Research Article The Dirichlet Problem on the Upper Half-Space

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1 Abstract and Applied Analysis Volume 2012, Article ID , 5 pages doi: /2012/ Research Article The Dirichlet Problem on the Upper Half-Space Jinjin Huang 1 and Lei Qiao 2 1 Department of Economics and Management, Zhouou Normal University, Zhouou , China 2 Department of Mathematics and Information Science, Henan University of Economics and Law, Zhengzhou , China Correspondence should be addressed to Lei Qiao, qiaocqu@163.com Received 18 April 2012; Accepted 19 September 2012 Academic Editor: Stefan Siegmund Copyright q 2012 J. Huang and L. 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 wor is properly cited. 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 R n n 3 denote the n-dimensional Euclidean space with points x x,x n, where x x 1,x 2,...,x n 1 R n 1 and x n R. The boundary and closure of an open set D of R n are denoted by D and D, respectively. The upper half space is the set H { x,x n R n : x n > 0}, whose boundary is H. We identify R n with R n 1 R and R n 1 with R n 1 {0}, writing typical points x, y R n as x x,x n,y y,y n, where y y 1,y 2,...,y n 1 R n 1 and putting n x y x j y j x y x n y n, x x x, j 1 x x x. 1.1 Let B r denote the open ball with center at the origin and radius r, andletσ denote n 1 -dimensional surface area measure. Let d denote the integer part of the positive real number d. In the sense of Lebesgue measure, dy dy 1 dy n 1 and dy dy dy n. Given a continuous function f on H, we say that h is a solution of the classical Dirichlet problem on H with f if Δh 0inH and lim x H,x z h x f z for every z H. The classical Poisson ernel for H is defined by P x, y 2x n ωn 1 x y n, where ω n 2π n/2 /Γ n/2 is the area of the unit sphere in R n.

2 2 Abstract and Applied Analysis To solve the Dirichlet problem on H, asin 1 6, we use the following modified Poisson ernel of order m defined by P m x, y ) P x, y ) when y 1, P ) x, y ) m 1 2x n x x y 0 ω Cn/2 n y n x y when y > 1, 1.2 where m is a nonnegative integer, and C n/2 t is the ultraspherical Gegenbauer polynomials 7. The expression arises from the generating function for Gegenbauer polynomials 1 2tr r 2) n/2 0 C n/2 t r, 1.3 where r < 1and t 1. The coefficient C n/2 t is called the ultraspherical Gegenbauer polynomial of degree associated with n/2, and the function C n/2 t is a polynomial of degree in t. Put ) U m f x P m x, y ) f y ) dy, 1.4 H where f y is a continuous function on H. Using the modified Poisson ernel P m x, y,yoshida cf. 6, Theorem 1 and Siegel and Talvila cf. 5, Corollary 2.1 gave classical solutions of the Dirichlet problem on H, 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 ρ r 1 on the interval 0,. We assume further that ρ r r log r ε 0 lim sup r ρ r Let F ρ, β be the set of continuous functions f on H such that ) f y dy H where β is a positive real number. Now, we have the following. < y ρ y n β 1 <, 1.6 Theorem 1.1. If f F ρ, β, then the integral U ρ y β f x is a solution of the Dirichlet problem on H with f. If one puts ρ y β m in Theorem 1.1, one immediately obtains the following cf. [6, Theorem 1] and [5, Corollary 2.1]).

