ON THE NATURE OF THE LAPLACE-BELTRAMI OPERATOR ON LIPSCHITZ MANIFOLDS

摘要

We study the basic properties of the Laplace-Beltrami operator on Lipschitz surfaces, as well as abstract Lipschitz manifolds, including mapping and invertibility properties on scales of Sobolev spaces, being the infinitesimal generator of an analytic semigroup, the nature of the spectrum and the regularity of eigenfunctions. Much of this analysis is carried out in the more general case of second order, divergence-form, strongly elliptic differential operators with bounded, measurable, complex matrix-valued coefficients.