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

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 $\mathrm{\Delta }(\mathrm{\Delta }f)=g$ ($g\in $\mathcal{C}$(\overline{$\mathbb{B}$^{n}},$\mathbb{R}$^{m})$)in$$\mathbb{B}$^{n}$, (2) the boundary conditions $f={\phi }_1$ $(\varphi_{1}\in $\mathcal{C}$($\mathbb{S}$^{n-1},$\mathbb{R}$^{m}))$on$$\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$$\mathbb{S}$^{n-1}$,where$\partial /\partial$\mathbf{n}$$standsfortheinwardnormalderivative,$$\mathbb{B}$^{n}$istheunitballin$$\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.