Abstract
We establish sharp $L^{\infty}$ bounds for functions defined on arbitrary open sets in $\Bbb R^2$ and $\Bbb R^3$, which vanish on the boundary and have $L^2$ Laplacians. All functions corresponding to the best possible constants are explicitly given. The proof is based on integral representations using the Green's function for the Helmholtz equation in arbitrary domains.
Citation
Wenzheng Xie. "Integral representations and $L^\infty$ bounds for solutions of the Helmholtz equation on arbitrary open sets in $\mathbb{R}^2$ and $\mathbb{R}^3$." Differential Integral Equations 8 (3) 689 - 698, 1995. https://doi.org/10.57262/die/1369316516
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