Dirichlet problems of harmonic functions

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R E S E A R C H

Open Access

Dirichlet problems of harmonic functions

Gang Xu

1*

_{, Pai Yang}

2

_{and Tao Zhao}

3

*_{Correspondence:}

[email protected] 1_{College of Computer and}

Information Engineering, Henan University of Economics and Law, Zhengzhou, 450000, P.R. China Full list of author information is available at the end of the article

Abstract

In this paper, a solution of the Dirichlet problem in the upper half-plane is constructed by the generalized Dirichlet integral with a fast growing continuous boundary function.

MSC: 31B05; 31B10

Keywords: Dirichlet problem; harmonic function; half-plane

1 Introduction and results

LetRbe the set of all real numbers, and letCdenote the complex plane with pointsz=

x+iy, wherex,y∈R. The boundary and closure of an open setare denoted by∂and

respectively. The upper half-plane is the setC+:={z=x+iy∈C:y> }, whose boundary

is∂C+=R. Let [d] denote the integer part of the positive real numberd.

Given a continuous functionf in∂C+, we say thathis a solution of the (classical)

Dirich-let problem inC+withf, ifh=  inC+andlimz∈C+,z→th(z) =f(t) for everyt∈∂C+.

The classical Poisson kernel inC+is deﬁned by

P(z,t) = y

π|z–t|,

wherez=x+iy∈C+andt∈R.

It is well known (see [, ]) that the Poisson kernelP(z,t) is harmonic forz∈C–{t}and has the expansion

P(z,t) = 

πIm

∞

k= zk

tk+,

which converges for|z|<|t|. We deﬁne a modiﬁed Cauchy kernel ofz∈C₊by

Cm(z,t) =

 π



t–z when|t| ≤,

 π



t–z–

 π

m k= z

k

tk+ when|t|> ,

wheremis a nonnegative integer.

To solve the Dirichlet problem inC+, as in [], we use the following modiﬁed Poisson

kernel deﬁned by

Pm(z,t) =ImCm(z,t) =

P(z,t) when|t| ≤,

P(z,t) –_π Imm_k_= zk

tk+ when|t|> .

We remark that the modiﬁed Poisson kernelPm(z,t) is harmonic inC+.

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Put

Um(f)(z) =

∞

–∞

Pm(z,t)f(t)dt,

wheref(t) is a continuous function in∂C+.

We say thatuis of orderλif

λ=lim sup r→∞

log(sup_H_∩_B₍_r₎|u|)

logr .

Ifλ<∞, thenuis said to be of ﬁnite order. See Hayman-Kennedy [, Deﬁnition .]. In caseλ<∞, about the solution of the Dirichlet problem with continuous data inH, we refer readers to the following result, which is due to Nevanlinna (see [–]).

Theorem A Let u be a nonnegative real-valued function harmonic inC+and continuous inC+.If

_∞

–∞

u(t)

 +tdt<∞,

then there exists a nonnegative real constant d such that

u(z) =dy+

∞

–∞P(z,t)u(t)dt

for all z=x+iy∈C+.

Inspired by Theorem A, we consider the Dirichlet problem for harmonic functions of infinite order inC+. To do this, we define a nondecreasing and continuously differentiable

functionρ(R)≥ on the interval [, +∞). We assume further that

ε=lim sup

R→∞

ρ(R)RlogR

ρ(R) < . (.)

Remark For any ( < <  – ), there exists a suﬃciently large positive numberRsuch

thatr>R, by (.) we have

ρ(r) <ρ(e)(lnr)+ _.

LetF(ρ,α) be the set of continuous functionsf on∂C+such that

∞

–∞

|f(t)|

 +|t|ρ(|t|)+α+dt<∞, (.)

whereαis a positive real number.

Now we show the solution of the Dirichlet problem with continuous data inC+. For

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Theorem  If f ∈F(ρ,α),then the integral U[ρ(|t|)+α](f)(z)is a solution of the Dirichlet problem inC+with f.

The following result is obtained by putting [ρ(|t|) +α] =min Theorem .

Corollary If f is a continuous function in∂C+satisfying

∞

–∞

|f(t)|

 +|t|m+dt<∞,

then Um(f)(z)is a solution of the Dirichlet problem inC+with f.

