f − LHα LHα αf However, we recall that (see formula (9.38)) Lu = μ(x )LN u + (μ(x ) − 1) (u|∂D ) ,

u ∈ D(L).

(9.45)

Hence it follows that the boundary condition (9.44) is equivalent to the following: L(Gα f ) = μ(x )LN (Gα f ) + (μ(x ) − 1) (Gα f |∂D ) = 0 on ∂D.

(9.46)

By condition (9.46), we have, for all f ∈ C0 (D \ M ), Gα f = 0 on M = {x ∈ ∂D : μ(x ) = 0} , and so A(Gα f ) = αGα f − f = 0 on M . Summing up, we have proved that f ∈ C0 (D \ M )

=⇒ Gα f ∈ D(A) = u ∈ C0 (D \ M ) : Au ∈ C0 (D \ M ), Lu = 0 ,

154


and further that (αI − A)Gα f = f,

f ∈ C0 (D \ M ).

This proves that (αI − A)Gα = I

on C0 (D \ M ).

Therefore, in order to prove formula (9.43), it suffices to show the injectivity of the operator αI − A for α > 0. Assume that, for α > 0, u ∈ D(A) and (αI − A)u = 0. Then, by Corollary 9.7 it follows that the function u can be written as follows: u = Hα (u|∂D ) , u|∂D ∈ D = D LHα . Thus we have the formula LHα (u|∂D ) = Lu = 0. In view of assertion (9.42), this implies that u|∂D = 0, so that u = Hα (u|∂D ) = 0 in D. This proves the injectivity of αI − A for α > 0. Step 5: Now we prove the following three assertions: (i) The operator Gα is non-negative on the space C0 (D \ M ): f ∈ C0 (D \ M ), f ≥ 0 on D \ M =⇒ Gα f ≥ 0 on D \ M .

(9.47)

(ii) The operator Gα is bounded on the space C0 (D \ M ) with norm Gα ≤

1 α

for all α > 0.

(9.48)

(iii) The domain D(A) is dense in the space C0 (D \ M ). Step 5-1: In order to prove assertion (i), we have only to show the nonnegativity of the operator Gα on the space C(D): f ∈ C(D), f ≥ 0 on D =⇒ Gα f ≥ 0 on D. Recall that the Dirichlet problem (α − A)u = f u=ϕ

in D, on ∂D

(9.49)

(9.2)


is uniquely solvable. Hence it follows that N 0 GN α f = Hα Gα f |∂D + Gα f

on D.

155

(9.50)

Indeed, the both sides satisfy the same equation (α − A)u = f in D and have the same boundary values GN α f on ∂D. Thus, by applying the boundary operator L to the both sides of formula (9.50) we obtain that N 0 LGN α f = LHα Gα f |∂D + LGα f. Since the operators −LHα

−1

and LG0α are non-negative, it follows that

f ≥ 0 on D =⇒ −1 −1 N 0f LGN −LHα f = −G f | + −LH LG ∂D α α α α ≥ −GN α f |∂D on ∂D. Therefore, by the non-negativity of Hα and G0α we find from formulas (9.40) and (9.50) that −1 LGN Gα f = GN α f + Hα −LHα αf N ≥ GN α f + Hα −Gα f |∂D = G0α f ≥0

on D.

This proves the desired assertion (9.49) and hence assertion (9.47). Step 5-2: Next we prove assertion (ii). To do this, it suffices to show the boundedness of the operator Gα on the space C(D): Gα 1 ≤

1 α

on D,

since Gα is non-negative on the space C(D). We remark (cf. formula (9.45)) that N N LGN α f = μ(x ) LN Gα f + (μ(x ) − 1) Gα f |∂D = (μ(x ) − 1) GN α f |∂D , so that

−1 LGN Gα f = GN α f − Hα LHα αf −1 N −LH (μ(x . f + H ) − 1)G f | = GN α α ∂D α α

(9.51)

156


Hence, by using this formula with f := 1 we obtain that −1 (1 − μ(x ))GN . Gα 1 = GN α 1 − Hα −LHα α 1|∂D However, we have, by inequality (9.27), 0 ≤ GN α1≤ and also

1 α

on D,

−1 (1 − μ(x ))GN ≥0 Hα −LHα α 1|∂D

since the operators Hα and −LHα on ∂D. Therefore, we obtain that

−1

on D,

are non-negative and since 1 − μ(x ) ≥ 0

0 ≤ Gα 1 ≤ GN α1 ≤

1 α

on D.

