arXiv subjects

Lucas Chesnel

Publications and source records attributed to Lucas Chesnel.

38 records · Page 3Linked to original sources

arXiv2013

A curious instability phenomenon for a rounded corner in presence of a negative material

We study a 2D scalar harmonic wave transmission problem between a classical dielectric and a medium with a real-valued negative permittivity/permeability which models a metal at optical frequency or an ideal negative metamaterial. We highlight an unusual instability phenomenon for this problem when the interface between the two media presents a rounded corner. To establish this result, we provide an asymptotic expansion of the solution, when it is well-defined, in the geometry with a rounded corner. Then, we prove error estimates. Finally, a careful study of the asymptotic expansion allows us to conclude that the solution, when it is well-defined, depends critically on the value of the rounding parameter. We end the paper with a numerical illustration of this instability phenomenon.

math.AP↗

arXiv2013

Bilaplacians problems with a sign-changing coefficient

We investigate the properties of the operator Δ(σΔ.):H^2_0(Ω)-> H^{-2}(Ω), where σis 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 w such that w=ν.\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 Δ(σΔ.):H^2_0(Ω)-> H^{-2}(Ω) is a Fredholm operator of index zero as soon as σ\in L^{\infty}(Ω), with σ^{-1}\in L^{\infty}(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.\nabla v=0 is replaced by σΔv=0.

math.AP↗