Research Article | Open Access
Baiyun Su, "Dirichlet Problem for the Schrödinger Operator in a Half Space", Abstract and Applied Analysis, vol. 2012, Article ID 578197, 14 pages, 2012. https://doi.org/10.1155/2012/578197
Dirichlet Problem for the Schrödinger Operator in a Half Space
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 .
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.
Remark 1.1. in the case .
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
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
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.
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
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 .
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.
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
Lemma 2.3. Let and , and let be any positive measure on having finite total mass. Then, has a covering satisfying
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 ), there exists , which is at most countable, such that are disjoint and .
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 )
For any fixed point satisfying , letting , , , , and , we write where
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
To estimate , we write where
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
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 .
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 .
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