Spectral properties of integral operators in problems of interface dynamics

TL;DR

This paper investigates the spectral characteristics of integral operators arising in interface dynamics and phase kinetics, providing bounds for their quadratic forms in relation to the $H^{-1}$ norm, which aids understanding of layered equilibria and front behavior.

Contribution

It introduces a spectral analysis of integral operators relevant to phase kinetics with conservation laws, offering new bounds for quadratic forms in $L^2$ and $H^{-1}$ spaces.

Findings

Derived lower bounds for quadratic forms of the operators

Analyzed spectra of integral operators in $L^2$ space

Connected spectral properties to interface and front dynamics

Abstract

We consider a family of integral operators which appears when analyzing layered equilibria and front dynamics of a phase kinetics equation with a conservation law. We study the spectra of these operators in $L^2$ and derive a lower bound for the associated quadratic forms in terms of the $H^{-1}$ norm.