# Dirichlet problems of harmonic functions

## Abstract

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 $z=x+iy$, where $x,y\in \mathbf{R}$. The boundary and closure of an open set Ω aredenoted by Ω and $\overline{\mathrm{\Omega }}$ respectively. The upper half-plane is the set${\mathbf{C}}_{+}:=\left\{z=x+iy\in \mathbf{C}:y>0\right\}$, whose boundary is $\partial {\mathbf{C}}_{+}=\mathbf{R}$. Let $\left[d\right]$ denote the integer part of the positive real numberd.

Given a continuous function f in $\partial {\mathbf{C}}_{+}$, we say that h is a solution of the(classical) Dirichlet problem in ${\mathbf{C}}_{+}$ with f, if $\mathrm{\Delta }h=0$ in ${\mathbf{C}}_{+}$ and ${lim}_{z\in {\mathbf{C}}_{+},z\to t}h\left(z\right)=f\left(t\right)$ for every $t\in \partial {\mathbf{C}}_{+}$.

The classical Poisson kernel in ${\mathbf{C}}_{+}$ is defined by

$P\left(z,t\right)=\frac{y}{\pi {|z-t|}^{2}},$

where $z=x+iy\in {\mathbf{C}}_{+}$ and $t\in \mathbf{R}$.

It is well known (see [1, 2]) that the Poisson kernel $P\left(z,t\right)$ is harmonic for $z\in \mathbf{C}-\left\{t\right\}$ and has the expansion

$P\left(z,t\right)=\frac{1}{\pi }Im\sum _{k=0}^{\mathrm{\infty }}\frac{{z}^{k}}{{t}^{k+1}},$

which converges for $|z|<|t|$. We define a modified Cauchy kernel of$z\in {\mathbf{C}}_{+}$ by

where m is a nonnegative integer.

To solve the Dirichlet problem in ${\mathbf{C}}_{+}$, as in , we use the following modified Poisson kernel defined by

We remark that the modified Poisson kernel ${P}_{m}\left(z,t\right)$ is harmonic in ${\mathbf{C}}_{+}$.

Put

${U}_{m}\left(f\right)\left(z\right)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{P}_{m}\left(z,t\right)f\left(t\right)\phantom{\rule{0.2em}{0ex}}dt,$

where $f\left(t\right)$ is a continuous function in $\partial {\mathbf{C}}_{+}$.

We say that u is of order λ if

$\lambda =\underset{r\to \mathrm{\infty }}{lim sup}\frac{log\left({sup}_{H\cap B\left(r\right)}|u|\right)}{logr}.$

If $\lambda <\mathrm{\infty }$, then u is said to be of finite order. SeeHayman-Kennedy [, Definition 4.1].

In case $\lambda <\mathrm{\infty }$, about the solution of the Dirichlet problem withcontinuous data in H, we refer readers to the following result, which isdue to Nevanlinna (see ).

Theorem A Let u be a nonnegative real-valued function harmonic in${\mathbf{C}}_{+}$and continuous in${\overline{\mathbf{C}}}_{+}$. If

${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{u\left(t\right)}{1+{t}^{2}}\phantom{\rule{0.2em}{0ex}}dt<\mathrm{\infty },$

then there exists a nonnegative real constant d such that

$u\left(z\right)=dy+{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}P\left(z,t\right)u\left(t\right)\phantom{\rule{0.2em}{0ex}}dt$

for all$z=x+iy\in {\mathbf{C}}_{+}$.

Inspired by Theorem A, we consider the Dirichlet problem for harmonic functions ofinfinite order in ${\mathbf{C}}_{+}$. To do this, we define a nondecreasing andcontinuously differentiable function $\rho \left(R\right)\ge 1$ on the interval $\left[0,+\mathrm{\infty }\right)$. We assume further that

${\epsilon }_{0}=\underset{R\to \mathrm{\infty }}{lim sup}\frac{{\rho }^{\prime }\left(R\right)RlogR}{\rho \left(R\right)}<1.$
(1.1)

Remark For any ϵ ($0<ϵ<1-{ϵ}_{0}$), there exists a sufficiently large positive numberR such that $r>R$, by (1.1) we have

$\rho \left(r\right)<\rho \left(e\right){\left(lnr\right)}^{{ϵ}_{0}+ϵ}.$

Let $F\left(\rho ,\alpha \right)$ be the set of continuous functions f on$\partial {\mathbf{C}}_{+}$ such that

${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{|f\left(t\right)|}{1+{|t|}^{\rho \left(|t|\right)+\alpha +1}}\phantom{\rule{0.2em}{0ex}}dt<\mathrm{\infty },$
(1.2)

where α is a positive real number.

Now we show the solution of the Dirichlet problem with continuous data in${\mathbf{C}}_{+}$. 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 ).

