Bilaplacian problems with a sign-changing coefficient

We investigate the properties of the operator ∆(σ∆·) : H0(Ω) → H−2(Ω), where σ is a given parameter whose sign can change on the bounded domain Ω. Here, H0(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∆·) : H0(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 σ∆v = 0 on ∂Ω.