H-Compactness of Elliptic Operators on Weighted Riemannian Manifolds

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 ℝⁿ with rapidly oscillating geometry.