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I'm dealing with the Helmholtz equation $\Delta u +k^2u=0$ in a exterior region $R^3/D$ ( $D$ opened and bounded) of a three dimensional space with Dirichlet boundary condition $u=g$ on $\partial D$ (and the usual radiation condition at infinity). It is known that the solution $u$ is unique and depends continuously on the boundary data $g$. I'm interested in finding the best bound $C$ such that: $\sup_{R^3/D}|u|\leq C·\sup_{\partial D}|g|$. I know that for the Laplace equation $C=1$ (maximum principle). And I guess that using boundary integral equations (double layer and single layer potentials) the resulting estimation for $C$ is very pessimistic in general. (Consider smooth boundary if required, and boundary data $g$ continuous on $\partial D$)