Bilaplacian problems with a sign-changing coefficient

Lucas Chesnel (1, 2)

1 DeFI - Shape reconstruction and identification
2 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique

We investigate the properties of the operator ∆(σ∆·) : H^2_0 (Ω) → H^{−2}(Ω), where σ is a given parameter whose sign can change on the bounded domain Ω. Here, H^2_0 (Ω) denotes the subspace of H^2(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∆·) : H^2_0 (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.

Mots clés: sign-changing coefficient; non smooth boundary; singularities; bilaplacian; interior transmission problem

Domaines: Equations aux dérivées partielles [math.AP]

Lucas Chesnel. Bilaplacian problems with a sign-changing coefficient. Mathematical Methods in the Applied Sciences, 2014, pp.10.1002/mma.3366. ⟨10.1002/mma.3366⟩. ⟨hal-01110878⟩