Integral representations and $L^\infty$ bounds for solutions of the Helmholtz equation on arbitrary open sets in $\mathbb{R}^2$ and $\mathbb{R}^3$

Abstract

We establish sharp $L^{\infty}$ bounds for functions defined on arbitrary open sets in $\Bbb R^2$ and $\Bbb R^3$, which vanish on the boundary and have $L^2$ Laplacians. All functions corresponding to the best possible constants are explicitly given. The proof is based on integral representations using the Green's function for the Helmholtz equation in arbitrary domains.