## 一族非齐次双调和方程解的边界Schwarz引理

1 武汉大学数学与统计学院 武汉 430072

2 华侨大学数学科学学院 福建泉州 362021

## Boundary Schwarz Lemma for Solutions to a Class of Inhomogeneous Biharmonic Equations

Bai Xiaojin,1,2, Zhu Jianfeng,2

1 School of Mathematics and Statistics, Wuhan University, Wuhan 430072

2 School of Mathematical Sciences, Huaqiao University, Fujian Quanzhou 362021

 基金资助: 国家自然科学基金.  12271189国家自然科学基金.  11971182福建省面上基金.  2021J01304福建省面上基金.  2019J0101

 Fund supported: the NSFC.  12271189the NSFC.  11971182the NSF of Fujian Province.  2021J01304the NSF of Fujian Province.  2019J0101

Abstract

Let $\mathbb{D}$ be the unit disk, ${\mathbb T}$ the unit circle. Assume that $f$ is a solution to inhomogeneous biharmonic equation: $\Delta f=g$, satisfying the boundary conditions: $(\Delta f)_{{\mathbb T}}=\psi$ and $f|_{{\mathbb T}}=f^*$, where $g\in {\cal C}(\overline{\mathbb{D}})$, and $\psi, f^*\in {\cal C}({\mathbb T})$ are continuous functions. In this paper, we establish the boundary Schwarz lemma for solutions $f$, this result enriches the related results of boundary Schwarz lemma on the plane.

Keywords： Inhomogeneous biharmonic equations ; Solution ; Dirichlet problem ; Boundary Schwarz lemma

Bai Xiaojin, Zhu Jianfeng. Boundary Schwarz Lemma for Solutions to a Class of Inhomogeneous Biharmonic Equations. Acta Mathematica Scientia[J], 2022, 42(6): 1633-1639 doi:

## 1 介绍

$$$\Delta(\Delta f)=g,$$$

$$$\left\{ \begin{array}{ll} \Delta f=\psi\ \ \ & \mbox{在}\ {\mathbb T}\ \mbox{上, }\\ f=f^*\ \ \ &\mbox{在}\ {\mathbb T}\ \mbox{上} \end{array} \right.$$$

(1) $\overline{\beta}f'(\alpha)\alpha\geq1$;

(2) $\overline{\beta}f'(\alpha)\alpha=1$当且仅当$f(z)\equiv {\rm e}^{{\rm i}\theta}z$, 其中${\rm e}^{{\rm i}\theta}=\beta\alpha^{-1}$并且$\theta\in{\mathbb R}$.

$$${\rm{Re}}[f_z(1)+f_{\bar{z}}(1)]\geq\frac{2}{\pi}-|{\mathcal P}_{f^*}(0)|-(8\log 2+6)\|\psi\|_\infty-\frac{85}{9}\|g\|_\infty,$$$

$$$\lim\limits_{r\rightarrow 1^-}\frac{|{\mathcal G}_\psi(r)|}{1-r}\leq(8\log 2+6)\|\psi\|_\infty.$$$

任取$\eta\in{\mathbb T}$, 由定义知${\mathcal G}_\psi(\eta)=0$显然成立. 为了证明(2.4) 式, 令$z=r$. 则有

$\zeta={\rm e}^{{\rm i}t}\in{\mathbb T}$可得

$\begin{eqnarray} \lim\limits_{r\rightarrow 1^-}\frac{|{\mathcal G}_\psi(r)|}{1-r}&=&\frac{1}{2\pi}\lim\limits_{r\rightarrow 1^-}\left| \int_{\mathbb T}\frac{g_1(z, \zeta)}{1-r}\psi(\zeta)\frac{{\rm d}\zeta}{\zeta}\right| \\ &\leq&\frac{\|\psi\|_\infty}{\pi}\int_{\mathbb T}(2|{\rm{Re}}(\bar{\zeta}\log(1-\zeta))|+1)|{\rm d}\zeta|\\ &=&\frac{2\|\psi\|_\infty}{\pi}\left( \int_{0}^{2\pi}|{\rm{Re}} ({\rm e}^{-{\rm i}t}\log(1-{\rm e}^{{\rm i}t})|{\rm d}t+\pi\right). \end{eqnarray}$

$z=r\in(0, 1)$并且令$r\rightarrow1^-$, 由引理2.1和引理2.2可得如下不等式

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