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

Abstract

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