This proves the desired assertion (9.51) and hence assertion (9.48). Step 5-3: Finally, we prove assertion (iii). In view of formula (9.43), it suffices to show that lim αGα f − f ∞ = 0,

α→+∞

f ∈ C0 (D \ M ) ∩ C ∞ (D),

(9.52)

since the space C0 (D \ M ) ∩ C ∞ (D) is dense in C0 (D \ M ). We remark that −1 N LG f − f − αH LH f αGα f − f = αGN α α α α N −1 α(1 − μ(x ))GN . (9.53) = αGα f − f + Hα LHα α f |∂D We estimate the last two terms of formula (9.53) as follows: (1) By assertion (9.36), it follows that the first term of formula (9.53) tends to zero as α → +∞: (9.54) lim αGN α f − f ∞ = 0. α→+∞

(2) To estimate the second term of formula (9.53), we remark that −1 Hα LHα α(1 − μ(x ))GN α f |∂D −1 = Hα LHα ((1 − μ(x ))f |∂D ) −1 (1 − μ(x ))(αGN . (9.55) +Hα LHα α f − f )|∂D However, it follows that the second term in the right-hand side of formula (9.55) tends to zero as α → +∞. Indeed, we have, by assertion (9.54),


−1 (1 − μ(x )) αGN Hα LHα α f − f |∂D ∞ −1 ≤ −LHα · (1 − μ(x )) αGN f − f | ∂D ∞ α 1 (1 − μ(x )) αGN ≤ α f − f |∂D ∞ kα 1 αGN f − f −→ 0. ≤ α ∞ k1

157

(9.56)

Here we have used the following facts (cf. the proof of assertion (9.42)): 1 −1 , α > 0. −LHα ≤ kα k1 = min (−LH1 1)(x ) ≤ kα = min (−LHα 1)(x ) x ∈∂D

x ∈∂D

for all α ≥ 1.

Thus we are reduced to the study of the first term of the right-hand side of formula (9.55) −1 Hα LHα ((1 − μ(x ))f |∂D ) . Now, for any given ε > 0, we can find a function h(x ) in C ∞ (∂D) such that h = 0 near M = {x ∈ ∂D : μ(x ) = 0}, (1 − μ(x ))f |∂D − h ∞ < ε. Then we have, for all α ≥ 1, −1 −1 Hα LHα ((1 − μ(x ))f |∂D ) − Hα LHα h ∞ −1 ≤ −LHα · (1 − μ(x ))f |∂D − h ∞ ε ≤ kα ε ≤ . k1 Furthermore, we can find a function θ(x ) in C0∞ (∂D) such that near M , θ(x ) = 1 (1 − θ(x ))h(x ) = h(x ) on ∂D. Then we have the assertion h(x ) = (1 − θ(x )) h(x ) 1 − θ(x ) = (−LHα 1(x )) h(x ) −LHα 1(x ) 1−θ ≤ −LHα 1 · h ∞ (−LHα 1(x )) . ∞

(9.57)

158


Since the operator −LHα that

−1

is non-negative on the space C(∂D), it follows 1−θ −1 · h ∞ on ∂D, −LHα h ≤ −LHα 1 ∞

so that −1 −1 Hα LHα h ≤ −LHα h

∞

1−θ ≤ −LHα 1

∞

· h ∞ .

(9.58)

However, there exists a positive constant δ0 such that 0≤

1 − θ(x ) ≤ δ0 μ(x )

for all x ∈ ∂D.

Thus it follows that 1 − θ(x ) 1 − θ(x ) = ∂ −LHα 1(x ) μ(x ) − ∂n (Hα 1(x )) + (1 − μ(x )) 1 − θ(x ) 1 ∂ ≤ μ(x ) − ∂n (Hα 1(x )) 1 ∂ , ≤ δ0 minx ∈∂D − ∂n (Hα 1(x )) and hence from Lemma 9.16 that 1−θ lim α→+∞ −LHα 1

= 0.

