A sharp pointwise bound for functions with 𝐿²-Laplacians on arbitrary domains and its applications

Xie, Wenzheng

doi:10.1090/S0273-0979-1992-00279-3

A sharp pointwise bound for functions with $L^ 2$-Laplacians on arbitrary domains and its applications
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Bull. Amer. Math. Soc. 26 (1992), 294-298 Request permission

Abstract:

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

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