A sharp pointwise bound for functions with $L^2$-Laplacians on arbitrary domains and its applications

For all functions on an arbitrary open set $\Omega\subset\R^3$ with zero boundary values, we prove the optimal bound $$\sup_{\Omega}|u| \leq (2\pi)^{-1/2} \left(\int_{\Omega}|\nabla u|^2 \,dx\, \int_{\Omega}|\Delta u|^2 \,dx\right)^{1/4}.$$ The method of proof is elementary and admits generalizations. The inequality is applied to establish an existence theorem for the Burgers equation.