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Dirichlet problems of harmonic functions
Boundary Value Problemsvolume 2013, Article number: 262 (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 .
The classical Poisson kernel in is defined by
where and .
which converges for . We define a modified Cauchy kernel of by
where m is a nonnegative integer.
To solve the Dirichlet problem in , as in , we use the following modified Poisson kernel defined by
We remark that the modified Poisson kernel is harmonic in .
where is a continuous function in .
We say that u is of order λ if
If , then u is said to be of finite order. SeeHayman-Kennedy [, Definition 4.1].
Theorem A Let u be a nonnegative real-valued function harmonic inand continuous in. If
then there exists a nonnegative real constant d such that
Inspired by Theorem A, we consider the Dirichlet problem for harmonic functions ofinfinite order in . To do this, we define a nondecreasing andcontinuously differentiable function on the interval . We assume further that
Remark For any ϵ (), there exists a sufficiently large positive numberR such that , by (1.1) we have
Let be the set of continuous functions f on such that
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.
Corollary If f is a continuous function in satisfying
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
By a simple calculation, we have the following inequality:
for any and satisfying , where M is a positive constant.
Take a number r satisfying , where R is a sufficiently large positivenumber. For any ϵ (), from Remark we have
which yields that there exists a positive constant dependent only on r such that
for any .
For any and , we have by (1.2), (2.1), (2.2), and Hölder’s inequality
Thus is finite for any . Since is a harmonic function of for any fixed , is also a harmonic function of.
Now we shall prove the boundary behavior of . For any fixed boundary point , we can choose a number T such that. Now we write
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.
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The authors are thankful to the referees for their helpful suggestions andnecessary corrections in the completion of this paper.
The authors declare that there is no conflict of interests regarding the publicationof this article.
All authors contributed equally to the manuscript and read and approved the finalmanuscript.