A generalized eigenproblem for the Laplacian which arises in lightning

Abstract

The following generalized eigenproblem is analyzed: Find u ∈ H 0 1 ( Ω ) , u ≠ 0 , and λ ∈ R such that 〈 ∇ u , ∇ v 〉 D = λ 〈 ∇ u , ∇ v 〉 Ω for all v ∈ H 0 1 ( Ω ) , where Ω ⊂ R n 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 d x . It is proved that any f ∈ H 0 1 ( Ω ) can be expanded in terms of orthogonal eigenfunctions for the generalized eigenproblem. During the analysis, we present a new inner product on H 1 / 2 ( ∂ D ) with the following properties: (a) the norm associated with the inner product is equivalent to the usual norm on H 1 / 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 H 1 / 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 H 0 1 ( Ω ) with support in Ω ∖ D ; the eigenvalue is 0. 3. Functions in H 0 1 ( 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.