### Abstract

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.

Original language | English |
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Pages (from-to) | 385-408 |

Number of pages | 24 |

Journal | Journal of the Ramanujan Mathematical Society |

Volume | 22 |

Issue number | 4 |

Publication status | Published - 2007 |

Externally published | Yes |

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## Cite this

Robinson, D. W., & Sikora, A. (2007). Degenerate elliptic operators: capacity, flux and separation.

*Journal of the Ramanujan Mathematical Society*,*22*(4), 385-408.