The limitations of the Poincaré inequality for Grušin type operators

Derek W. Robinson, Adam Sikora*

*Corresponding author for this work

    Research output: Contribution to journalArticlepeer-review

    4 Citations (Scopus)

    Abstract

    We examine the validity of the Poincaré inequality for degenerate, second-order, elliptic operatorsH in divergence form on L2(Rn × Rm). We assume the coefficients are real symmetric and a1Hδ ≥ H ≥ a2Hδ for some a1, a2 > 0 where Hd is a generalized Grušin operator,(Formula presented.).Here (Formula presented.) and (Formula presented.) and(Formula presented.). We prove that the Poincaré inequality, formulated in terms of thegeometry corresponding to the control distance of H, is valid if n ≥ 2, or if n = 1 and δ1 ∨ δ′1 ∈ [0, 1/2〉 but it fails if n = 1 and δ1 ∨ δ′1 ∈ [1/2, 1 〉. The failure is caused by the leading term. If δ1 ∈ [1/2, 1〉, it is an effect of the local degeneracy {pipe}x1{pipe}2δ1, but if δ1 ∈ [0, 1/2〉 and δ1 ∈ [1/2, 1〉, it is an effect of the growth at infinity of {pipe}x1{pipe}2δ′1. If n = 1 and δ1 ∈ [1/2, 1〉, then the semigroup S generated by the Friedrichs' extensionof H is not ergodic. The subspaces x1 ≥ 0 and x1≤ 0 are S-invariant, and the Poincaré inequality is validon each of these subspaces. If, however, n = 1, δ1 ∈ [0, 1/2〉 and δ′1 ∈ [1/2, 1〉, then the semigroup S isergodic, but the Poincaré inequality is only valid locally. Finally, we discuss the implication of these resultsfor the Gaussian and non-Gaussian behaviour of the semigroup S.

    Original languageEnglish
    Pages (from-to)535-563
    Number of pages29
    JournalJournal of Evolution Equations
    Volume14
    Issue number3
    DOIs
    Publication statusPublished - 2014

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