Advertisement

Fractional De Giorgi Classes and Applications to Nonlocal Regularity Theory

  • Matteo CozziEmail author
Chapter
Part of the Springer INdAM Series book series (SINDAMS, volume 33)

Abstract

We present some recent results obtained by the author on the regularity of solutions to nonlocal variational problems. In particular, we review the notion of fractional De Giorgi class, explain its role in nonlocal regularity theory, and propose some open questions in the subject.

Keywords

Fractional De Giorgi classes Nonlocal Caccioppoli inequality Hölder continuity Harnack inequality Nonlocal functionals Nonlinear integral operators 

2010 Mathematics Subject Classification

49N60 35B45 35B50 35B65 35R11 47G20 

Notes

Acknowledgements

The author wishes to thank Serena Dipierro, the Università degli Studi di Bari, and INdAM for their kind invitation, warm hospitality, and financial support. The author also thanks the anonymous referee for her/his keen comments on a previous version of this note. The author is supported by the “Mara de Maeztu” MINECO grant MDM-2014-0445, by the MINECO grant MTM2017-84214-C2-1-P, and by a Royal Society Newton International Fellowship.

References

  1. 1.
    L. Brasco, E. Parini, The second eigenvalue of the fractional p-Laplacian. Adv. Calc. Var. 9(4), 323–355 (2016)MathSciNetCrossRefGoogle Scholar
  2. 2.
    L. Brasco, E. Lindgren, A. Schikorra, Higher Hölder regularity for the fractional p-Laplacian in the superquadratic case. Adv. Math. 338, 782–846 (2018)MathSciNetCrossRefGoogle Scholar
  3. 3.
    X. Cabré, M. Cozzi, A gradient estimate for nonlocal minimal graphs. Duke Math. J. 168(5), 775–848 (2019)MathSciNetCrossRefGoogle Scholar
  4. 4.
    L.A. Caffarelli, A. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation. Ann. Math. (2) 171(3), 1903–1930 (2010)MathSciNetCrossRefGoogle Scholar
  5. 5.
    L.A. Caffarelli, A. Vasseur, The De Giorgi method for nonlocal fluid dynamics, in Nonlinear Partial Differential Equations. Advanced Courses in Mathematics. CRM Barcelona (Birkhäuser/Springer Basel AG, Basel, 2012), pp. 1–38Google Scholar
  6. 6.
    L.A. Caffarelli, J.-M. Roquejoffre, Y. Sire, Variational problems for free boundaries for the fractional Laplacian. J. Eur. Math. Soc. 12(5), 1151–1179 (2010)MathSciNetCrossRefGoogle Scholar
  7. 7.
    L.A. Caffarelli, C.H. Chan, A. Vasseur, Regularity theory for parabolic nonlinear integral operators. J. Am. Math. Soc. 24(3), 849–869 (2011)MathSciNetCrossRefGoogle Scholar
  8. 8.
    M. Cozzi, Regularity results and Harnack inequalities for minimizers and solutions of nonlocal problems: a unified approach via fractional De Giorgi classes. J. Funct. Anal. 272(11), 4762–4837 (2017)MathSciNetCrossRefGoogle Scholar
  9. 9.
    M. Cozzi, E. Valdinoci, Plane-like minimizers for a non-local Ginzburg-Landau-type energy in a periodic medium. J. Éc. Polytech. Math. 4, 337–388 (2017)MathSciNetCrossRefGoogle Scholar
  10. 10.
    E. De Giorgi, Sulla differenziabilità e l’analiticità delle estremali degli integrali multipli regolari. Mem. Accad. Sci. Torino. Cl. Sci. Fis. Mat. Nat. (3) 3, 25–43 (1957)Google Scholar
  11. 11.
    A. Di Castro, T. Kuusi, G. Palatucci, Nonlocal Harnack inequalities. J. Funct. Anal. 267(6), 1807–1836 (2014)MathSciNetCrossRefGoogle Scholar
  12. 12.
    A. Di Castro, T. Kuusi, G. Palatucci, Local behavior of fractional p-minimizers. Ann. Inst. H. Poincaré Anal. Non Linéaire 33(5), 1279–1299 (2016)MathSciNetCrossRefGoogle Scholar
  13. 13.
    E. Di Nezza, G. Palatucci, E. Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136(5), 521–573 (2012)MathSciNetCrossRefGoogle Scholar
  14. 14.
    E. DiBenedetto, N.S. Trudinger, Harnack inequalities for quasiminima of variational integrals. Ann. Inst. H. Poincaré Anal. Non Linéaire 1(4), 295–308 (1984)MathSciNetCrossRefGoogle Scholar
  15. 15.
    M. Giaquinta, E. Giusti, On the regularity of the minima of variational integrals. Acta Math. 148, 31–46 (1982)MathSciNetCrossRefGoogle Scholar
  16. 16.
    M. Giaquinta, L. Martinazzi, An Introduction to the Regularity Theory for Elliptic Systems, Harmonic Maps and Minimal Graphs. Appunti, Scuola Normale Superiore di Pisa (Nuova Serie) [Lecture Notes. Scuola Normale Superiore di Pisa (New Series)], vol. 11 (Edizioni della Normale, Pisa, 2012)Google Scholar
  17. 17.
    E. Giusti, Direct Methods in the Calculus of Variations (World Scientific Publishing Co., Inc., River Edge, 2003)CrossRefGoogle Scholar
  18. 18.
    M. Kassmann, A priori estimates for integro-differential operators with measurable kernels. Calc. Var. Partial Differ. Equ. 34(1), 1–21 (2009)MathSciNetCrossRefGoogle Scholar
  19. 19.
    M. Kassmann, Harnack inequalities and Hölder regularity estimates for nonlocal operators revisited (2011). PreprintGoogle Scholar
  20. 20.
    T. Kuusi, G. Mingione, Y. Sire, Nonlocal self-improving properties. Anal. PDE 8(8), 57–114 (2015)MathSciNetCrossRefGoogle Scholar
  21. 21.
    O.A. Ladyzhenskaya, N.N. Uraltseva, Linear and Quasilinear Elliptic Equations (Academic, New York, 1968). Translated from the Russian by Scripta Technica, Inc., Translation editor: Leon EhrenpreisGoogle Scholar
  22. 22.
    G. Mingione, Gradient potential estimates. J. Eur. Math. Soc. 13(2), 459–486 (2011)MathSciNetzbMATHGoogle Scholar
  23. 23.
    X. Ros-Oton, J. Serra, The boundary Harnack principle for nonlocal elliptic operators in non-divergence form. Potential Anal. (to appear). https://doi.org/10.1007/s11118-018-9713-7
  24. 24.
    A. Schikorra, Integro-differential harmonic maps into spheres. Commun. Partial Differ. Equ. 40(3), 506–539 (2015)MathSciNetCrossRefGoogle Scholar
  25. 25.
    L. Silvestre, Hölder estimates for solutions of integro-differential equations like the fractional Laplace. Indiana Univ. Math. J. 55(3), 1155–1174 (2006)MathSciNetCrossRefGoogle Scholar
  26. 26.
    K.-O. Widman, Hölder continuity of solutions of elliptic systems. Manuscripta Math. 5, 299–308 (1971)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.University of Bath, Department of Mathematical SciencesBathUK

Personalised recommendations