Skip to main content
SpringerLink
Log in

Equivalence of weak and viscosity solutions in fractional non-homogeneous problems

  • Published:
Mathematische Annalen Aims and scope Submit manuscript

Abstract

We establish the equivalence between the notions of weak and viscosity solutions for non-homogeneous equations whose main operator is the fractional p-Laplacian and the lower order term depends on x, u and \(D_s^p u\), being the last one a type of fractional derivative.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Adams, R.A.: Sobolev Spaces, Pure and Applied Mathematics, vol.65, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York (1975)

  2. Abdellaoui, B., Fernández, A.J.: Nonlinear fractional Laplacian problems with nonlocal “gradient terms”, to appear in Proc. Royal Soc. Edinburgh Sect. A

  3. Banerjee, A., Dávila, G., Sire, Y.: Regularity for parabolic systems with critical growth in the gradient and applications. arXiv:2005.04004

  4. Barrios, B., Peral, I., Soria, F., Valdinoci, E.: A Widder’s type theorem for the heat equation with nonlocal diffusion. Arch. Ration. Mech. Anal. 213(2), 629–650 (2014)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  6. Bjorland, C., Caffarelli, L., Figalli, A.: Non-local gradient dependent operators. Adv. Math. 230(4–6), 1859–1894 (2012)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  8. Bucur, C.: A symmetry result in R2 for global minimizers of a general type of nonlocal energy. Calc. Var. Partial Differ. Equ. 59(2), 52 (2020)

    Article  Google Scholar 

  9. Bucur, C., Squassina, M.: An asymptotic expansion for the fractional -Laplacian and gradient dependent nonlocal operators. arXiv:2001.09892

  10. Caffarelli, L., Dávila, G.: Interior regularity for fractional systems. Annales de l’Institut Henri Poincaré C, Analyse non linéaire 36(1), 165–180 (2019)

    Article  MathSciNet  Google Scholar 

  11. Comi, G., Stefani, G.: A distributional approach to fractional Sobolev spaces and fractional variation: existence of blow-up. J. Funct. Anal. 277, 3373–3435 (2019)

    Article  MathSciNet  Google Scholar 

  12. Del Pezzo, L.M., Quaas, A.: A Hopf’s lemma and a strong minimum principle for the fractional \(p\)-Laplacian. J. Differ. Equ. 263(1), 765–778 (2017)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  15. Di Nezza, E., Palatucci, G., Valdinoci, E.: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. math. 136(5), 521–573 (2012)

    Article  MathSciNet  Google Scholar 

  16. Iannizzotto, A., Mosconi, S., Squassina, M.: Global Holder regularity for the fractional \(p\)-Laplacian. Rev. Mat. Iberoam 32(4), 1353–1392 (2016)

    Article  MathSciNet  Google Scholar 

  17. Ishii, H., Nakamura, G.: A class of integral equations and approximation of p-Laplace equations. Calc. Var. Partial Differ. Equ. 37(3–4), 485–522 (2010)

    Article  MathSciNet  Google Scholar 

  18. Jarohs, S.: Strong comparison principle for the fractional \(p\)-Laplacian and applications to starshaped rings. Adv. Nonlinear Stud. 18(4), 691–704 (2018)

    Article  MathSciNet  Google Scholar 

  19. Julin, V., Juutinen, P.: A new proof for the equivalence of weak and viscosity solutions for the p-Laplace equation. Commun. PDE 37(5), 934–946 (2012)

    Article  MathSciNet  Google Scholar 

  20. Juutinen, P., Lindqvist, P.: A theorem of Radó’s type for the solutions of a quasi-linear equation. Math. Res. Lett. 11, 31–34 (2004)

    Article  MathSciNet  Google Scholar 

  21. Juutinen, P., Lindqvist, P., Manfredi, J.: On the equivalence of viscosity solutions and weak solutions for a quasilinear equation. SIAM J. Math. Anal. 33(3), 699–717 (2001)

    Article  MathSciNet  Google Scholar 

  22. Korvenpää, J., Kuusi, T., Lindgren, E.: Equivalence of solutions to fractional \(p\)-Laplace type equations. J. Math. Pures Appl. (9), 132 (2019)

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

  24. Korvenpää, J., Kuusi, T., Palatucci, G.: Fractional superharmonic functions and the Perron method for nonlinear integro-differential equations. Math. Ann. 369(3–4), 1443–1489 (2017)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  27. Leonori, T., Porretta, A., Riey, G.: Comparison principles for p-Laplace equations with lower order terms. Annali di Matematica 1–27 (2016)

  28. Lindgren, E., Lindqvist, P.: Fractional eigenvalues. Calc. Var. PDE 49, 795–826 (2014)

    Article  MathSciNet  Google Scholar 

  29. Medina, M., Ochoa, P.: On viscosity and weak solutions for non-homogeneous p-Laplace equations. Adv. Nonlinear Anal. 8, 468–481 (2019)

    Article  MathSciNet  Google Scholar 

  30. Palatucci, G.: The Dirichlet problem for the p-fractional Laplace equation. Nonlinear Anal. 177, 699–732 (2018)

    Article  MathSciNet  Google Scholar 

  31. Pokrovskii, A.V.: Removable singularities of solutions of elliptic equations. J. Math. Sci. 160, 61–83 (2008)

    Article  MathSciNet  Google Scholar 

  32. Pucci, P., Serrin, J.: The Maximum Principle, Progress in Non-linear Differential Equations and their Applications, vol. 73. Birkhäuser, Boston (2007)

    MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  34. Schikorra, A., Shieh, T., Spector, D.: Regularity for a fractional p-Laplace equation. Commun. Contemp. Math. 20 (01) (2018)

Download references

Author information

Authors and Affiliations

  1. Departamento de Análisis Matemático, Universidad de La Laguna, C/Astrofísico Francisco Sánchez s/n, 38271, La Laguna, Spain

    Begoña Barrios

  2. Departamento de Matemáticas, Universidad Autónoma de Madrid, Ciudad Universitaria de Cantoblanco, 28049, Madrid, Spain

    Maria Medina

Authors
  1. Begoña Barrios
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Maria Medina
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Maria Medina.

Ethics declarations

Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.

Additional information

Communicated by Y. Giga.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

B. Barrios was partially supported by the MEC project PGC2018-096422-B-I00 (Spain) and the Ramón y Cajal fellowship RYC2018-026098-I (Spain). M. Medina was supported by the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie Grant Agreement N 754446 and UGR Research and Knowledge Transfer Fund - Athenea3i. M. Medina wants to acknowledge Universidad de la Laguna’s hospitality, where this work was initiated during a visit in July 2019.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Barrios, B., Medina, M. Equivalence of weak and viscosity solutions in fractional non-homogeneous problems. Math. Ann. 381, 1979–2012 (2021). https://doi.org/10.1007/s00208-020-02119-w

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00208-020-02119-w

Search

Navigation