Generalized spectrum of second order differential operators

Abstract

We analyze the spectrum of the operator $\Delta^{-1} [\nabla \cdot (K\nabla u)]$, where $\Delta$ denotes the Laplacian and $K=K(x,y)$ is a symmetric tensor. Our main result shows that this spectrum can be derived from the spectral decomposition $K=Q \Lambda Q^T$, where $Q=Q(x,y)$ is an orthogonal matrix and $\Lambda=\Lambda(x,y)$ is a diagonal matrix. More precisely, provided that $K$ is continuous, the spectrum equals the convex hull of the ranges of the diagonal function entries of $\Lambda$. The involved domain is assumed to be bounded and Lipschitz, and both homogeneous Dirichlet and homogeneous Neumann boundary conditions are considered. We study operators defined on infinite dimensional Sobolev spaces. Our theoretical investigations are illuminated by numerical experiments, using discretized problems. The results presented in this paper extend previous analyses which have addressed elliptic differential operators with scalar coefficient functions. Our investigation is motivated by both preconditioning issues (efficient numerical computations) and the need to further develop the spectral theory of second order PDEs (core analysis).