Higher Order Elliptic Equations on Nonsmooth Domains

TL;DR

This paper extends $W^{ ext{ell,p}}$ estimates for higher-order elliptic equations on nonsmooth domains, broadening the range of $p$ and improving upon previous results for biharmonic equations using a new method.

Contribution

Introduces a novel approach to establish $W^{ ext{ell,p}}$ estimates for higher-order elliptic equations on Lipschitz and convex domains, expanding the valid $p$ range.

Findings

Established uniform $W^{ ext{ell,p}}$ estimates for higher-order elliptic equations.

Extended the $p$ range for estimates compared to previous biharmonic results.

Provided new techniques that do not rely on maximum principle methods.

Abstract

In 1995, D. Jerison and C. Kenig in \cite{JK-1995} considered the the inhomogeneous Dirichlet problem $\Delta u= f$ on $\Omega$, $u=0$ on $\partial\Omega$ in Lipschitz domains. One of their main results shows that the $W^{1,p}$ estimate holds for the sharp range $\frac{3}{2}-\varepsilon<p<3+\varepsilon$ for $d\geq 3$ and $\frac{4}{3}-\varepsilon<p<4+\varepsilon$ if $d=2$. Although the argument employed in \cite{JK-1995} yields optimal results, they rely on an essential fashion on the maximum principle and, as such, do not readily adapt to higher-order case. By using a new method, the aim of this paper is to establish an extension of their theorem for higher order inhomogeneous elliptic equations on bounded Lipschitz and convex domains, uniform $W^{\ell,p}$ estimates are obtained for $p$ in certain ranges. Especially, compare to the result in \cite{MM-2013} for biharmonic equation, a larger, sharp, range of $p's$ was obtained in this paper.