The plasmonic eigenvalue problem

A plasmon of a bounded domain Ω ⊂ ℝn is a non-trivial bounded harmonic function on ℝn\∂Ω which is continuous at ∂Ω and whose exterior and interior normal derivatives at ∂Ω 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.

Domination of the supremum of a bounded harmonic function by its supremum over a countable subset

For what sequences {an} of points of the open unit disc D does there exist a constant k such thatfor all bounded harmonic functions f on D?

Inequalities for the derivatives of a bounded harmonic function

If h(r, θ) is harmonic in the unit circle | r | < 1 and satisfies the condition | h | ≤ 1, then there is a function u(ø) which satisfies | u | ≤ 1 such thatand conversely. Hence, any properties of such harmonic functions should be deducible from equation (1). A number of such properties have been proved by Koebe (Math. Z. 6 (1920), 52–84, 69), using Schwarz's lemma and the geometry of simple conformal transformations. They can be deduced from (1) together with an elementary lemma on the rearrangement of a function (Lemma 1 below). As, however, students of this subject will regard Koebe's method as the one best adapted to establish his theorems, we shall illustrate the alternative method by considering two new problems, namely to find max ∂h/∂r, max ∂h/∂θ, where the maximum in each case is taken for all harmonic functions h which satisfy