Spectral Stability underLp-Perturbation of the Second-Order Coefficients

Abstract

Let H be a second order elliptic operator acting on a domain 0/R. There has been a lot of work on the stability of the spectrum and the resolvent of H under various sorts of perturbations. Perturbations of the 0th-order term are well studied. See for example [7]. There are also several results on perturbations of the higher-order terms. Classical perturbation theory can be applied in the case of uniform or asymptotic perturbations (see [5]). P. Deift [4] has obtained results for measurable perturbations in the context of scattering theory. More recent results ([3]) deal with boundary perturbations. In this paper we study L-perturbations of the second-order terms. We work on a bounded Euclidean domain 0/R which we assume initially to have a C boundary. The operators involved are uniformly elliptic with real measurable coefficients and satisfy Dirichlet boundary conditions. We first prove eigenvalue stability, but our main aim is the stability of the resolvent in trace classes. If one is only interested in the fact of convergence rather than controlling the rate, then a simple approach based upon the monotone or dominated convergence theorem exists. However, quantitative control by such methods is not possible. We emphasize the fact that we deal with operators with measurable coefficients. If one restricts attention to the smooth coefficient case, then standard methods of perturbation theory involve assuming uniform bounds on the first derivatives of the coefficients. Such bounds are not needed in our approach and are not always available in applications: apart from being more general, the measurable coefficients hypothesis is necessary for the study of the heat transport in a body with randomly distributed impurities. In particular we study the asymptotic form of the heat diffusion article no. 0011