(9.59)

∞

Summing up, we obtain from inequalities (9.57) and (9.58) and assertion (9.59) that −1 lim sup Hα LHα ((1 − μ(x ))f |∂D ) ∞ α→+∞ −1 ≤ lim sup Hα LHα h α→+∞ −1 −1 + Hα LHα ((1 − μ(x ))f |∂D ) − Hα LHα h ∞ 1−θ ε · h ∞ + ≤ lim α→+∞ −LHα 1 k1 ∞ ε ≤ . k1 Since ε is arbitrary, this proves that the first term of the right-hand side of formula (9.55) tends to zero as α → +∞: −1 lim Hα LHα ((1 − μ(x ))f |∂D ) = 0. (9.60) α→+∞

∞

9.4 Proof of Part (ii) of Theorem 1.3

159

By assertions (9.56) and (9.60), we obtain that the last term of formula (9.53) also tends to zero: −1 α(1 − μ(x ))GN lim Hα LHα f | (9.61) = 0. ∂D α α→+∞

∞

Therefore, the desired assertion (9.52) follows by combining assertions (9.54) and (9.61). The proof of assertion (iii) is complete. Step 6: Summing up, we have proved that the operator A, defined by formula (9.39), satisfies conditions (a) through (d) in Theorem 2.16. Hence, in view of assertion (8.2), it follows from an application of part (ii) of the same theorem that the operator A is the infinitesimal generator of some Feller semigroup {Tt }t≥0 on D \ M . The proof of Theorem 9.18 and hence that of Theorem 1.4 is now complete.

9.4 Proof of Part (ii) of Theorem 1.3 We apply Theorem 2.2 to the operator A. In the proof of Theorem 9.18, we have proved that the domain D(A) is dense in the space C0 (D \ M ). Furthermore, part (i) of Theorem 1.3 verifies condition (2.1). Therefore, it follows from an application of Theorem 2.2 that: The semigroup Tt can be extended to a semigroup Tz which is analytic in the sector Δε = {z = t + is : z = 0, | arg z| < π/2 − ε} for any 0 < ε < π/2. This (together with Theorem 1.4) proves part (ii) of Theorem 1.3.

10 Application to Semilinear Initial-Boundary Value Problems

This chapter is devoted to the semigroup approach to a class of initialboundary value problems for semilinear parabolic differential equations. We prove Theorem 1.5 by using the theory of fractional powers of analytic semigroups (Theorems 10.1 and 10.2). To do this, we verify that all the conditions of Theorem 2.8 are satisfied. Our semigroup approach here can be traced back to the pioneering work of Fujita–Kato [FK]. For detailed studies of semilinear parabolic equations, the reader is referred to Friedman [Fr1], Henry [He] and also [Ta4].

10.1 Local Existence and Uniqueness Theorems Let D be a bounded domain of Euclidean space RN , with smooth boundary ∂D; its closure D = D ∪ ∂D is an N -dimensional, compact smooth manifold with boundary. We let A=

N i,j=1

∂2 ∂ + bi (x) + c(x) ∂xi ∂xj i=1 ∂xi N

aij (x)

be a second-order, elliptic differential operator with real smooth coefficients on D such that: (1) aij (x) = aji (x) for all x ∈ D and 1 ≤ i, j ≤ N . (2) There exists a positive constant a0 such that N


for all (x, ξ) ∈ D × RN .

i,j=1

(3) c(x) ≤ 0 on D.


161

162


As an application of Theorem 1.2, we consider the following semilinear initial-boundary value problem: Given functions f (x, t, u, ξ) and u0 (x) defined in D×[0, T )×R×RN and in D, respectively, find a function u(x, t) in D×[0, T ) such that ⎧ ∂ ⎨ ∂t − A u(x, t) = f (x, t, u, grad u) in D × (0, T ), ∂u (1.7) Lu(x , t) := μ(x ) ∂n + γ(x )u = 0 on ∂D × [0, T ), ⎩ u(x, 0) = u0 (x) in D. Here: (4) μ ∈ C ∞ (∂D) and μ(x ) ≥ 0 on ∂D. (5) γ ∈ C ∞ (∂D) and γ(x ) ≤ 0 on ∂D. (6) n = (n1 , n2 , . . . , nN ) is the unit interior normal to the boundary ∂D (see Figure 1.1). Recall that the operator Ap is a unbounded linear operator from Lp (D) into itself given by the following formulas: (a) The domain of definition D(Ap ) of Ap is the space

