Research Article | Open Access
The Modification of Kernel Function and Its Applications
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 .
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 ).
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
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 ).
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 .
Lemma 6. (1) If , then .
(2) If , then .
(3) If , then .
(4) If and , then .
The following Lemma is due to Qiao (see ).
Lemma 7. If , , and is a positive measure in satisfying , then has a covering such that
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 .
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 , then
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
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
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
Sincefor any , we havefrom Theorem 4.
Moreover, (18) gives that
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work was supported by the National Natural Science Foundation of China under Grant nos. 11301140 and U1304102.
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Copyright © 2014 Tao Zhao. 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.