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Spectral properties of operators and wave propagation in high-contrast media

Yuri A. Godin, Leonid Koralov, Boris Vainberg

TL;DR

The paper develops a unified, analyticity-based framework for spectral problems of elliptic operators with high-contrast, piecewise-constant coefficients, covering Dirichlet, Neumann, and Bloch settings. By analyzing the inverse operator and reducing to non-local boundary-value problems on interfaces, it proves the analytic continuation of eigenvalues/eigenfunctions in a neighborhood of $\varepsilon=0$ and explicitly characterizes the limit problem and dispersion relations. The results extend to multiple inclusions and variable coefficients, with resolvent convergence and a detailed limit-spectrum description. Rich examples, including 1D and 3D geometries, illustrate the theory and provide explicit limit spectra, highlighting the method's practical applicability for wave control in high-contrast media.

Abstract

The paper aims to study the spectral properties of elliptic operators with highly inhomogeneous coefficients and related issues concerning wave propagation in high-contrast media. A unified approach to solving problems in bounded domains with Dirichlet or Neumann boundary conditions, as well as in infinite periodic media, is proposed. For a small parameter $\varepsilon > 0$ characterizing the contrast of the components of the medium, the analyticity of the eigenvalues and eigenfunctions is established in a neighborhood of $\varepsilon = 0$. Effective operators corresponding to $\varepsilon = 0$ are described.

Spectral properties of operators and wave propagation in high-contrast media

TL;DR

The paper develops a unified, analyticity-based framework for spectral problems of elliptic operators with high-contrast, piecewise-constant coefficients, covering Dirichlet, Neumann, and Bloch settings. By analyzing the inverse operator and reducing to non-local boundary-value problems on interfaces, it proves the analytic continuation of eigenvalues/eigenfunctions in a neighborhood of and explicitly characterizes the limit problem and dispersion relations. The results extend to multiple inclusions and variable coefficients, with resolvent convergence and a detailed limit-spectrum description. Rich examples, including 1D and 3D geometries, illustrate the theory and provide explicit limit spectra, highlighting the method's practical applicability for wave control in high-contrast media.

Abstract

The paper aims to study the spectral properties of elliptic operators with highly inhomogeneous coefficients and related issues concerning wave propagation in high-contrast media. A unified approach to solving problems in bounded domains with Dirichlet or Neumann boundary conditions, as well as in infinite periodic media, is proposed. For a small parameter characterizing the contrast of the components of the medium, the analyticity of the eigenvalues and eigenfunctions is established in a neighborhood of . Effective operators corresponding to are described.

Paper Structure

This paper contains 7 sections, 11 theorems, 58 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The Dirichlet problem
  3. The Neumann problem
  4. Waves in periodic media, dispersion surface
  5. Multiple inclusions
  6. Operators with variable coefficients
  7. Examples

Key Result

Theorem 2.1

The operator $\widehat{\bm {\msfsl {B}}}_\varepsilon$ exists for $0< \varepsilon\ll1$ and can be extended analytically for $|\varepsilon|\ll1$. If then $u^{-}_0=c_0$ and $u^{+}_0$ is the solution of (psiP0) with $\phi=c_0$, where constant $c_0$ is defined by the equation

Figures (2)

  • Figure 1: The cell of periodicity $\Pi$ of a homogeneous medium with an inclusion $\Omega_-$.
  • Figure 2: The domain $\Omega$ in ${\mathbb R}^2$ containing three disjoint inclusions $\Omega_{-}^1, \Omega_{-}^2,$ and $\Omega_{-}^3$ with the boundaries $\Gamma_1, \Gamma_2,$ and $\Gamma_3$, respectively. $\Omega_- =\cup_{i=1}^m \Omega_{-}^i$, $\Gamma =\cup_{i=1}^m \Gamma_i$, $m=3$.

Theorems & Definitions (27)

  • Theorem 2.1
  • Remark 1
  • Remark 2
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • proof : Proof of Theorem \ref{['thm10']}
  • Theorem 2.2
  • ...and 17 more