Regularity results and Harnack inequalities for minimizers and solutions of nonlocal problems: a unified approach via fractional De Giorgi classes

2016/09/30 by Matteo Cozzi · 2 citations
Mathematics · #math.AP #msc:49N60 #msc:47G20 #msc:35R11 #msc:35D10 #msc:35B65

paper · pdf · doi:10.1016/j.jfa.2017.02.016

published as J. Funct. Anal. 272 (11) (2017) 4762-4837 · 59 pages

arxiv created 2018/11/21 · arxiv updated 2018/11/22

Abstract

We study energy functionals obtained by adding a possibly discontinuous potential to an interaction term modeled upon a Gagliardo-type fractional seminorm. We prove that minimizers of such non-differentiable functionals are locally bounded, Hölder continuous, and that they satisfy a suitable Harnack inequality. Hence, we provide an extension of celebrated results of M. Giaquinta and E. Giusti to the nonlocal setting. To do this, we introduce a particular class of fractional Sobolev functions, reminiscent of that considered by E. De Giorgi in his seminal paper of 1957. The flexibility of these classes allows us to also establish regularity of solutions to rather general nonlinear integral equations.

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