A generalized eigenproblem for the Laplacian which arises in lightning

The following generalized eigenproblem is analyzed: Find u∈H01(Ω), u≠0u≠0, and λ∈Rλ∈R such that〈∇u,∇v〉D=λ〈∇u,∇v〉Ω〈∇u,∇v〉D=λ〈∇u,∇v〉Ω for all v∈H01(Ω), where Ω⊂RnΩ⊂Rn is a bounded domain, D is a subdomain with closure contained in Ω  , and 〈⋅,⋅〉Ω〈⋅,⋅〉Ω is the inner product〈∇u,∇v〉Ω=∫Ω∇u⋅∇vdx. It is proved that any f∈H01(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)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)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)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∖DΩ∖D; the eigenvalue is 0.2.Functions in H01(Ω) with support in Ω∖DΩ∖D; the eigenvalue is 0.3.Functions in H01(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)(0,1). The only possible accumulation point is λ=1/2λ=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.