The plasmonic eigenvalue problem

A plasmon of a bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^n$ is a non-trivial bounded harmonic function on ${\mathbb{R}}^n\setminus \mathrm{\partial }\mathrm{\Omega }$ which is continuous at $\mathrm{\partial }\mathrm{\Omega }$ and whose exterior and interior normal derivatives at $\mathrm{\partial }\mathrm{\Omega }$ have a constant ratio. We call this ratio a plasmonic eigenvalue of $\mathrm{\Omega }$. Plasmons arise in the description of electromagnetic waves hitting a metallic particle $\mathrm{\Omega }$. We investigate these eigenvalues and prove that they form a sequence of numbers converging to one. Also, we prove regularity of plasmons, derive a variational characterization, and prove a second order perturbation formula. The problem can be reformulated in terms of Dirichlet-Neumann operators, and as a side result we derive a formula for the shape derivative of these operators.