Dirichlet problems of harmonic functions
© Xu et al.; licensee Springer. 2013
Received: 6 September 2013
Accepted: 5 November 2013
Published: 2 December 2013
In this paper, a solution of the Dirichlet problem in the upper half-plane isconstructed by the generalized Dirichlet integral with a fast growing continuousboundary function.
MSC: 31B05, 31B10.
1 Introduction and results
Let R be the set of all real numbers, and let C denote the complexplane with points , where . The boundary and closure of an open set Ω aredenoted by ∂ Ω and respectively. The upper half-plane is the set, whose boundary is . Let denote the integer part of the positive real numberd.
Given a continuous function f in , we say that h is a solution of the(classical) Dirichlet problem in with f, if in and for every .
where and .
where m is a nonnegative integer.
We remark that the modified Poisson kernel is harmonic in .
where is a continuous function in .
If , then u is said to be of finite order. SeeHayman-Kennedy [, Definition 4.1].
where α is a positive real number.
Now we show the solution of the Dirichlet problem with continuous data in. For similar results in a cone, we refer readers tothe paper by Qiao (see [7, 8]). For similar results with respect to the Schrödinger operator in ahalf-space, we refer readers to the paper by Ren, Su and Yang (see [9–12]).
Theorem 1 If, then the integralis a solution of the Dirichlet problem inwith f.
The following result is obtained by putting in Theorem 1.
thenis a solution of the Dirichlet problem inwith f.
Theorem 2 Let u be a real-valued function harmonic inand continuous in. If, then we havefor all, whereis an entire function inand vanishes continuously in.
2 Proof of Theorem 1
for any and satisfying , where M is a positive constant.
for any .
Thus is finite for any . Since is a harmonic function of for any fixed , is also a harmonic function of.
Note that is the Poisson integral of , where is the characteristic function of the interval. So it tends to as . Clearly, vanishes on . Further, , which tends to zero as . Thus the function can be continuously extended to such that for any . Theorem 1 is proved.
3 Proof of Theorem 2
Consider that the function , which is harmonic in , can be continuously extended to and vanishes in .
The Schwarz reflection principle [, p.68] applied to shows that there exists an entire harmonic function in satisfying such that for .
Thus for all , where is an entire function in and vanishes continuously in . Then we complete the proof of Theorem 2.
The authors are thankful to the referees for their helpful suggestions andnecessary corrections in the completion of this paper.
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