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Fractional superharmonic functions and the Perron method for nonlinear integro-differential equations

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Abstract

We deal with a class of equations driven by nonlocal, possibly degenerate, integro-differential operators of differentiability order \(s\in (0,1)\) and summability growth \(p>1\), whose model is the fractional p-Laplacian with measurable coefficients. We state and prove several results for the corresponding weak supersolutions, as comparison principles, a priori bounds, lower semicontinuity, and many others. We then discuss the good definition of (s, p)-superharmonic functions, by also proving some related properties. We finally introduce the nonlocal counterpart of the celebrated Perron method in nonlinear Potential Theory.

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Notes

  1. For an elementary introduction to this topic and for a quite wide, but still limited, list of related references we refer to [10].

  2. As we were finishing this manuscript, we became aware of very recent manuscript [29] having an independent and different approach to the problem.

  3. We take the liberty to call superharmonic functions appearing in this context as (s, p)-superharmonic emphasizing the (s, p)-order of the involved Gagliardo kernel.

  4. When needed, our definition of Tail can also be given in a more general way by replacing the ball \(B_r\) and the corresponding \(r^{sp}\) term by an open bounded set \(E\subset {\mathbb {R}}^n\) and its rescaled measure \(|E|^{sp/n}\), respectively. This is not the case in the present paper.

References

  1. Barrios, B., Peral, I., Vita, S.: Some remarks about the summability of nonlocal nonlinear problems. Adv. Nonlinear Anal. 4(2), 91–107 (2015)

    MathSciNet  MATH  Google Scholar 

  2. Brasco, L., Lindgren, E.: Higher Sobolev regularity for the fractional \(p\)-Laplace equation in the superquadratic case. Adv. Math. 304, 300–354 (2017)

  3. Brasco, L., Parini, E., Squassina, M.: Stability of variational eigenvalues for the fractional \(p\)-Laplacian. Discrete Contin. Dyn. Syst. 36, 1813–1845 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  4. Chambolle, A., Lindgren, E., Monneau, R.: A Hölder infinity Laplacian. ESAIM Control Optim. Calc. Var. 18, 799–835 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  5. Caffarelli, L., Silvestre, L.: An extension problem related to the fractional Laplacian. Commun. Partial Differ. Equ. 32, 1245–1260 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  6. Da Lio, F., Rivière, T.: Three-term commutators estimates and the regularity of \(1/2\)-harmonic maps into spheres. Anal. PDE 4, 149–190 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  7. Da Lio, F., Rivière, T.: Sub-criticality of non-local Schrödinger systems with antisymmetric potentials and applications to half-harmonic maps. Adv. Math. 227(3), 1300–1348 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  8. Di Castro, A., Kuusi, T., Palatucci, G.: Nonlocal Harnack inequalities. J. Funct. Anal. 267(6), 1807–1836 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  9. Di Castro, A., Kuusi, T., Palatucci, G.: Local behavior of fractional \(p\)-minimizers. Ann. Inst. H. Poincaré Anal. Non Linéaire 33, 1279–1299 (2016)

  10. Di Nezza, E., Palatucci, G., Valdinoci, E.: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136, 521–573 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  11. Dyda, B.: Fractional calculus for power functions and eigenvalues of the fractional Laplacian. Fract. Calc. Appl. Anal. 15(4), 536–555 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  12. Franzina, G., Palatucci, G.: Fractional \(p\)-eigenvalues. Riv. Mat. Univ. Parma 5(2), 373–386 (2014)

    MathSciNet  MATH  Google Scholar 

  13. Granlund, S., Lindqvist, P., Martio, O.: Note on the PWB-method in the nonlinear case. Pac. J. Math. 125(2), 381–395 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  14. Hamel, F., Ros-Oton, X., Sire, Y., Valdinoci, E.: A one-dimensional symmetry result for a class of nonlocal semilinear equations in the plane. Ann. Inst. H. Poincaré Anal. Non Linéaire (2016). doi:10.1016/j.anihpc.2016.01.001

  15. Heinonen, J., Kilpeläinen, T., Martio, O.: Nonlinear Potential Theory of Degenerate Elliptic Equations. Dover Publications Inc., Mineola (2006)

    MATH  Google Scholar 

  16. Kassmann, M.: Harnack inequalities and Hölder regularity estimates for nonlocal operators revisited. Fakultät für Mathematik, Univ. Bielefeld Preprint 11015 (2011). http://www.math.uni-bielefeld.de/sfb701/preprints/view/523

  17. Korvenpää, J., Kuusi, T., Lindgren, E.: Equivalence of solutions to fractional \(p\)-Laplace type equations. J. Math. Pures Appl. (2016). (To appear)

  18. Korvenpää, J., Kuusi, T., Palatucci, G.: Hölder continuity up to the boundary for a class of fractional obstacle problems. Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl.27, 355–367 (2016)

  19. Korvenpää, J., Kuusi, T., Palatucci, G.: The obstacle problem for nonlinear integro-differential operators. Calc. Var. Partial Differ. Equ. 55(3), Art. 63 (2016)

  20. Korvenpää, J., Kuusi, T., Palatucci, G.: A note on fractional supersolutions. Electron. J. Differ. Equ. 2016(263), 1–9 (2016)

    MathSciNet  MATH  Google Scholar 

  21. Kuusi, T., Mingione, G., Sire, Y.: A fractional Gehring lemma, with applications to nonlocal equations. Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 25(4), 345–358 (2014)

  22. Kuusi, T., Mingione, G., Sire, Y.: Nonlocal self-improving properties. Anal. PDE 8(1), 57–114 (2015)

  23. Kuusi, T., Mingione, G., Sire, Y.: Nonlocal equations with measure data. Commun. Math. Phys. 337(3), 1317–1368 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  24. Iannizzotto, A., Mosconi, S., Squassina, M.: Global Hölder regularity for the fractional \(p\)-Laplacian. Rev. Mat. Iberoamericana (2017). https://www.researchgate.net/publication/268150481. (To appear)

