A generalized eigenproblem for the Laplacian which arises in lightning

The following generalized eigenproblem is analyzed: Find u ∈ H1 0 (Ω), u = 0, and λ ∈ R such that ∇u,∇v D = λ ∇u,∇v Ω for all v ∈ H1 0 (Ω), where Ω ⊂ Rn is a bounded domain, D is a subdomain with closure contained in Ω, and ·,· Ω is the inner product ∇u,∇v Ω = Ω ∇u · ∇v dx. It is proved that any f ∈ H1 0 (Ω) can be expanded in terms of orthogonal eigenfunctions for the generalized eigenproblem. During the analysis, we present a new inner product on H1/2(∂D) with the following properties: (a) the norm associated with the inner product is equivalent to the usual norm on H1/2(∂D), and (b) the double layer potential operator is self adjoint with respect to the new inner product and compact as a mapping from H1/2(∂D) into itself. The analysis identifies four classes of eigenfunctions for the generalized eigenproblem: 1. The function Π which is 1 on D and harmonic on Ω \D; the eigenvalue is 0. 2. Functions in H1 0 (Ω) with support in Ω \D; the eigenvalue is 0. 3. Functions in H1 0 (Ω) with support in D; the eigenvalue is 1. 4. Excluding Π, the harmonic extension of the eigenfunctions of a double layer potential on ∂D. The eigenvalues are contained in the open interval (0, 1). The only possible accumulation point is λ = 1/2. A positive lower bound for the smallest positive eigenvalue is obtained. These results can be used to evaluate the change in the electric potential due to a lightning discharge.