Degenerate elliptic operators: capacity, flux and separation

Let S = {S}t≥o be the semigroup generated on L2(Rd) by a self-adjoint, second-order, divergence-form, elliptic operator H with Lipschitz continuous coefficients. Further let Ω be an open subset of Rd with Lipschitz continuous boundary dΩ. 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.