Abstract and Applied Analysis

Volume 2012 (2012), Article ID 203096, 5 pages

doi:10.1155/2012/203096

Research Article

The Dirichlet Problem on the Upper Half-Space

Jinjin Huang¹ and Lei Qiao²

¹Department of Economics and Management, Zhoukou Normal University, Zhoukou 466001, China

²Department of Mathematics and Information Science, Henan University of Economics and Law, Zhengzhou 450002, China

Received 18 April 2012; Accepted 19 September 2012

Academic Editor: Stefan Siegmund

Copyright © 2012 Jinjin Huang and Lei Qiao. 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.

A solution of the Dirichlet problem on the upper half-space is constructed by the generalized Dirichlet integral with a fast-growing continuous boundary function.

1. Introduction and Results

Let denote the -dimensional Euclidean space with points , where and . The boundary and closure of an open set of are denoted by and , respectively. The upper half space is the set , whose boundary is . We identify with and with , writing typical points as , where and putting

Let denote the open ball with center at the origin and radius , and let denote -dimensional surface area measure. Let denote the integer part of the positive real number . In the sense of Lebesgue measure, and .

Given a continuous function on , we say that is a solution of the (classical) Dirichlet problem on with if in and for every .

The classical Poisson kernel for is defined by , where is the area of the unit sphere in .

To solve the Dirichlet problem on , as in [1–6], we use the following modified Poisson kernel of order defined by

where is a nonnegative integer, and is the ultraspherical (Gegenbauer) polynomials [7]. The expression arises from the generating function for Gegenbauer polynomials

where and . The coefficient is called the ultraspherical (Gegenbauer) polynomial of degree associated with , and the function is a polynomial of degree in .

Put

where is a continuous function on .

Using the modified Poisson kernel , Yoshida (cf. [6, Theorem 1]) and Siegel and Talvila (cf. [5, Corollary 2.1]) gave classical solutions of the Dirichlet problem on , respectively. Motivated by their results, we consider the Dirichlet problem for harmonic functions of infinite order (e.g., see [8, Definition 4.1, page 2, Line 12] for the definition of harmonic functions).

To do this, we define a nondecreasing and continuously differentiable function on the interval . We assume further that

Let be the set of continuous functions on such that

where is a positive real number.

Now, we have the following.

Theorem 1.1. If , then the integral is a solution of the Dirichlet problem on with .

If one puts in Theorem 1.1, one immediately obtains the following (cf. [6, Theorem 1] and [5, Corollary 2.1]).

Corollary 1.2. If is a continuous function on satisfying , then is a solution of the Dirichlet problem on with .

Theorem 1.3. Let be harmonic in and continuous on . If , then one has

for all , where is harmonic in and vanishes continuously on .

2. Proof of Theorem 1.1

We need to use the following inequality (see [5, page 3]):

for any and satisfying , where is a positive constant.

For any , there exists a sufficiently large positive number such that , and by (1.5), we have

which yields that there exists a positive constant dependent only on such that

for any .

For any and , we have by (1.6), (2.1), (2.3), , and Hölder’s inequality

Thus, is finite for any . Since is a harmonic function of for any fixed , is also a harmonic function of .

To verify the boundary behavior of , we fix a boundary point , choose a large , and write

where

Notice that is the Poisson integral of , where is the characteristic function of the ball . So it tends to as . Since are polynomial times and , both of them tend to zero as . Thus, the function can be continuously extended to such that , for any . Theorem 1.1 is proved.

3. Proof of Theorem 1.3

Consider that the function , which is harmonic in , can be continuously extended to and vanishes on .

The Schwarz reflection principle [9, page 68] applied to shows that there exists a harmonic function in such that for , where denotes reflection in just as .

Thus, for all , where is a harmonic function on vanishing continuously on . We complete the proof of Theorem 1.3.

Acknowledgments

The authors wish to express their appreciation to Professor Guantie Deng for some very useful conversations related to this problem. They are grateful to the referee for her or his careful reading and helpful suggestions which led to an improvement of their original manuscript. This work is supported by The National Natural Science Foundation of China under Grant 11271045 and Specialized Research Fund for the Doctoral Program of Higher Education under Grant 20100003110004.

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