∂u + γ(x )u = 0 . D(Ap ) = u ∈ H 2,p (D) = W 2,p (D) : Lu(x ) = μ(x ) ∂n (b) Ap u = Au, u ∈ D(Ap ). By using the operator Ap , we can formulate problem (1.7) in terms of the abstract Cauchy problem in the Banach space Lp (D) as follows: du 0 < t < T, dt = Ap u(t) + F (t, u(t)) , (1.8) u|t=0 = u0 . Here u(t) = u(·, t) and F (t, u(t)) = f (·, t, u(t), grad u(t)) are functions defined on the interval [0, T ), taking values in the space Lp (D). First, we consider the case N < p < ∞: Theorem 10.1. Let N < p < ∞, and assume that conditions (A) and (B) are satisfied: (A) μ(x ) ≥ 0 on ∂D. (B) γ(x ) < 0 on M = {x ∈ D : μ(x ) = 0}. If the nonlinear term f (x, t, u, ξ) is a locally Lipschitz continuous function with respect to all its variables (x, t, u, ξ) ∈ D×[0, T )×R×RN with the possible exception of the x variables, then, for every function u0 of D(Ap ), problem (1.8) has a unique local solution u ∈ C ([0, T1 ]; Lp (D)) ∩ C 1 ((0, T1 ); Lp (D)) where T1 = T1 (p, u0 ) is a positive constant.

10.2 Fractional Powers and Imbedding Theorems

163

Here C ([0, T ]; Lp (D)) denotes the space of continuous functions on the closed interval [0, T ] taking values in Lp (D), and C 1 ((0, T ); Lp (D)) denotes the space of continuously differentiable functions on the open interval (0, T ) taking values in Lp (D), respectively. In the case 1 < p < N , the domain D(Ap ) is large compared with the case N < p < ∞. Hence we must impose some growth conditions on the nonlinear term f (x, t, u, ξ): Theorem 10.2. Let N/2 < p < N , and assume that conditions (A) and (B) are satisfied. Furthermore, we assume that there exist a non-negative continuous function ρ(t, r) on R × R and a constant 1 ≤ γ < N/(N − p) such that the following four conditions are satisfied: (a) |f (x, t, u, ξ)| ≤ ρ(t, |u|)(1 + |ξ|γ ). (b) |f (x, t, u, ξ) − f (x, s, u, ξ)| ≤ ρ(t, |u|) (1 + |ξ|γ ) |t − s|.

γ−1 |ξ − η|. (c) |f (x, t, u, ξ) − f (x, t, u, η)| ≤ ρ(t, |u|) 1 + |ξ|γ−1 + |η| γ (d) |f (x, t, u, ξ) − f (x, t, v, ξ)| ≤ ρ(t, |u| + |v|) (1 + |ξ| ) |u − v|. Then, for every function u0 of D(Ap ), problem (1.8) has a unique local solution u ∈ C ([0, T2 ]; Lp (D)) ∩ C 1 ((0, T2 ); Lp (D)) where T2 = T2 (p, u0 ) is a positive constant. Theorems 10.1 and 10.2 prove Theorem 1.5, and are a generalization of Pazy [Pa, Section 8.4, Theorems 4.4 and 4.5] to the degenerate case.

10.2 Fractional Powers and Imbedding Theorems First, we study the imbedding properties of the domains of the fractional powers (−Ap )α (0 < α < 1) into Sobolev spaces of Lp type. By virtue of Theorem 2.8, this allows us to solve, by successive approximations, problem (1.8), proving Theorems 10.1 and 10.2. By Theorem 7.1, we may assume that the operator Ap satisfies condition (2.21) in Subsection 2.1.2 (see Figure 7.1): (1) The resolvent set of Ap contains the region Σ as in Figure 10.1: (2) There exists a positive constant M such that the resolvent (Ap − λI)−1 satisfies the estimate (Ap − λI)−1 ≤

M (1 + |λ|)

for all λ ∈ Σ.