Theorem 1 If$f\in F\left(\rho ,\alpha \right)$, then the integral${U}_{\left[\rho \left(|t|\right)+\alpha \right]}\left(f\right)\left(z\right)$is a solution of the Dirichlet problem in${\mathbf{C}}_{+}$with f.

The following result is obtained by putting $\left[\rho \left(|t|\right)+\alpha \right]=m$ in Theorem 1.

Corollary If f is a continuous function in $\partial {\mathbf{C}}_{+}$ satisfying

${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{|f\left(t\right)|}{1+{|t|}^{m+2}}\phantom{\rule{0.2em}{0ex}}dt<\mathrm{\infty },$

then${U}_{m}\left(f\right)\left(z\right)$is a solution of the Dirichlet problem in${\mathbf{C}}_{+}$with f.

Theorem 2 Let u be a real-valued function harmonic in${\mathbf{C}}_{+}$and continuous in${\overline{\mathbf{C}}}_{+}$. If$u\in F\left(p,\rho ,\alpha \right)$, then we have$u\left(z\right)={U}_{\left[\rho \left(|t|\right)+\alpha \right]}\left(u\right)\left(z\right)+Im\mathrm{\Pi }\left(z\right)$for all$z\in {\overline{\mathbf{C}}}_{+}$, where$\mathrm{\Pi }\left(z\right)$is an entire function in${\mathbf{C}}_{+}$and vanishes continuously in$\partial {\mathbf{C}}_{+}$.

## 2 Proof of Theorem 1

By a simple calculation, we have the following inequality:

$|{C}_{m}\left(z,t\right)|\le M{|z|}^{m+1}{|t|}^{-m-2}$
(2.1)

for any $z\in {\mathbf{C}}_{+}$ and $t\in \partial {\mathbf{C}}_{+}$ satisfying $|t|\ge max\left\{1,2|z|\right\}$, where M is a positive constant.

Take a number r satisfying $r>R$, where R is a sufficiently large positivenumber. For any ϵ ($0<ϵ<1-{ϵ}_{0}$), from Remark we have

$\rho \left(r\right)<\rho \left(e\right){\left(lnr\right)}^{\left({ϵ}_{0}+ϵ\right)},$

which yields that there exists a positive constant $M\left(r\right)$ dependent only on r such that

${k}^{-\alpha /2}{\left(2r\right)}^{\rho \left(k+1\right)+\alpha +1}\le M\left(r\right)$
(2.2)

for any $k>{k}_{r}=\left[2r\right]+1$.

For any $z\in {\mathbf{C}}_{+}$ and $|z|\le r$, we have by (1.2), (2.1), (2.2),$1/p+1/q=1$ and Hölder’s inequality

$\begin{array}{r}M\sum _{k={k}_{r}}^{\mathrm{\infty }}{\int }_{k\le |t|

Thus ${U}_{\left[\rho \left(|t|\right)+\alpha \right]}\left(f\right)\left(z\right)$ is finite for any $z\in {\mathbf{C}}_{+}$. Since ${P}_{\left[\rho \left(|t|\right)+\alpha \right]}\left(z,t\right)$ is a harmonic function of $z\in {\mathbf{C}}_{+}$ for any fixed $t\in \partial {\mathbf{C}}_{+}$, ${U}_{\left[\rho \left(|t|\right)+\alpha \right]}\left(f\right)\left(z\right)$ is also a harmonic function of$z\in {\mathbf{C}}_{+}$.

Now we shall prove the boundary behavior of ${U}_{\left[\rho \left(|t|\right)+\alpha \right]}\left(f\right)\left(z\right)$. For any fixed boundary point ${t}^{\prime }\in \partial {\mathbf{C}}_{+}$, we can choose a number T such that$T>|{t}^{\prime }|+1$. Now we write

${U}_{\left[\rho \left(|t|\right)+\alpha \right]}\left(f\right)\left(z\right)={I}_{1}\left(z\right)-{I}_{2}\left(z\right)+{I}_{3}\left(z\right),$

where

$\begin{array}{c}{I}_{1}\left(z\right)={\int }_{|t|\le 2T}P\left(z,t\right)f\left(t\right)\phantom{\rule{0.2em}{0ex}}dt,\phantom{\rule{2em}{0ex}}{I}_{2}\left(z\right)=Im\sum _{k=0}^{\left[\rho \left(|t|+\alpha \right)\right]}{\int }_{1<|t|\le 2T}\frac{{z}^{k}}{\pi {t}^{k+1}}f\left(t\right)\phantom{\rule{0.2em}{0ex}}dt,\hfill \\ {I}_{3}\left(z\right)={\int }_{|t|>2T}{P}_{\left[\rho \left(|t|+\alpha \right)\right]}\left(z,t\right)f\left(t\right)\phantom{\rule{0.2em}{0ex}}dt.\hfill \end{array}$

