Degenerate elliptic operators in one dimension

Derek W. Robinson, Adam Sikora

    Research output: Contribution to journalArticlepeer-review

    8 Citations (Scopus)

    Abstract

    Let H be the symmetric second-order differential operator on L2(R) with domain C c and action Hφ = -(cφ)′ where c ε W1,2 loc(R) is a real function that is strictly positive on R\{0} but with c(0) = 0. We give a complete characterization of the self-adjoint extensions and the submarkovian extensions of H. In particular if ν = ν+ ∨ ν where ν±(x) = ± ∫±1 ±x c-1 then H has a unique self-adjoint extension if and only if ν ∉ L2(0,1) and a unique submarkovian extension if and only if ν ∉ L(0,1). In both cases, the corresponding semigroup leaves L2(0,∞) and L2(-∞,0) invariant. In addition, we prove that for a general non-negative c ε W1,∞ loc(R) the corresponding operator H has a unique submarkovian extension.

    Original languageEnglish
    Pages (from-to)731-759
    Number of pages29
    JournalJournal of Evolution Equations
    Volume10
    Issue number4
    DOIs
    Publication statusPublished - 2010

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