000099 Robinson D W;Sikora A (NO, Centre for Mathematics and its Applications, Mathematical Sciences Ins, Australian National Univ, Canberra, ACT 0200, Australia, Email: Derek.Robinson@anu.edu.au) : Degenerate elliptic operators: capacity, flux and separation. J Ramanujan Mathl Soc 2007, 22(4), 385-408.
Let S = (St}t ≥ 0 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 ∂Ω. Authors proved 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.