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Let $\Omega \subset \mathbb{R}^n$ bounded with smooth boundary $\partial \Omega$. Let k>0 and $\partial \Omega = \Gamma_1 \cup \Gamma_2$. Consider the problem

$\Delta u + k^2 u = 0$ in $\Omega$

$\frac{\partial u}{\partial n} = f $ on $\Gamma_1$

$\frac{\partial u}{\partial n} + M u = g $ on $\Gamma_2$.

Does anyone know uniquness/existence properties/theorems of these kind of problems. I know that pure Neumann problems might have a non-unique solution for some $k$ and that the pure Robin problem is uniquely solvable (if $\Im(M) > 0$). (See for example Ivan G. Graham, Ulrich Langer, Jens M. Melenk, Mourad Sini: Direct and Inverse Problems in Wave Propagation and Applications)

I cannot find anything about the interior problem with these mixed boundary conditions (only Dirichlet/Neumann-mix). Now I am asking myself, if this problem behaves more like a Neumann problem, where I might have non-uniqueness for some $k$ or if it is always uniquely solvable.

Does anyone have some answers or references for this?

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  • $\begingroup$ With fixed $M$, you have non-uniqueness when $-k^2$ is an eigenvalue (possibly complex) of the Laplacian with these boundary conditions. So if the solution is always always unique (for any complex $k$), this would imply that this Laplacian has empty spectrum, which is rather unlikely. $\endgroup$
    – Igor Khavkine
    Commented Aug 21, 2017 at 13:18

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