Continuity aspects for traces of Dirichlet forms with respect to monotone weak convergence of G-Kato measures

TL;DR

This paper studies the convergence and continuity properties of traces of Dirichlet forms with respect to measures that are limits of G-Kato measures, providing spectral convergence results and applications to elliptic operators and graph Laplacians.

Contribution

It establishes spectral and resolvent convergence for traces of Dirichlet forms under monotone weak limits of G-Kato measures, with applications to elliptic PDEs and graph Laplacians.

Findings

Spectral convergence of eigenvalues and eigenfunctions.

Continuity of stationary solutions with measure-valued coefficients.

Approximation methods for eigenvalues of graph Laplacians and Laplacians on thin sets.

Abstract

We investigate some analytic properties of traces of Dirichlet forms with respect to measures satisfying Hardy-type inequality. Among other results we prove convergence of spectra, ordered eigenvalues, eigenfunctions as well as convergence of resolvents on appropriate spaces, for traces of Dirichlet forms when the speed measure is the monotone weak limit of G-Kato measures. Some quantitative estimates are also given. As applications we show continuity of stationary solutions of some elliptic operators with measure-valued coefficients with respect to the coefficients and give an approximation procedure for the eigenvalues of one-dimensional graph Laplacian as well as for the Laplacian on annular thin sets with mixed Neumann- Wentzell boundary conditions.