Theorem  Let u be a real-valued function harmonic inC+and continuous inC+.If u∈ F(p,ρ,α),then we have u(z) =U[ρ(|t|)+α](u)(z) +Im(z)for all z∈C+,where(z)is an entire function inC+and vanishes continuously in∂C+.

2 Proof of Theorem 1

By a simple calculation, we have the following inequality:

Cm(z,t)≤M|z|m+|t|–m– (.)

for anyz∈C+andt∈∂C+satisfying|t| ≥max{, |z|}, whereMis a positive constant.

Take a numberrsatisfyingr>R, whereRis a suﬃciently large positive number. For any ( < <  – ), from Remark we have

ρ(r) <ρ(e)(lnr)( + )_,

which yields that there exists a positive constantM(r) dependent only onrsuch that

k–α/(r)ρ(k+)+α+≤M(r) (.)

for anyk>kr= [r] + .

For anyz∈C+ and|z| ≤r, we have by (.), (.), (.), /p+ /q=  and Hölder’s

in-equality

M

∞

k=kr

k≤|t|<k+

|z|[ρ(|t|)+α]+

|t|[ρ(|t|)+α]+f(t)dt

≤M

∞

k=kr

rρ(k+)+α+ kα/

k≤|t|<k+

|f(t)| |t|ρ(|t|)++pα/dt

≤MM(r)

|t|≥kr

|f(t)|  +|t|ρ(|t|)++pα/dt

<∞.

ThusU[ρ(|t|)+α](f)(z) is ﬁnite for anyz∈C+. SinceP[ρ(|t|)+α](z,t) is a harmonic function

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Now we shall prove the boundary behavior ofU[ρ(|t|)+α](f)(z). For any ﬁxed boundary

pointt∈∂C+, we can choose a number T such thatT>|t|+ . Now we write U[ρ(|t|)+α](f)(z) =I(z) –I(z) +I(z),

where

I(z) =

|t|≤T

P(z,t)f(t)dt, I(z) =Im [ρ(|t|+α)]

k=

<|t|≤T

zk

πtk+f(t)dt,

I(z) =

|t|>T

P[ρ(|t|+α)](z,t)f(t)dt.

Note thatI(z) is the Poisson integral ofu(t)χ[–T,T](t), whereχ[–T,T]is the

characteris-tic function of the interval [–T, T]. So it tends tof(t) asz→t. Clearly,I(z) vanishes on ∂C+. Further,I(z) =O(y), which tends to zero asz→t. Thus the functionU[ρ(|t|)+α](f)(z)

can be continuously extended toC+such thatU[ρ(|t|)+α](f)(t) =f(t) for anyt∈∂C+.

The-orem  is proved.

3 Proof of Theorem 2

Consider that the functionu(z) –U[ρ(|t|)+α](u)(z), which is harmonic inC+, can be

contin-uously extended toC+and vanishes in∂C+.

The Schwarz reﬂection principle [, p.] applied tou(z) –U[ρ(|t|)+α](u)(z) shows that

there exists an entire harmonic function (z) in C+ satisfying(z) =(z) such that

Im(z) =u(z) –U[ρ(|t|)+α](u)(z) forz∈C+.

Thusu(z) =U[ρ(|t|)+α](u)(z) +Im(z) for allz∈C+, where(z) is an entire function in

C+and vanishes continuously in∂C+. Then we complete the proof of Theorem .

Competing interests

The authors declare that there is no conﬂict of interests regarding the publication of this article.

Authors’ contributions

All authors contributed equally to the manuscript and read and approved the ﬁnal manuscript.

Author details

1_{College of Computer and Information Engineering, Henan University of Economics and Law, Zhengzhou, 450000, P.R.}

China.2_{College of Applied Mathematics, Chengdu University of Information Technology, Chengdu, 610225, P.R. China.} 3_{Department of Mathematics, Henan University of Economics and Law, Zhengzhou, 450000, P.R. China.}

Acknowledgements

The authors are thankful to the referees for their helpful suggestions and necessary corrections in the completion of this paper.

Received: 6 September 2013 Accepted: 5 November 2013 Published:02 Dec 2013

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