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

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