**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 = (S_{t}}_{t} ≥ 0 be the semigroup generated on L_{2}(R^{d}) by a self-adjoint, second-order, divergence-form, elliptic operator H with Lipschitz continuous coefficients. Further let Ω be an open subset of R^{d} with Lipschitz continuous boundary ∂Ω. Authors proved that S leaves L_{2}(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.

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