(10.1)

By using estimate (10.1), we can define the fractional powers (−Ap )α for 0 < α < 1 on the space Lp (D) as follows (cf. formula (2.23)): sin απ ∞ −α (−Ap )−α = − s (Ap − sI)−1 ds, π 0

164


Σ

0

Fig. 10.1.

and

(−Ap )α = the inverse of (−Ap )−α .

We recall that (−Ap )α is a closed operator with domain D ((−Ap )α ) ⊃ D(Ap ). In this section we study the imbedding characteristics of D((−Ap )α ), which will make these spaces so useful in the study of semilinear parabolic differential equations. We let Xα = the space D ((−Ap )α ) endowed with the graph norm · α of (−Ap )α . Here

1/2 2 u α = u 2p + (−Ap )α u p ,

u ∈ D ((−Ap )α ) .

Then we have the following three assertions: (1) The space Xα is a Banach space. (2) The graph norm u α is equivalent to the norm (−Ap )α u p . (3) If 0 < α < β < 1, then we have Xβ ⊂ Xα with continuous injection. Furthermore, since the domain D(Ap ) is contained in W 2,p (D), we can obtain the following imbedding properties of the spaces Xα into Sobolev spaces (cf. [He, Theorem 1.6.1]): Theorem 10.3. Let 1 < p < ∞. Then we have the following continuous injections: (i) Xα ⊂ W 1,q (D) if (ii) Xα ⊂ C ν (D) if

1 2 N 2p

< α < 1, 1p − 2α−1 < 1q ≤ 1p , 1 < p < N . N < α < 1, 0 ≤ ν < 2α − Np , p = N .

10.3 Proof of Theorems 10.1 and 10.2

165

10.3 Proof of Theorems 10.1 and 10.2 This section is devoted to the proof of our local existence and uniqueness theorems for problem (1.8) (Theorems 10.1 and 10.2). To do this, we verify that all the conditions of Theorem 2.8 are satisfied. 10.3.1 Proof of Theorem 10.1 We verify that all the conditions of Theorem 2.8 are satisfied; then Theorem 10.1 follows from an application of the same theorem. Since p > N , we can choose a positive constant α such that 1 N + 1 < α < 1, 2 p so that 1 < 2α −

N . p

Then, by part (ii) of Theorem 10.3 with ν := 1, we have Xα ⊂ C 1 (D) and Xα ⊂ W 1,p (D),

(10.2)

with continuous injections. Thus we find that the function F (t, u) := f (x, t, u(x), grad u(x)) is well defined on [0, T ] × Xα . Furthermore, since the function f (x, t, u, ξ) is locally Lipschitz continuous, in view of assertion (10.2) it follows that we have, for all t, s ∈ [0, t0 ] and for all u, v ∈ Xα with u − u0 α ≤ R, v − u0 α ≤ R F (t, u) − F (s, v) p ≤ F (t, u) − F (t, v) p + F (t, v) − F (s, v) p ! N ∂ ≤ C u − v p + ∂xi (u − v) + |t − s| p i=1 ≤ C ( u − v 1,p + |t − s|) ≤ CC ( u − v α + |t − s|) .

(10.3)

Here C = C(t0 , R) is a (local) Lipschitz positive constant for the function f , and C is an imbedding (positive) constant for the imbedding: Xα ⊂ W 1,p (D). By inequality (10.3), we obtain that the function F (t, u) is locally Lipschitz continuous in t and u. The proof of Theorem 10.1 is complete.

166


10.3.2 Proof of Theorem 10.2 The proof is similar to that of Theorem 10.1; we verify that all the conditions of Theorem 2.8 are satisfied. Since N/2 < p < N and 1 ≤ γ < N/(N − p), we can choose a positive constant α such that N 1 N γ−1 , + max < α < 1, (10.4) 2p 2 2p γ so that 0 < 2α −

N p

and

1 2α − 1 1 1 − < ≤ . p N pγ p

Then, by Theorem 10.3 with ν := 0 and q := pγ, we have Xα ⊂ L∞ (D)

and Xα ⊂ W 1,pγ (D),

(10.5)

with continuous injections. We let F (t, u) := f (x, t, u(x), grad u(x)),

t ∈ [0, T ], u ∈ Xα .

