Flows and invariance for degenerate elliptic operators

A. F M Ter Elst*, Derek W. Robinson, Adam Sikora

*Corresponding author for this work

    Research output: Contribution to journalArticlepeer-review

    3 Citations (Scopus)


    Let S be a sub-Markovian semigroup on L 2(ℝ d) generated by a self-adjoint, second-order, divergence-form, elliptic operator H with W 1,∞(ℝ d) coefficients c kl, and let Ω be an open subset of ℝ d. We prove that if either C c(ℝ d) is a core of the semigroup generator of the consistent semigroup on L p(ℝ d) for some p ∈ [1,∞] or Ω has a locally Lipschitz boundary, then S leaves L 2 (Ω) invariant if and only if it is invariant under the flows generated by the vector fields Σ d l=1c kll for all k. Further, for all p ∈ [1,2] we derive sufficient conditions on the coefficients for the core property to be satisfied. Then by combination of these results we obtain various examples of invariance in terms of boundary degeneracy both for Lipschitz domains and domains with fractal boundaries.

    Original languageEnglish
    Pages (from-to)317-339
    Number of pages23
    JournalJournal of the Australian Mathematical Society
    Issue number3
    Publication statusPublished - Jun 2011


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