A sharp pointwise bound for functions with $L\sp 2$-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

$$\mathop {{\text{sup}}}\limits_\Omega \vert u\vert \leq {(2\pi )^{... ... }\vert\nabla u{\vert^2}dx{\smallint _\Omega }\vert\Delta u{\vert^2}dx)^{1/4}}.$$

The method of proof is elementary and admits generalizations. The inequality is applied to establish an existence theorem for the Burgers equation.