A sharp pointwise bound for functions with 𝐿²-Laplacians on arbitrary domains and its applications
Wenzheng Xie · Bulletin of the American Mathematical Society · 1992
For all functions on an arbitrary open set Ω ⊂ R 3 \Omega \subset {R^3} with zero boundary values, we prove the optimal bound \[ sup Ω | u | ≤ ( 2 π ) − 1 / 2 ( ∫ Ω | ∇ u | 2 d x ∫ Ω | Δ u | 2 d x ) 1 / 4 . \sup _\Omega |u| \leq (2\pi )^{-1/2} (\smallint _\Omega | abla u|^2\,dx \smallint _\Omega |\Delta u|^2\,dx)^{1/4}. \] The method of proof is elementary and admits generalizations. The inequality is applied to establish an existence theorem for the Burgers equation.