Then we have, by condition (a), F (t, u) pp ≤ 2p−1 ρ(t, u ∞ )p

(1 + |grad u|pγ )dx ≤ 2p−1 ρ(t, u ∞ )p |D| + u pγ 1,pγ , D

where |D| denotes the volume of the domain D. By assertion (10.5), it follows that the function F (t, u) is well defined on [0, T ] × Xα for all α satisfying condition (10.4). Step 1: First, we verify the local Lipschitz continuity of F (t, u) with respect to the variable t. By condition (b), it follows that F (t, u) − F (s, u) pp = |f (x, t, u(x), grad u(x)) − f (x, s, u(x), grad u(x))|p dx D p−1 ≤ 2 ρ(t, u ∞ )p |t − s|p (1 + |grad u|pγ )dx D p ≤ 2p−1 ρ(t, u ∞ )p |D| + u pγ 1,pγ |t − s| . In view of assertion (10.5), this proves that F (t, u) − F (s, u) p ≤ C1 ( u α ) |t − s|, where C1 ( u α ) is a positive constant depending on the norm u α.

(10.6)

10.3 Proof of Theorems 10.1 and 10.2

167

Step 2: Secondly, we verify the local Lipschitz continuity of F (t, u) with respect to the variable u. To do this, we remark that F (t, u) − F (t, v) pp = |f (x, t, u(x), grad u(x)) − f (x, t, v(x), grad v(x))|p dx D p−1 ≤2 |f (x, t, u(x), grad u(x)) − f (x, t, u(x), grad v(x))|p dx D +2p−1 |f (x, t, u(x), grad v(x)) − f (x, t, v(x), grad v(x))|p dx. (10.7) D

We estimate the two terms on the right-hand side of inequality (10.7): Step 2-1: By condition (c), it follows that |f (x, t, u(x), grad u(x)) − f (x, t, u(x), grad v(x))|p dx D

≤ 3p−1 ρ(t, u ∞ )p 1 + |grad u|p(γ−1) + |grad v|p(γ−1) |grad (u − v)|p dx. (10.8) × D

However, by H¨ older’s inequality it follows that

|grad (u − v)| dx ≤ p

D

(γ−1)/γ 1/γ pγ 1dx |grad (u − v)| dx

D (γ−1)/γ

≤ |D|

u −

D p v 1,pγ ,

(10.9)

|grad u|

p(γ−1)

(γ−1)/γ

|grad (u − v)| dx ≤

D

|grad u|

p

pγ

D

× D p(γ−1)

≤ u 1,pγ

p(γ−1)

|grad v|p(γ−1) |grad (u − v)|p dx ≤ v 1,pγ

1/γ |grad (u − v)|pγ dx u − v p1,pγ

(10.10)

u − v p1,pγ .

(10.11)

D

Thus, by carrying these three inequalities (10.9), (10.10) and (10.11) into inequality (10.8) we obtain that |f (x, t, u(x), grad u(x)) − f (x, t, u(x), grad v(x))|p dx D p(γ−1) p(γ−1) ≤ 3p−1 ρ(t, u ∞ )p |D|(γ−1)/γ + u 1,pγ + v 1,pγ × u − v p1,pγ .

(10.12)

168


Step 2-2: By condition (d), it follows that |f (x, t, u(x), grad v(x)) − f (x, t, v(x), grad v(x))|p dx D p−1 ≤ 2 ρ(t, u ∞ + v ∞ )p (1 + |grad v|pγ ) |u − v|p dx D p−1 p p ≤ 2 ρ(t, u ∞ + v ∞ ) u − v ∞ (1 + |grad v|pγ ) dx D p ≤ 2p−1 ρ(t, u ∞ + v ∞ )p |D| + v pγ (10.13) 1,pγ u − v ∞ . Therefore, by combining inequalities (10.7), (10.12) and (10.13) we obtain that F (t, u) − F (t, v) pp p(γ−1) p(γ−1) ≤ 6p−1 ρ(t, u ∞)p |D|(γ−1)/γ + u 1,pγ + v 1,pγ u − v p1,pγ p +4p−1 ρ(t, u ∞ + v ∞ )p |D| + v pγ 1,pγ u − v ∞ . In view of assertion (10.5), this proves that F (t, u) − F (t, v) p ≤ C2 ( u α , v α ) u − v α ,

(10.14)

where C2 ( u α , v α ) is a positive constant depending on the norms u α and v α . Summing up, we find from inequalities (10.6) and (10.14) that the function F (t, u) is locally Lipschitz continuous in t and u. The proof of Theorem 10.2 is now complete.

