Bilaplacian problems with a sign-changing coefficient

We investigate the properties of the operator ∆(σ ∆.):H₀¹( Ω)→ H^-2( Ω), where σ is a given parameter whose sign can change on the bounded domain Ω. Here, H₀¹( Ω) denotes the subspace of H²(Ω) made of the functions v such that v = ν·∇v = 0 on ∂Ω. The study of this problem arises when one is interested in some configurations of the interior transmission eigenvalue problem. We prove that ∆(σ ∆.):H₀¹( 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 σ remains uniformly positive (or uniformly negative) in a neighbourhood of ∂Ω. We also study configurations where σ changes sign on ∂Ω, 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 on ∂Ω.