Bilaplacians problems with a sign-changing coefficient

TL;DR

This paper analyzes the mathematical properties of a bilaplacian operator with a sign-changing coefficient, focusing on conditions for it to be a Fredholm operator and implications for the Interior Transmission Eigenvalue Problem.

Contribution

It establishes conditions under which the bilaplacian operator with sign-changing coefficients is Fredholm of index zero, and explores how boundary sign changes affect this property.

Findings

Operator is Fredholm of index zero when \sigma is uniformly positive or negative near the boundary.

Fredholm property can be lost when \sigma changes sign on the boundary.

Detailed analysis of a simplified boundary condition problem.

Abstract

We investigate the properties of the operator \Delta(\sigma\Delta.):H^2_0(\Omega)-> H^{-2}(\Omega), where \sigma is a given parameter whose sign can change on the bounded domain \Omega. Here, H^2_0(\Omega) denotes the subspace of H^2(\Omega) made of the functions w such that w=\nu.\nabla 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 \Delta(\sigma\Delta.):H^2_0(\Omega)-> H^{-2}(\Omega) is a Fredholm operator of index zero as soon as \sigma\in L^{\infty}(\Omega), with \sigma^{-1}\in L^{\infty}(\Omega), is such that \sigma remains uniformly positive (or uniformly negative) in a neighbourhood of the boundary. We also study configurations where \sigma changes 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 \nu.\nabla v=0 is replaced by \sigma\Delta v=0.