Regularity results and Harnack inequalities for minimizers and solutions of nonlocal problems: A unified approach via fractional De Giorgi classes
By Matteo Cozzi. 2016
147 citations recorded by the source corpus.
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, Holder 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.
Subfields: Applied Mathematics
View Paper: https://www.semanticscholar.org/paper/dd151c034d1ea8350fc46813dcb50241b3c8c940
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In Applied Mathematics
- Lorenzo Brasco
- Armin Schikorra
- Erik Lindgren
- Nicola Garofalo
- Giampiero Palatucci
- Cristiana De Filippis
- P. Garain
- Marco Squassina
Citations in the corpus
- Fractional thoughts (2017, background): “One should also see the preprint [Co16] which contains related results for minimizers of nonlocal functionals of the calculus of variations.”
- Higher Hölder regularity for the fractional p-Laplacian in the superquadratic case (2017, background): “In [13], the results of [16, 17] are generalized to functions belonging to a fractional analogue of the so-called De Giorgi classes (we refer to [19, Chapter 7] for the classical definition).”
- Hölder regularity for weak solutions to nonlocal double phase problems (2021, unknown): “Cozzi [12] extended these results to non-homogeneous problems with lower order terms, by using fractional De Giorgi classes.”