$L^{\infty }$-estimates for the Neumann problem on general domains

Abstract

Let $\Omega \subset {\mathbb{R}}^d$ be bounded open and connected. Suppose that $W^{1,2}(\Omega )\subset L^r(\Omega )$ for some $r>2$. Let $A$ be a pure second-order elliptic differential operator with bounded real measurable coefficients on $\Omega$. Let $q>d$ with $\frac{1}{2}-\frac{1}{q}>\frac{1}{r}$. If $p$ is the dual exponent of $q$, then we show that the pre-image of the space $(W^{1,p}(\Omega ){)}^{\ast }$ under the map $A$ is contained in the space of bounded functions on $\Omega$. The considerations are complemented by results on optimal Sobolev regularity for $A$.