Note that ${I}_{1}\left(z\right)$ is the Poisson integral of $u\left(t\right){\chi }_{\left[-2T,2T\right]}\left(t\right)$, where ${\chi }_{\left[-2T,2T\right]}$ is the characteristic function of the interval$\left[-2T,2T\right]$. So it tends to $f\left({t}^{\prime }\right)$ as $z\to {t}^{\prime }$. Clearly, ${I}_{2}\left(z\right)$ vanishes on $\partial {\mathbf{C}}_{+}$. Further, ${I}_{3}\left(z\right)=O\left(y\right)$, which tends to zero as $z\to {t}^{\prime }$. Thus the function ${U}_{\left[\rho \left(|t|\right)+\alpha \right]}\left(f\right)\left(z\right)$ can be continuously extended to${\overline{\mathbf{C}}}_{+}$ such that ${U}_{\left[\rho \left(|t|\right)+\alpha \right]}\left(f\right)\left({t}^{\prime }\right)=f\left({t}^{\prime }\right)$ for any ${t}^{\prime }\in \partial {\mathbf{C}}_{+}$. Theorem 1 is proved.

## 3 Proof of Theorem 2

Consider that the function $u\left(z\right)-{U}_{\left[\rho \left(|t|\right)+\alpha \right]}\left(u\right)\left(z\right)$, which is harmonic in ${\mathbf{C}}_{+}$, can be continuously extended to${\overline{\mathbf{C}}}_{+}$ and vanishes in $\partial {\mathbf{C}}_{+}$.

The Schwarz reflection principle [, p.68] applied to $u\left(z\right)-{U}_{\left[\rho \left(|t|\right)+\alpha \right]}\left(u\right)\left(z\right)$ shows that there exists an entire harmonic function$\mathrm{\Pi }\left(z\right)$ in ${\mathbf{C}}_{+}$ satisfying $\mathrm{\Pi }\left(\overline{z}\right)=\overline{\mathrm{\Pi }\left(z\right)}$ such that $Im\mathrm{\Pi }\left(z\right)=u\left(z\right)-{U}_{\left[\rho \left(|t|\right)+\alpha \right]}\left(u\right)\left(z\right)$ for $z\in {\overline{\mathbf{C}}}_{+}$.

Thus $u\left(z\right)={U}_{\left[\rho \left(|t|\right)+\alpha \right]}\left(u\right)\left(z\right)+Im\mathrm{\Pi }\left(z\right)$ for all $z\in {\overline{\mathbf{C}}}_{+}$, where $\mathrm{\Pi }\left(z\right)$ is an entire function in ${\mathbf{C}}_{+}$ and vanishes continuously in $\partial {\mathbf{C}}_{+}$. Then we complete the proof of Theorem 2.

## References

1. Hayman WK, Kennedy PB 1. In Subharmonic Functions. Academic Press, London; 1976.

2. Ransford T: Potential Theory in the Complex Plane. Cambridge University Press, Cambridge; 1995.

3. Finkelstein M, Scheinberg S: Kernels for solving problems of Dirichlet typer in a half-plane. Adv. Math. 1975, 1(18):108-113.

4. Axler S, Bourdon P, Ramey W: Harmonic Function Theory. 2nd edition. Springer, New York; 1992.

5. Nevanlinna R: Über die Eigenschaften meromorpher Funktionen in einem Winkelraum. Acta Soc. Sci. Fenn. 1925, 12(50):1-45.

6. Stein EM: Harmonic Analysis. Princeton University Press, Princeton; 1993.

7. Qiao L: Integral representations for harmonic functions of infinite order in acone. Results Math. 2012, 61: 63-74. 10.1007/s00025-010-0076-7

8. Qiao L, Pan G-S: Integral representations of generalized harmonic functions. Taiwan. J. Math. 2013, 17: 1503-1521.

9. Ren YD: Solving integral representations problems for the stationary Schrödingerequation. Abstr. Appl. Anal. 2013., 2013: Article ID 715252

10. Su BY: Dirichlet problem for the Schrödinger operator in a half space. Abstr. Appl. Anal. 2012., 2012: Article ID 578197

11. Su BY: Growth properties of harmonic functions in the upper half space. Acta Math. Sin. 2012, 55(6):1095-1100. (in Chinese)

12. Yang, DW, Ren, YD: A Dirichlet problem on the upper half space. Proc. IndianAcad. Sci. Math. Sci. (to appear)

## Acknowledgements

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

## Author information

Authors

### Corresponding author

Correspondence to Gang Xu.

### Competing interests

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

### Authors’ contributions

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

## Rights and permissions

Reprints and Permissions

Xu, G., Yang, P. & Zhao, T. Dirichlet problems of harmonic functions. Bound Value Probl 2013, 262 (2013). https://doi.org/10.1186/1687-2770-2013-262 