Growth estimates for modified Neumann integrals in a half-space
© Ren and Yang; licensee Springer. 2013
Received: 13 August 2013
Accepted: 23 October 2013
Published: 4 December 2013
Our aim in this paper is to deal with the growth properties for modified Neumann integrals in a half-space of . As an application, the solutions of Neumann problems in it for a slowly growing continuous function are also given.
KeywordsDirichlet problem harmonic function half-space
1 Introduction and main results
Let R and be the sets of all real numbers and of all positive real numbers, respectively. Let () denote the n-dimensional Euclidean space with points , where and . The boundary and closure of an open set Ω of are denoted by ∂ Ω and , respectively. For and , let denote the open ball with center at x and radius r in .
The upper half-space is the set , whose boundary is ∂H. For a set F, , we denote and by HF and , respectively. We identify with and with , writing typical points as , , where . Let θ be the angle between x and , i.e., and , where is the i th unit coordinate vector and is normal to ∂H.
We shall say that a set has a covering if there exists a sequence of balls with centers in H 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 M. Throughout this paper, let M denote various constants independent of the variables in question. Further, we use the standard notations, is the integer part of d and , where d is a positive real number.
for every point .
where is the ultraspherical (Gegenbauer) polynomials . The series converges for , and each term in it is a harmonic function of x.
where f is a continuous function on ∂H, and is the volume of the unit n-ball.
for and .
If f is a measurable function on ∂H satisfying (1.1), we remark that the total mass of μ is finite.
The set is denoted by .
for a non-negative integer m.
where f is a continuous function on ∂H. Here, note that is nothing but the Neumann integral .
The next result deals with a type of uniqueness of solutions for the Neumann problem on H (see [, Theorem 3]).
and is a polynomial of of degree less than .
Our first aim is to be concerned with the growth property of at infinity and establish the following theorem.
As an application of Theorem 1, we now show the solution of the Neumann problem with continuous data on H.
Theorem 2 Let p, β, α and m be defined as in Theorem 1. If f is a continuous function on ∂H satisfying (1.1), then the function is a solution of the Neumann problem on H with f and (1.4) holds, where the exceptional set (⊂H) has a covering satisfying (1.3).
Remark In the case , and , then (1.3) is a finite sum and the set is a bounded set. So (1.4) holds in H. That is to say, (1.2) holds. This is just the result of Theorem A.
If f is a continuous function on ∂H satisfying (1.1), then the function is a solution of the Neumann problem on H with f and (1.5) holds.
The following result extends Theorem B, which is our result in the case and .
for any and is a polynomial of of degree less than .
If , then .
If , then .
If , then .
If and , then .
The following lemma is due to Qiao (see ).
Lemma 3 ([, Lemma 4])
for any fixed .
Lemma 4 ([, Lemma 1])
3 Proof of Theorem 1
and divide H into two sets and .
which is similar to the estimate of .
where is a positive integer satisfying .
Combining (3.2), (3.4)-(3.7), we obtain that if is sufficiently large and ϵ is a sufficiently small number, then as , where . Finally, there exists an additional finite ball covering , which together with Lemma 2, gives the conclusion of Theorem 1.
4 Proof of Theorem 2
Hence is absolutely convergent and finite for any . Thus is harmonic on H.
for any point , we only need to apply Lemma 3 to and .
We complete the proof of Theorem 2.
5 Proof of Theorem 3
Consider the function . Then it follows from Theorems 2 and 3 that is a solution of the Neumann problem on H with f and it is an even function of (see [, p.92]).
from Theorem 2.
This implies that is a polynomial of degree less than (see [, Appendix]), which gives the conclusion of Theorem 3 from Lemma 4.
The authors are thankful to the referees for their helpful suggestions and necessary corrections in the completion of this paper.
- Szegö G American Mathematical Society: Colloquium Publications 23. In Orthogonal Polynomials. Am. Math. Soc., Providence; 1975.Google Scholar
- Armitage DH:The Neumann problem for a function in . Arch. Ration. Mech. Anal. 1976, 63: 89–105. 10.1007/BF00280145MATHMathSciNetView ArticleGoogle Scholar
- Huang JJ, Qiao L: The Dirichlet problem on the upper half-space. Abstr. Appl. Anal. 2012., 2012: Article ID 203096Google Scholar
- Qiao L: Modified Poisson integral and Green potential on a half-space. Abstr. Appl. Anal. 2012., 2012: Article ID 765965Google Scholar
- Siegel D, Talvila E: Sharp growth estimates for modified Poisson integrals in a half space. Potential Anal. 2001, 15(4):333–360. 10.1023/A:1011817130061MATHMathSciNetView ArticleGoogle Scholar
- Qiao L, Deng GT: Growth estimates for modified Green potentials in the upper-half space. Bull. Sci. Math. 2011, 135: 279–290. 10.1016/j.bulsci.2010.08.002MATHMathSciNetView ArticleGoogle Scholar
- Armitage DH: On harmonic polynomials. Proc. Lond. Math. Soc. 1979, 38: 53–71.MATHMathSciNetView ArticleGoogle Scholar
- Su BY: Dirichlet problem for the Schrödinger operator in a half space. Abstr. Appl. Anal. 2012., 2012: Article ID 578197Google Scholar
- Su BY: Growth properties of harmonic functions in the upper half space. Acta Math. Sin. 2012, 55(6):1095–1100. (in Chinese)MATHGoogle Scholar
- Hayman WK, Kennedy PB 1. In Subharmonic Functions. Academic Press, London; 1976.Google Scholar
- Brelot M: Éléments de la théorie classique du potential. Centre de Documentation Universitaire, Paris; 1965:53–71.Google Scholar
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.