Regularity of stable solutions of p-Laplace equations through geometric Sobolev type inequalities

Daniele Castorina, Manel Sanchon

    Research output: Contribution to journalArticleResearchpeer-review

    5 Citations (Scopus)

    Abstract

    © 2015 European Mathematical Society. We prove a Sobolev and a Morrey type inequality involving the mean curvature and the tangential gradient with respect to the level sets of the function that appears in the inequalities. Then, as an application, we establish a priori estimates for semistable solutions of 1pu D g.u/ in a smooth bounded domain Rn. In particular, we obtain new Lr and W1;r bounds for the extremal solution u when the domain is strictly convex. More precisely, we prove that u 2 L1./ if n p C 2 and u 2 L np np2 ./ \ W 1;p0 ./ if n p C 2.
    Original languageEnglish
    Pages (from-to)2949-2975
    JournalJournal of the European Mathematical Society
    Volume17
    Issue number11
    DOIs
    Publication statusPublished - 1 Jan 2015

    Keywords

    • Geometric inequalities
    • Mean curvature of level sets
    • P-laplace equations
    • Regularity of stable solutions
    • Schwarz symmetrization

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