Spectral analysis and shape optimization of Steklov--Dirichlet eigenvalues on shell-type domains
We investigate the Steklov--Dirichlet eigenvalues on shell-type domains in $\mathbb{R}^n$, $n\geq 2$, where the Steklov and Dirichlet conditions are imposed on the outer and the inner boundaries, respectively. We introduce the Green function associated with the problem, which is well-defined for spectral parameters outside the spectrum and exhibits blow-up behavior at eigenvalues, and establish its spectral representation. We also examine its differentiability and derive its shape derivative under translations of the inner boundary. Using these results and a layer potential approach, we propose a computational framework for optimizing the eigenvalue with respect to the position of the inner boundary, incorporating an eigenvalue estimator for which a local error bound is established. Numerical experiments for various inner and outer boundary shapes demonstrate the effectiveness of the proposed approach in identifying configurations that maximize the first eigenvalue.