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Homogenization of nonlocal spectral problems

Andrey Piatnitski, Volodymyr Rybalko

Abstract

We study asymptotic behavior of the bottom point of the spectrum of convolution type operators in environments with locally periodic microstructure. We show that its limit is described by an additive eigenvalue problem for Hamilton-Jacobi equation. In the periodic case we establish a more accurate two-term asymptotic formula.

Homogenization of nonlocal spectral problems

Abstract

We study asymptotic behavior of the bottom point of the spectrum of convolution type operators in environments with locally periodic microstructure. We show that its limit is described by an additive eigenvalue problem for Hamilton-Jacobi equation. In the periodic case we establish a more accurate two-term asymptotic formula.
Paper Structure (8 sections, 8 theorems, 95 equations)

This paper contains 8 sections, 8 theorems, 95 equations.

Table of Contents

  1. Introduction
  2. Problem setup.Convergence result for the bottom point of the spectrum
  3. Periodic case
  4. Problem reduction
  5. Resolvent convergence
  6. Proof of Theorem \ref{['Golovna_periodic']}
  7. Locally periodic case
  8. Acknowledgments

Key Result

Theorem 2.1

Suppose that $J$ satisfies umova_dva and $\kappa$, $a$ satisfy akappa. Then where $\Lambda$ is a unique additive eigenvalue of the problem

Theorems & Definitions (15)

  • Theorem 2.1
  • Theorem 3.1
  • Remark 3.2
  • Proposition 3.3
  • proof
  • Lemma 3.4
  • proof
  • Theorem 3.5
  • proof
  • Lemma 3.6
  • ...and 5 more