**Are you an EPFL student looking for a semester project?**

Work with us on data science and visualisation projects, and deploy your project as an app on top of Graph Search.

Publication# Spectral analysis for transmission eigenvalue problems with and without the complementing conditions

Abstract

The interior transmission eigenvalue problem is a system of partial differential equations equipped with Cauchy data on the boundary: the transmission conditions. This problem appears in the inverse scattering theory for inhomogeneous media when, for some frequency, the incident wave is not scattered by the medium. The relative Cauchy problem associated to the interior transmission eigenvalue problem plays also a crucial role in the stability of mathematical models describing phenomena using negative-index metamaterials. The first chapter is devoted to the analysis of the interior transmission eigenvalue problem in the acoustic setting. We establish the Weyl law for the interior transmission eigenvalues and the completeness of the generalized eigenfunctions for a system without complementing conditions, i.e., the two equations of the system have the same coefficients for the second order terms, and thus being degenerate. These coefficients are allowed to be anisotropic and are assumed to be two times continuously differentiable. One of the keys of the analysis is to establish the well-posedness and the regularity in $L^p$-scale for the relative Cauchy problem. To this end we mainly use the Fourier analysis, which allows obtaining results for non-smooth coefficients. As a result, we extend and rediscover known results for which the coefficients for the second order terms are required to be isotropic and smooth, but using a new approach. The second chapter is related to the interior transmission eigenvalue problem in the electromagnetic setting. Cakoni and Nguyen recently proposed very general conditions on the coefficients of Maxwell equations for which they established the discreteness of the set of interior transmission eigenvalues and studied their location. In this chapter, we establish the completeness of the generalized eigenfunctions and derive an optimal upper bound for the counting function under these conditions and assuming additionally that the coefficients are twice continuously differentiable. The approach is based on the spectral theory of Hilbert-Schmidt operators. Eventually, we present in chapter 3 partial results on the interior transmission eigenvalue problem in the electromagnetic setting under non-complementing conditions. We only prove a key point of our analysis leading to a priori estimates of the solutions to the relative Cauchy problem.

Official source

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related concepts (42)

Related MOOCs (21)

Related publications (61)

In mathematics, spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix to a much broader theory of the structure of operators in a variety of mathematical spaces. It is a result of studies of linear algebra and the solutions of systems of linear equations and their generalizations. The theory is connected to that of analytic functions because the spectral properties of an operator are related to analytic functions of the spectral parameter.

In mathematics, particularly linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix can be diagonalized (that is, represented as a diagonal matrix in some basis). This is extremely useful because computations involving a diagonalizable matrix can often be reduced to much simpler computations involving the corresponding diagonal matrix. The concept of diagonalization is relatively straightforward for operators on finite-dimensional vector spaces but requires some modification for operators on infinite-dimensional spaces.

Negative-index metamaterial or negative-index material (NIM) is a metamaterial whose refractive index for an electromagnetic wave has a negative value over some frequency range. NIMs are constructed of periodic basic parts called unit cells, which are usually significantly smaller than the wavelength of the externally applied electromagnetic radiation. The unit cells of the first experimentally investigated NIMs were constructed from circuit board material, or in other words, wires and dielectrics.

Introduction to optimization on smooth manifolds: first order methods

Learn to optimize on smooth, nonlinear spaces: Join us to build your foundations (starting at "what is a manifold?") and confidently implement your first algorithm (Riemannian gradient descent).

Plasma Physics: Introduction

Learn the basics of plasma, one of the fundamental states of matter, and the different types of models used to describe it, including fluid and kinetic.

Plasma Physics: Introduction

Learn the basics of plasma, one of the fundamental states of matter, and the different types of models used to describe it, including fluid and kinetic.

Mark Pauly, Francis Julian Panetta, Tian Chen, Christopher Brandt, Jean Jouve

Mechanical metamaterials enable customizing the elastic properties of physical objects by altering their fine-scale structure. A broad gamut of effective material properties can be produced even from a single fabrication material by optimizing the geometry ...

2023Romain Christophe Rémy Fleury, Bakhtiyar Orazbayev, Rayehe Karimi Mahabadi, Taha Goudarzi

Tunable metamaterials functionalities change in response to external stimuli. Mechanical deformation is known to be an efficient approach to tune the electromagnetic response of a deformable metamaterial. However, in the case of large mechanical deformatio ...

Kirigami is the art of paper cutting, and it is emerging as an elegant design and manufacturing solution in mechanical metamaterials. Currently, the majority of kirigami designs focus on shape-morphing, but there is little attention on the remarkable mecha ...