Abstract and Applied Analysis

Volume 2012 (2012), Article ID 578197, 14 pages

doi:10.1155/2012/578197

Research Article

Dirichlet Problem for the Schrödinger Operator in a Half Space

Baiyun Su

Department of Mathematics and Information Science, Henan University of Economics and Law, Zhengzhou 450002, China

Received 22 April 2012; Accepted 12 July 2012

Academic Editor: Jean Pierre Gossez

Copyright © 2012 Baiyun Su. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

For continuous boundary data, the modified Poisson integral is used to write solutions to the half space Dirichlet problem for the Schrödinger operator. Meanwhile, a solution of the Poisson integral for any continuous boundary function is also given explicitly by the Poisson integral with the generalized Poisson kernel depending on this boundary function.

1. Introduction and Results

Let and be the sets of all real numbers and of all positive real numbers, respectively. Let denote the -dimensional Euclidean space with points , where and . The unit sphere and the upper half unit sphere in are denoted by and , respectively. The boundary and closure of an open set of are denoted by and , respectively. The upper half space is the set , whose boundary is .

For a set , , we denote and by and , respectively. We identify with and with , writing typical points as , where , and putting

For and , let denote the open ball with center at and radius in . We will say that a set has a covering if there exists a sequence of balls with centers in such that , where is the radius of and is the distance between the origin and the center of .

Let denote the class of nonnegative radial potentials , that is, , , such that with some if and with if or .

This paper is devoted to the stationary Schrödinger equation

where , is the Laplace operator and . These solutions are called -harmonic functions or generalized harmonic functions associated with the operator . Note that they are (classical) harmonic functions in the case . Under these assumptions the operator can be extended in the usual way from the space to an essentially self-adjoint operator on (see [1–3]). We will denote it by as well. This last one has a Green function . Here, is positive on and its inner normal derivative . We denote this derivative by , which is called the Poisson -kernel with respect to . We remark that and are the Green function and Poisson kernel of the Laplacian in , respectively.

Let be a Laplace-Beltrami operator (spherical part of the Laplace) on the unit sphere. It is known (see, e.g., [4, page 41]) that the eigenvalue problem

has the eigenvalues . Corresponding eigenfunctions are denoted by , where is the multiplicity of . We norm the eigenfunctions in and .

Hence, well-known estimates (see, e.g., [5, page 14]) imply the following inequality:

where the symbol denotes a constant depending only on .

Let and stand, respectively, for the increasing and nonincreasing, as , solutions of the equation

normalized under the condition .

We will also consider the class , consisting of the potentials such that there exists a finite limit . Moreover, . If , then solutions of (1.2) are continuous (see [6]).

In the rest of paper, we assume that , and we will suppress this assumption for simplicity. Further, we use the standard notations , , is the integer part of and , where is a positive real number.

Denote

Remark 1.1. in the case .

It is known (see [7]) that in the case under consideration the solutions to (1.5) have the asymptotics

where and are some positive constants.

If , it is known that the following expansion for the Green function (see [8, Chapter 11], [1, 9])

where and is its Wronskian. The series converges uniformly if either or .

For a nonnegative integer and two points , we put

where

We introduce another function of

The generalized Poisson kernel with respect to is defined by

In fact

We remark that the kernel function coincides with ones in Finkelstein and Scheinberg [10] and Siegel and Talvila [11] (see [8, Chapter 11]).

Put

where is a continuous function on .

If is a real number and , and

If these conditions all hold, we write .

Let and be functions on satisfying

For and , we define the positive measure (resp., ) on by

We remark that the total mass of and is finite.

Let and , and let be any positive measure on having finite mass. For each , the maximal function is defined by

The set is denoted by .

About classical solutions of the Dirichlet problem for the Laplacian, Siegel and Talvila (cf. [11, Corollary 2.1]) proved the following result.

Theorem A. If is a continuous function on satisfying

then, the function satisfies

Our first aim is to give the growth properties at infinity for .

Theorem 1.2. If , and is a measurable function on satisfying (1.16), then there exists a covering of satisfying

such that

If is a measurable function on satisfying

where is a real number, for this and , we define

Obviously, the total mass of is also finite.

