The plasmonic eigenvalue problem

A plasmon of a bounded domain $Ω\subset\mathbb{R}^n$ is a non-trivial bounded harmonic function on $\mathbb{R}^n\setminus\partialΩ$ which is continuous at $\partialΩ$ and whose exterior and interior normal derivatives at $\partialΩ$ have a constant ratio. We call this ratio a plasmonic eigenvalue of $Ω$. Plasmons arise in the description of electromagnetic waves hitting a metallic particle $Ω$. 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.

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