Modulus of continuity and Heinz-Schwarz type inequalities of solutions to biharmonic equations

Abstract

For positive integers $n\ge 2$ and $m\ge 1$, suppose that function $f\in {\mathcal{C}}^4({\mathbb{B}}^n,{\mathbb{R}}^m)$ satisfying the following: $(1)$ the inhomogeneous biharmonic equation $\Delta (\Delta f)=g$ ($g\in \mathcal{C}(\overline{{\mathbb{B}}^n},{\mathbb{R}}^m)$) in ${\mathbb{B}}^n$, (2) the boundary conditions $f={\phi }_1$ $({\phi }_1\in \mathcal{C}({\mathbb{S}}^{n-1},{\mathbb{R}}^m))$ on ${\mathbb{S}}^{n-1}$ and $\partial f/\partial \mathbf{n}={\phi }_2$ ( ${\phi }_2\in \mathcal{C}({\mathbb{S}}^{n-1},{\mathbb{R}}^m)$) on ${\mathbb{S}}^{n-1}$, where $\partial /\partial \mathbf{n}$ stands for the inward normal derivative, ${\mathbb{B}}^n$ is the unit ball in ${\mathbb{R}}^n$ and ${\mathbb{S}}^{n-1}$ is the unit sphere of ${\mathbb{B}}^n$. First, we establish the representation formula of solutions to the above inhomogeneous biharmonic Dirichlet problem, and then discuss the Heinz-Schwarz type inequalities and the modulus of continuity of the solutions.

23 pages