11 Concluding Remarks

This book is devoted to a careful and accessible exposition of the functional analytic approach to the problem of construction of Markov processes with Ventcel’ boundary conditions in probability theory. More precisely, we prove that there exists a Feller semigroup corresponding to such a diffusion phenomenon that a Markovian particle moves continuously in the state space D \ M until it “dies” at the time when it reaches the set M where the particle is definitely absorbed (see Figure 11.1). Our approach here is distinguished by the extensive use of the ideas and techniques characteristic of the recent developments in the theory of pseudo-differential operators which may be considered as a modern theory of the classical potential theory. More generally, it is known (see [BCP], [SU], [Ta2], [We]) that the infinitesimal generator W of a Feller semigroup {Tt }t≥0 is described analytically by a Waldenfels operator W and a Ventcel’ boundary condition L, which we formulate precisely. Let W be a second-order, elliptic integro-differential operator with real coefficients such that W u(x) = Au(x) + Sr u(x) ⎞ ⎛ N N 2 u ∂ ∂u aij (x) (x) + bi (x) (x) + c(x)u(x)⎠ := ⎝ ∂x ∂x ∂x i j i i,j=1 i=1 ⎡ ⎤ N ∂u s(x, y) ⎣u(y) − u(x) − (yj − xj ) (x)⎦ dy. + ∂x j D j=1 Here: (1) aij ∈ C ∞ (D), aij (x) = aji (x) for all x ∈ D and 1 ≤ i, j ≤ N , and there exists a positive constant a0 such that N


for all (x, ξ) ∈ Ω × RN .

i,j=1


169

170


∂D

D

M = {¹ = 0} Fig. 11.1.

(2) bi ∈ C ∞ (D) for all 1 ≤ i ≤ N . (3) c ∈ C ∞ (D) and c(x) ≤ 0 in D. (4) The integral kernel s(x, y) is the distribution kernel of a properly supN ported, pseudo-differential operator S ∈ L2−κ 1,0 (R ), κ > 0, which has the transmission property with respect to ∂D (see [Bt]), and s(x, y) ≥ 0 off the diagonal {(x, x) : x ∈ RN } in RN × RN . The measure dy is the Lebesgue measure on RN . The operator W is called a second-order, Waldenfels operator (cf. [Wa]). The differential operator A is called a diffusion operator which describes analytically a strong Markov process with continuous paths (diffusion process) in the interior D. The operator Sr is called a second-order Lévy operator which is supposed to correspond to the jump phenomenon in the interior D. More precisely, a Markovian particle moves by jumps to a random point, chosen with kernel s(x, y), in the interior D. Therefore, the Waldenfels operator W = A + Sr is supposed to correspond to such a diffusion phenomenon that a Markovian particle moves both by jumps and continuously in the state space D (see Figure 11.2).

D

Fig. 11.2.


171

Let L be a second-order boundary condition such that, in local coordinates (x1 , x2 , . . . , xN −1 ), ∂u Lu(x ) = Qu(x ) + μ(x ) (x ) − δ(x )W u(x ) + Γ u(x ) ∂n ⎛ ⎞ N −1 N −1 2 ∂ u ∂u := ⎝ αij (x ) (x ) + β i (x ) (x ) + γ(x )u(x )⎠ ∂x ∂x ∂x i j i i,j=1 i=1 ∂u + μ(x ) (x ) − δ(x )W u(x ) ∂n ⎛ ⎡ ⎤ N −1 ∂u +⎝ r(x , y ) ⎣u(y ) − u(x ) − (yj − xj ) (x )⎦ dy ∂x j ∂D j=1 ⎡ ⎤ ⎞ N −1 ∂u + t(x , y) ⎣u(y) − u(x ) − (yj − xj ) (x )⎦ dy ⎠ . ∂xj D j=1 Here: (1) The operator Q is a second-order, degenerate elliptic differential operator on the boundary ∂D with non-positive principal symbol. In other words, the αij are the components of a smooth symmetric contravariant tensor of type 20 on ∂D satisfying the condition N −1

