2013/09/11 by Lucas Chesnel, Chesnel, Lucas
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical methods in inverse problems #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.1309.2891
openalex publication_date 2013/09/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We investigate the properties of the operator Δ(σΔ.):H20(Ω)-> H-2(Ω), where σis a given parameter whose sign can change on the bounded domain Ω. Here, H20(Ω) denotes the subspace of H2(Ω) made of the functions w such that w=ν.∇ w=0 on the boundary. The study of this problem arises when one is interested in some configurations of the Interior Transmission Eigenvalue Problem. We prove that Δ(σΔ.):H20(Ω)-> H-2(Ω) is a Fredholm operator of index zero as soon as σ∈ L∞(Ω), with σ-1∈ L∞(Ω), is such that σremains uniformly positive (or uniformly negative) in a neighbourhood of the boundary. We also study configurations where σchanges sign on the boundary and we prove that Fredholm property can be lost for such situations. In the process, we examine in details the features of a simpler problem where the boundary condition ν.∇ v=0 is replaced by σΔv=0.