Degenerate elliptic operators

Abstract

Let S = {St}t≥0 be the semigroup generated on L2(Rd) by a self-adjoint, second-order, divergence-form, elliptic operator H with Lips-chitz continuous coefficients. Further let Ω be an open subset of Rd with Lipschitz continuous boundary ∂Ω. We prove that S leaves L2(Ω) invariant if, and only if, the capacity of the boundary with respect to H is zero or if, and only if, the energy flux across the boundary is zero. The global result is based on an analogous local result. Show more