Regularity and rigidity theorems for a class of anisotropic nonlocal operators

Alberto Farina, Enrico Valdinoci

    Research output: Contribution to journalArticle

    9 Citations (Scopus)

    Abstract

    We consider here operators which are sum of (possibly) fractional derivatives, with (possibly different) order. The main constructive assumption is that the operator is of order 2 in one variable. By constructing an explicit barrier, we prove a Lipschitz estimate which controls the oscillation of the solutions in such direction with respect to the oscillation of the nonlinearity in the same direction. As a consequence, we obtain a rigidity result that, roughly speaking, states that if the nonlinearity is independent of a coordinate direction, then so is any global solution (provided that the solution does not grow too much at infinity). A Liouville type result then follows as a byproduct.

    Original languageEnglish
    Pages (from-to)53-70
    Number of pages18
    JournalManuscripta Mathematica
    Volume153
    Issue number1-2
    DOIs
    Publication statusPublished - 1 May 2017

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    Rigidity
    Regularity
    Operator
    Theorem
    Nonlinearity
    Oscillation
    Fractional Derivative
    Global Solution
    Lipschitz
    Infinity
    Estimate
    Class

    Cite this

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    Regularity and rigidity theorems for a class of anisotropic nonlocal operators. / Farina, Alberto; Valdinoci, Enrico.

    In: Manuscripta Mathematica, Vol. 153, No. 1-2, 01.05.2017, p. 53-70.

    Research output: Contribution to journalArticle

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    AU - Valdinoci, Enrico

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    AB - We consider here operators which are sum of (possibly) fractional derivatives, with (possibly different) order. The main constructive assumption is that the operator is of order 2 in one variable. By constructing an explicit barrier, we prove a Lipschitz estimate which controls the oscillation of the solutions in such direction with respect to the oscillation of the nonlinearity in the same direction. As a consequence, we obtain a rigidity result that, roughly speaking, states that if the nonlinearity is independent of a coordinate direction, then so is any global solution (provided that the solution does not grow too much at infinity). A Liouville type result then follows as a byproduct.

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