H-compactness of elliptic operators on weighted Riemannian Manifolds

TL;DR

This paper investigates the asymptotic behavior of elliptic operators on weighted Riemannian manifolds, establishing H-compactness results and analyzing spectral properties in oscillating geometric settings.

Contribution

It introduces an H-compactness theorem for elliptic operators with measurable coefficients on weighted manifolds, extending homogenization techniques to geometric contexts.

Findings

Established H-compactness for elliptic operators on weighted manifolds.

Analyzed spectral behavior for operators with locally periodic coefficients.

Studied asymptotics of operators on oscillating submanifolds.

Abstract

In this paper we study the asymptotic behavior of second-order uniformly elliptic operators on weighted Riemannian manifolds. They naturally emerge when studying spectral properties of the Laplace-Beltrami operator on families of manifolds with rapidly oscillating metrics. We appeal to the notion of H-convergence introduced by Murat and Tartar. In our main result we establish an H-compactness result that applies to elliptic operators with measurable, uniformly elliptic coefficients on weighted Riemannian manifolds. We further discuss the special case of ``locally periodic'' coefficients and study the asymptotic spectral behavior of compact submanifolds of $\mathbb R^n$ with rapidly oscillating geometry.