If we take in Theorem 1.2, then we immediately have the following growth property based on (1.5) and Remark 1.1.

Corollary 1.3. Let , and . If is defined as previously, then the function is a harmonic function on and there exists a covering of satisfying

such that

Remark 1.4. In the case , (1.26) is a finite sum, and the set is a bounded set and (1.27) holds in .

Next we are concerned with solutions of the Dirichlet problem for the Schrödinger operator on . For related results, we refer the readers to the paper by Kheyfits [1].

Theorem 1.5. If and is a continuous function on satisfying (1.16), then

If we take , then we immediately have the following corollary, which is just Theorem A in the case .

Corollary 1.6. If is a continuous function on satisfying

then (1.28) hold and

As an application of Corollary 1.6, we can give a solution of the Dirichlet problem for any continuous function on .

Theorem 1.7. If is a continuous function on satisfying (1.31) and is a solution of the Dirichlet problem for the Schrödinger operator on with satisfying

then

where and are constants.

2. Lemmas

Throughout this paper, let denote various constants independent of the variables in questions, which may be different from line to line.

Lemma 2.1. If , then

If and , then

If (, then

Proof. Equations (2.1) and (2.2) are obtained by Kheyfits (see [8, Chapter 11] or [1, Lemma 1]). Equation (2.3) follows from Hayman and Kennedy (see [12, Lemma 4.2]).

Lemma 2.2 (see [2, Theorem 1]). If is a solution of (1.2) on satisfying

then

Lemma 2.3. Let and , and let be any positive measure on having finite total mass. Then, has a covering satisfying

Proof. Set

If , then there exists a positive number such that

Here, can be covered by the union of a family of balls . By the Vitali lemma (see [13]), there exists , which is at most countable, such that are disjoint and .

So

On the other hand, note that , so that

Hence, we obtain

Since , then is finally covered by a sequence of balls satisfying

where is the ball that covers .

3. Proof of Theorem 1.2

We only prove the case , the remaining case can be proved similarly.

For any , there exists such that

The relation implies this inequality (see [14])

For any fixed point satisfying , letting , , , , and , we write

where

By , (1.16), (2.1), and (3.1), we have the following growth estimates

Next, we will estimate .

Take a sufficiently small positive number such that for any , where

and divide into two sets and .

If , then there exists a positive such that for any , and hence

We will consider the case . Now put

where .

Since , we have

where is a positive integer satisfying .

Since , we obtain

for .

Since , we have

So

By , (1.7), (2.2), and (3.1), we have

We only consider in the case , since for . By the definition of , (1.4), and (2.2), we see that

where

To estimate , we write

where

Notice that

Thus, by , (1.7), and (1.16), we conclude

Analogous to the estimate of , we have

Thus, we can conclude that

which yields

Combining (3.5)–(3.22), we obtain that if is sufficiently large and is sufficiently small, then as , where . Finally, there exists an additional finite ball covering , which together with Lemma 2.3 gives the conclusion of Theorem 1.2.

4. Proof of Theorem 1.5

For any fixed , take a number satisfying . By , (1.5), (1.16), and (2.2), we have

Then, is absolutely convergent and finite for any . Thus is a solution of (1.2) on .

Now we study the boundary behavior of . Let be any fixed point and any positive number satisfying .

Set as the characteristic function of , and write

where

Notice that is the Poisson -integral of , We have . Since as , we have from the definition of the kernel function . and therefore tends to zero.

So the function can be continuously extended to such that

for any from the arbitrariness of .

Finally, (1.29) and (1.30) follow from (1.22) and (1.23), respectively, in the case . Thus, we complete the proof of Theorem 1.5.

5. Proof of Theorem 1.7

From Corollary 1.6, we have the solution of the Dirichlet problem on with satisfying (1.31). Consider the function . Then, it follows that this is a solution of (1.2) in and vanishes continuously on .

Since

for any , we have

from (1.32) and (1.33). Then, the conclusion of Theorem 1.7 follows immediately from Lemma 2.2.

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