The Modification of Kernel Function and Its Applications

Abstract

By virtue of the modified Riesz kernel introduced by Qiao (2012), we give the integral representations for solutions of the Neumann problems in a half space.

1. Introduction and Main 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 boundary and closure of an open set of are denoted by and , respectively. For and , let denote the open ball with center at and radius in . Let .

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 . Let be the angle between and , that is, and , where is the th unit coordinate vector and is normal to .

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 .

For positive functions and , we say that if for some positive constant . Throughout this paper, let denote various constants independent of the variables in question. Further, we use the standard notations , is the integer part of , and , where is a positive real number.

Given a continuous function on , we say that is a solution of the Neumann problem on with , if is a harmonic function on and for every point .

For and , consider the kernel function where and is the surface area of the -dimensional unit sphere.

The Neumann integral on is defined by where is a continuous function on .

The Neumann integral is a solution of the Neumann problem on with if (see [1, Theorem 1 and Remarks])

In this paper, we consider functions satisfying for and .

For and , we define the positive measure on by If is a measurable function on satisfying (5), we remark that the total mass of is finite.

Let and . For each , the maximal function is defined by

The set is denoted by .

To obtain the Neumann solution for the boundary data on , as in [2, 3], we use the following modified Riesz kernel defined by where is a nonnegative integer.

For and , the generalized Neumann kernel is defined by

Put where is continuous function on . Here note that is nothing but the Neumann integral .

The following result is due to Su (see [4]).

Theorem A. If is a continuous function on satisfying (5) with and , then

Our first aim is to be concerned with the growth property of at infinity in a half space and establish the following theorem.

Theorem 1. Let , , and If is a measurable function on satisfying (5), then there exists a covering of satisfying such that

Remark 2. In the case that , , and , then (13) is a finite sum and the set is a bounded set. So (14) holds in . That is to say, (11) holds. This is just the result of Theorem A.

Corollary 3. Let , and If is a measurable function on satisfying (5), then

As an application of Theorem 1, we now show the solution of the Neumann problem with continuous data on . About the solutions of the Dirichlet problem with respect to the Schrödinger operator in a half space, we refer readers to the paper by Su (see [5]).

Theorem 4. Let , , , and be defined as in Theorem 1. If is a continuous function on satisfying (5), then the function is a solution of the Neumann problem on with and (14) holds, where the exceptional set has a covering satisfying (13).

Finally we have the following result.

Theorem 5. Let , , be a positive integer and If is a continuous function on satisfying (5) and is a solution of the Neumann problem on with such that then for any , where and is a polynomial of of degree less than .

2. Lemmas

In our discussions, the following estimates for the kernel function are fundamental (see [6, Lemma 4.2] and [3, Lemmas 2.1 and 2.4]).

Lemma 6. (1) If , then .
(2) If , then .
(3) If , then .
(4) If and , then .

The following Lemma is due to Qiao (see [3]).

Lemma 7. If , , and is a positive measure in satisfying , then has a covering    such that

Lemma 8. Let , , , and be defined as in Theorem 1. If is a local integral and upper semicontinuous function on satisfying (5), then for any fixed point .

Proof. Let be any fixed point an and let be any positive number. Take a positive number , , such that for any .
By Lemma 6(4) and (5), we can choose a number , , such that for any .
Put
Since for any and , we have for any .
Since for any , we observe that
Finally (23), (27), and (29) yield
From Lemma 6(4) we obtain for any .
These and (24) yield

Now the conclusion immediately follows.

Lemma 9 (see [1, Lemma 1]). If is a harmonic polynomial of of degree and vanishes on , then there exists a polynomial of degree such that

3. Proof of Theorem 1

We prove only the case ; the proof of the case is similar.

For any , there exists such that

Take any point such that and write where

First note that so that

If and , then . By Lemma 6(1), (34), and Hölder inequality, we have

Put where

If , then

Moreover, by (34) and (39) we get

That is,

By Lemma 6(3), (34), and Hölder inequality, we have

If , then . We obtain Lemma 6(4), (34), and Hölder inequality:

Finally, 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 number such that for any , and hence which is similar to the estimate of .

We will consider the case . Now put where .

Since , we have where is a positive integer satisfying .

Similar to the estimate of we obtain for .

Since , we have for and

So

Combining (38) and (44)–(54), we obtain that if is sufficiently large and is a sufficiently small number, then as , where . Finally, there exists an additional finite ball covering , which, together with Lemma 7, gives the conclusion of Theorem 1.

4. Proof of Theorem 4

For any fixed , take a number satisfying . If , then . By (5), Lemma 6(4), and Hölder inequality, we have Hence, is absolutely convergent and finite for any . Thus, is harmonic on .

To prove for any point , we only need to apply Lemma 8 to and .

We complete the proof of Theorem 4.

5. Proof of Theorem 5

Consider the function . Then it follows from Theorems 4 and 5 that is a solution of the Neumann problem on with and it is an even function of (see [1, page 92]).

Since for any , we have from Theorem 4.

Moreover, (18) gives that

This implies that is a polynomial of degree less than (see [7, Appendix]), which gives the conclusion of Theorem 5 from Lemma 9.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This work was supported by the National Natural Science Foundation of China under Grant nos. 11301140 and U1304102.