$L_p$ regularity theorem for elliptic equations in less smooth domains

Abstract

We consider a $2m$th-order strongly elliptic operator $A$ subject to Dirichlet boundary conditions in a domain $\Omega$ of $\mathbb{R}^{n}$, and show the $L_{p}$ regularity theorem, assuming that the domain has less smooth boundary. We derive the regularity theorem from the following isomorphism theorems in Sobolev spaces. Let $k$ be a nonnegative integer. When $A$ is a divergence form elliptic operator, $A-\lambda$ has a bounded inverse from the Sobolev space $W^{k-m}_{p}(\Omega)$ into $W^{k+m}_{p}(\Omega)$ for $\lambda$ belonging to a suitable sectorial region of the complex plane, if $\Omega$ is a uniformly $C^{k,1}$ domain. When $A$ is a non-divergence form elliptic operator, $A-\lambda$ has a bounded inverse from $W^{k}_{p}(\Omega)$ into $W^{k+2m}_{p}(\Omega)$, if $\Omega$ is a uniformly $C^{k+m,1}$ domain. Compared with the known results, we weaken the smoothness assumption on the boundary of $\Omega$ by $m-1$.