Question

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 $\partial D$)

Answer 1

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).