Stability of weighted Laplace-Beltrami operators under \(L^ p\)-perturbation of the Riemannian metric (Q1922220)

Let \((M,G)\) be a Riemannian manifold, where \(G\) is a suitable measurable metric. The author considers operators of the form \(H:f\mapsto- \sigma^{-2}\nabla_G\cdot (\sigma^2\nabla_G f)+f\), where \(\sigma\) is a measurable weight. When \(M\) has no boundary, he shows that \(H^{-1}\) maps \(L^p\) continuously into \(W^{1,p}\) if \(p\) is close to 2; as \(G\), \(\sigma\) vary in \(L^p\), quantitative estimates for the stability of \(H^{-1}\) in trace norms are given. This result is extended to the case when \(M\) has a boundary and mixed Dirichlet-Neumann boundary conditions are imposed. Actually, this paper extends earlier results by the same author published in a previous paper [J. Differ. Equations 124, No. 2, 302-323 (1996; Zbl 0848.35082)], where \(M\) was a bounded domain with Lipschitz boundary in \(\mathbb{R}^n\) and the Dirichlet boundary condition was imposed.