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

2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/152797For 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.5 pagesAnalysis of PDEsA sharp pointwise bound for functions with $L^2$-Laplacians on arbitrary domains and its applicationstext