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

## TL;DR

This paper derives representation formulas for solutions to inhomogeneous biharmonic equations with boundary conditions, and investigates Heinz-Schwarz inequalities and the modulus of continuity of these solutions.

## Contribution

It introduces new representation formulas for biharmonic solutions and extends Heinz-Schwarz inequalities to this context, analyzing their continuity properties.

## Key findings

- Established solution representation formulas for biharmonic equations.
- Extended Heinz-Schwarz type inequalities to biharmonic solutions.
- Analyzed the modulus of continuity of solutions.

## Abstract

For positive integers $n\geq2$ and $m\geq1$, 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 $f=\varphi_{1}$ $(\varphi_{1}\in \mathcal{C}(\mathbb{S}^{n-1},\mathbb{R}^{m}))$ on $\mathbb{S}^{n-1}$ and $\partial f/\partial\mathbf{n}=\varphi_{2}$ ( $\varphi_{2}\in \mathcal{C}(\mathbb{S}^{n-1},\mathbb{R}^{m})$) 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.