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arXiv · 1710.09352

H-compactness of elliptic operators on weighted Riemannian Manifolds

Helmer Hoppe·Jun Masamune·Stefan Neukamm
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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.

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Helmer Hoppe, Jun Masamune, Stefan Neukamm. 2017-10-25. H-compactness of elliptic operators on weighted Riemannian Manifolds. https://arxiv.org/abs/1710.09352

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