αij (x )ξi ξj ≥ 0 for all x ∈ ∂D and ξ =

&N −1 j=1

ξj dxj ∈ Tx∗ (∂D).

i,j=1

(2) (3) (4) (5) (6)

(7)

Here Tx∗ (∂D) is the cotangent space of ∂D at x . Q1 = γ ∈ C ∞ (∂D) and γ(x ) ≤ 0 on ∂D. μ ∈ C ∞ (∂D) and μ(x ) ≥ 0 on ∂D. δ ∈ C ∞ (∂D) and δ(x ) ≥ 0 on ∂D. n = (n1 , n2 , . . . , nN ) is the unit interior normal to the boundary ∂D. The integral kernel r(x , y ) is the distribution kernel of a pseudo-differen 1 tial operator R ∈ L2−κ 1,0 (∂D), κ1 > 0, and r(x , y ) ≥ 0 off the diagonal Δ∂D = {(x , x ) : x ∈ ∂D} in ∂D × ∂D. The density dy is a strictly positive density on ∂D. The integral kernel t(x, y) is the distribution kernel of a properly supN 2 ported, pseudo-differential operator T ∈ L2−κ 1,0 (R ), κ2 > 0, which has the transmission property with respect to the boundary ∂D (see [Bt]), and t(x, y) ≥ 0 off the diagonal {(x, x) : x ∈ RN } in RN × RN .

The boundary condition L is called a second-order Ventcel’ boundary condition (cf. [We]). The six terms of L N −1 i,j=1

αij (x )

N −1 ∂2u ∂u (x ) + β i (x ) (x ), ∂xi ∂xj ∂x i i=1

172


γ(x )u(x ),

∂u μ(x ) (x ), ∂n ⎡

δ(x )W u(x ),

⎤

N −1

∂u ⎦ (x ) dy , ∂x j j=1 ⎡ ⎤ N −1 ∂u t(x , y) ⎣u(y) − u(x ) − (yj − xj ) (x )⎦ dy ∂x j D j=1 r(x , y ) ⎣u(y ) − u(x ) −

(yj − xj )

∂D

are supposed to correspond to the diffusion along the boundary, the absorption phenomenon, the reflection phenomenon, the sticking (viscosity) phenomenon and the jump phenomenon on the boundary and the inward jump phenomenon from the boundary, respectively (see Figures 11.3 through 11.5). D

D

@D

@D

absorption

reflection Fig. 11.3.

D

D

@D

@D

diffusion along the boundary

sticking (viscosity)

Fig. 11.4.

Finally, we give an overview for general results on generation theorems for Feller semigroups proved mainly by the author using the theory of pseudodifferential operators ([Ho1], [Se1], [Se2]) and the theory of singular integral operators ([CZ]):


D

∂D

173

D

∂D

jump into the interior

jump on the boundary Fig. 11.5.

diffusion operator A

Lévy operator Sr

Ventcel’ condition L

using the theory of

proved by

smooth case

null

second-order case

pseudodifferential operators

[Ta2]

smooth case

general case

general case


[Ta3]

smooth case

Hölder continuous case

degenerate case


[Ta5]

discontinuous case

general case

Dirichlet case

singular integral operators

[Ta6]

discontinuous case

null

first-order case

singular integral operators

[Ta7]

174


It should be emphasized that the Calder´ on–Zygmund theory of singular integral operators with non-smooth kernels provides a powerful tool to deal with smoothness of solutions of elliptic boundary value problems, with minimal assumptions of regularity on the coefficients. The theory of singular integrals continues to be one of the most influential works in modern history of analysis, and is a very refined mathematical tool whose full power is yet to be exploited (see [St2]).

Boundary Value Problems and Markov Processes

Boundary Value Problems and Markov Processes

Polyharmonic boundary value problems

Mixed boundary value problems

Polyharmonic boundary value problems