Journal of the Ramanujan Mathematical Society

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Degenerate Elliptic Operators:Capacity, Flux and Separation


Affiliations
1 Centre for Mathematics and its Applications, Mathematical Sciences Institute, Australian National University, Canberra, ACT 0200, Australia
2 Department of Mathematical Sciences, New Mexico State University, P.O. Box 30001, Las Cruces, NM 88003-8001, United States
     

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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. 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.
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  • Degenerate Elliptic Operators:Capacity, Flux and Separation

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Authors

Derek W. Robinson
Centre for Mathematics and its Applications, Mathematical Sciences Institute, Australian National University, Canberra, ACT 0200, Australia
Adam Sikora
Department of Mathematical Sciences, New Mexico State University, P.O. Box 30001, Las Cruces, NM 88003-8001, United States

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 Lipschitz 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.