  25. Iannizzotto, A., Squassina, M.: Weyl-type laws for fractional \(p\)-eigenvalue problems. Asympt. Anal. 88, 233–245 (2014)

    MathSciNet  MATH  Google Scholar 

  26. Leonori, T., Peral, I., Primo, A., Soria, F.: Basic estimates for solutions of a class of nonlocal elliptic and parabolic equations. Discr. Cont. Dyn. Syst. 35(12), 6031–6068 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  27. Lindgren, E.: Hölder estimates for viscosity solutions of equations of fractional \(p\)-Laplace type (2015). http://arxiv.org/pdf/1405.6612. (Preprint)

  28. Lindgren, E., Lindqvist, P.: Fractional eigenvalues. Calc. Var. Partial Differ. Equ. 49(1–2), 795–826 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  29. Lindgren, E., Lindqvist, P.: Perron’s method and Wiener’s theorem for a nonlocal equation. Potential Anal. (2016). http://arxiv.org/abs/1603.09184. (To appear)

  30. Lindqvist, P.: On the definition and properties of \(p\)-superharmonic functions. J. Reine Angew. Math. (Crelles J.) 365, 67–79 (1986)

    MathSciNet  MATH  Google Scholar 

  31. Lindqvist, P.: Notes on the \(p\)-Laplace equation. Rep. Univ. Jyvaskyla Dep. Math. Stat. 102 (2006)

  32. Mingione, G.: The Calderón–Zygmund theory for elliptic problems with measure data. Ann. Sc. Norm. Super. Pisa Cl. Sci. 6, 195–261 (2007)

  33. Mingione, G.: Gradient potential estimates. J. Eur. Math. Soc. 13, 459–486 (2011)

    MathSciNet  MATH  Google Scholar 

  34. Molica Bisci, G., Radulescu, V.D., Servadei, R.: Variational Methods for Nonlocal Fractional Equations. Cambridge University Press, Cambridge (2016)

  35. Palatucci, G., Pisante, A.: Improved Sobolev embeddings, profile decomposition, and concentration-compactness for fractional Sobolev spaces. Calc. Var. Partial Differ. Equ. 50(3–4), 799–829 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  36. Palatucci, G., Pisante, A.: A global compactness type result for Palais–Smale sequences in fractional Sobolev spaces. Nonlinear Anal. 117, 1–7 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  37. Palatucci, G., Savin, O., Valdinoci, E.: Local and global minimizers for a variational energy involving a fractional norm. Ann. Mat. Pura Appl. 192(4), 673–718 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  38. Salsa, S.: The problems of the obstacle in lower dimension and for the fractional Laplacian. Regularity estimates for nonlinear elliptic and parabolic problems. In: Lecture Notes in Mathematics, vol. 2045, pp. 153–244. Springer, Heidelberg (2012)

  39. Savin, O., Valdinoci, E.: Density estimates for a variational model driven by the Gagliardo norm. J. Math. Pures Appl. 101(1), 1–26 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  40. Schikorra, A.: Nonlinear commutators for the fractional \(p\)-Laplacian and applications. Math. Ann. 366(1), 695–720 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  41. Servadei, R., Valdinoci, E.: Weak and viscosity solutions of the fractional Laplace equation. Publ. Mat. 58(1), 133–154 (2014)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

This paper was partially carried out while Giampiero Palatucci was visiting the Department of Mathematics and Systems Analysis at Aalto University School of Science in Helsinki, supported by the Academy of Finland. The authors would like to thank Professor Juha Kinnunen for the hospitality and the stimulating discussions. A special thank also to Agnese Di Castro for her useful observations on a preliminary version of this paper. The authors would like to thank Erik Lindgren, who has kindly informed us of his paper [29] in collaboration with Peter Lindqvist, where they deal with a general class of fractional Laplace equations with bounded boundary data, in the case when the operators \(\mathcal {L}\) in (2) does reduce to the pure fractional p-Laplacian \((-\Delta )^s_p\) without coefficients. This very relevant paper contains several important results, as a fractional Perron method and a Wiener resolutivity theorem, together with the subsequent classification of the regular points, in such a nonlinear fractional framework. It could be interesting to compare those results together with the ones presented here. Finally, the authors would like to thank the referees for their useful suggestions, which allowed to improve the manuscript.

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Authors and Affiliations

  1. Department of Mathematics and Systems Analysis, Aalto University, P.O. Box 1100, 00076, Aalto, Finland

    Janne Korvenpää & Tuomo Kuusi

  2. Dipartimento di Matematica e Informatica, Università degli Studi di Parma, Campus-Parco Area delle Scienze, 53/A, 43124, Parma, Italy

    Giampiero Palatucci

Authors
  1. Janne Korvenpää
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  2. Tuomo Kuusi
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  3. Giampiero Palatucci
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Corresponding author

Correspondence to Tuomo Kuusi.

Additional information

Communicated by Y. Giga.

J. Korvenpää has been supported by the Magnus Ehrnrooth Foundation (Grant No. ma2014n1, ma2015n3). T. Kuusi has been supported by the Academy of Finland. G. Palatucci is a member of Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of Istituto Nazionale di Alta Matematica “F. Severi” (INdAM), whose support is acknowledged.

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Korvenpää, J., Kuusi, T. & Palatucci, G. Fractional superharmonic functions and the Perron method for nonlinear integro-differential equations. Math. Ann. 369, 1443–1489 (2017). https://doi.org/10.1007/s00208-016-1495-x

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