3 Abstract and Applied Analysis 3 Corollary 1.2. If f is a continuous function on H satisfying H f y 1 y n m dy <,then U m f x is a solution of the Dirichlet problem on H with f. Theorem 1.3. Let u be harmonic in H and continuous on H. Ifu F ρ, β, then one has u x U ρ y β u x h x, 1.7 for all x H, whereh x is harmonic in H and vanishes continuously on H. 2. Proof of Theorem 1.1 We need to use the following inequality see 5, page 3 : Pm x, y ) Mxn x m y n m, 2.1 for any x H and y H satisfying y max{1, 2 x }, where M is a positive constant. For any ɛ 0 <ɛ<1 ɛ 0, there exists a sufficiently large positive number R such that r>r,andby 1.5, we have ρ r <ρ e ln r ɛ 0 ɛ, 2.2 which yields that there exists a positive constant M r dependent only on r such that β/2 2r ρ 1 β 1 M r, 2.3 for any > r 2r 1. For any x H and x r, we have by 1.6, 2.1, 2.3, 1/p 1/q 1, and Hölder s inequality M r M {y H: y < 1} r 2r ρ 1 β 1 2r M r 2MM r {y H: y < 1} ρ 1 β 1 β/2 2 x ρ y β 1 y f y ) dy ρ y β n {y H: y < 1} ) f y p y ρ y n pβ/2 1 dy ) 1/p ) 1/q y q{ ρ y β n ρ y n 1 /p β/2} dy {y H: y r } {y H: y < 1} f y ) ) f y y ρ y n β/2 1 dy 1 y ρ y n β/2 1 dy <. 2.4

4 4 Abstract and Applied Analysis Thus, U ρ y β f x is finite for any x H. Since P ρ y β x, y is a harmonic function of x H for any fixed y H, U ρ y β f x is also a harmonic function of x H. To verify the boundary behavior of U ρ y β f x, we fix a boundary point z H, choose a large t> z 1, and write U ρ y β f ) x X x Y x Z x, 2.5 where Y x ρ y β 1 0 X x 2x n x ω n Z x {y H: y t} {y H:1< y t} {y H: y >t} P x, y ) f y ) dy, ) 1 x y Cn/2 y n x f y y ) dy, P ρ y β x, y ) f y ) dy. 2.6 Notice that X x is the Poisson integral of f y χ B t y, where χ B t is the characteristic function of the ball B t.soittendstof z as x z. Since Y x are polynomial times x n and Z x O x n, both of them tend to zero as x z. Thus, the function U ρ y β f x can be continuously extended to H such that U ρ y β f z f z, for any z H. Theorem 1.1 is proved. 3. Proof of Theorem 1.3 Consider that the function u x U ρ y β u x, which is harmonic in H, can be continuously extended to H and vanishes on H. The Schwarz reflection principle 9, page 68 applied to u x U ρ y β u x shows that there exists a harmonic function h x in H such that h x h x u x U ρ y β u x for x H, where denotes reflection in H just as x x, x n. Thus, u x h x U ρ y β u x for all x H, where h x is a harmonic function on H vanishing continuously on H. We complete the proof of Theorem 1.3. Acnowledgments 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 wor is supported by The National Natural Science Foundation of China under Grant and Specialized Research Fund for the Doctoral Program of Higher Education under Grant

5 Abstract and Applied Analysis 5 References 1 D. H. Armitage, Representations of harmonic functions in half-spaces, Proceedings of the London Mathematical Society, vol. 38, no. 3, pp , M. Finelstein and S. Scheinberg, Kernels for solving problems of Dirichlet type in a half-plane, Advances in Mathematics, vol. 18, no. 1, pp , S. J. Gardiner, The Dirichlet and Neumann problems for harmonic functions in half-spaces, The the London Mathematical Society, vol. 24, no. 3, pp , D. Siegel and E. O. Talvila, Uniqueness for the n-dimensional half space Dirichlet problem, Pacific Mathematics, vol. 175, no. 2, pp , D. Siegel and E. Talvila, Sharp growth estimates for modified Poisson integrals in a half space, Potential Analysis, vol. 15, no. 4, pp , H. Yoshida, A type of uniqueness for the Dirichlet problem on a half-space with continuous data, Pacific Mathematics, vol. 172, no. 2, pp , G. Szegő, Orthogonal Polynomials, vol. 23 of American Mathematical Society Colloquium Publications, American Mathematical Society, Providence, RI, USA, W. K. Hayman and P. B. Kennedy, Subharmonic Functions, Academic Press, London, UK, 1 edition, S. Axler, P. Bourdon, and W. Ramey, Harmonic Function Theory, Springer, New Yor, NY, USA, 2nd edition, 1992.

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