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 $\partial D$)



Stability of Dirichlet data for Helmholtz equation
The solutions of the Helmholtz equation are 
somewhat difficult to estimate. In particular, 
the constants depend heavily on the geometry of 
the domain. To see what I mean let us imagine for a moment
the set $D$ to be with an interior void. The boundary value
problem inside the void would be ill-posed for those $k$ 
which are square roots of the (positive) eigenvalues 
of the Laplacian. 


Now, if there is a hole connecting the void with the exterior,
the problem become well-posed (with appropriate boundary conditions, 
including the infinity). However, if the hole is small enough, 
the constant $C$ can be made as big as you wish.
(The things are better if $D$ is convex. Unfortunately,
I do not remember any appropriate reference now).


