Shape optimization with surface interactions

Following are brief statements of some open problems raised during the AIM Workshop Shape optimization with surface interactions, June 17–21, 2019. The name of the participant who suggested the problem is stated, along with references to further details. Naturally, the participant is not always the original proposer of the problem in the literature.

The majority of the problems concern eigenvalues of Laplace operators on a bounded domain $\Omega \subset \mathbb{R}^n$. Throughout, we shall use the following notational conventions:

1. The eigenvalues of the Dirichlet Laplacian on a bounded domain $\Omega$, enumerated in non-decreasing order and repeated according to their multiplicity, will be denoted by $\lambda_k(\Omega)$, with $k=1, 2, 3, \ldots$.
2. The eigenvalues of the Neumann Laplacian on a bounded Lipschitz domain $\Omega$, enumerated in non-decreasing order and repeated according to their multiplicity, will be denoted by $\mu_k(\Omega)$, with $k=0, 1, 2, 3, \ldots$. In particular, we note that in this convention $\mu_0(\Omega)=0$ and the first non-trivial Neumann eigenvalue of a connected domain $\Omega$ is $\mu_1(\Omega)$.
3. The eigenvalues of the Robin Laplacian on a bounded Lipschitz domain $\Omega$ with boundary condition given by $$ \frac{\partial u(x)}{\partial n} + \alpha u(x)= 0, \quad \mbox{on }\partial\Omega, $$ where $n$ is the outer unit normal to $\partial\Omega$ and $\alpha$ is a real \REMOVE?{coupling} constant, will be denoted by $\lambda_k(\Omega, \alpha)$, with $k=1, 2, 3, \ldots$. Again the eigenvalues are enumerated in a non-decreasing order and repeated according to their multiplicity. Within the chosen orientation of the normal $n$, the principal eigenvalue $\lambda_1(\Omega,\alpha)$ has the same sign as $\alpha$.