Skip to main content
SpringerLink
Log in

Global regularity results for a class of singular/degenerate fully nonlinear elliptic equations

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

Abstract

We provide the Alexandroff–Bakelman–Pucci estimate and global \(C^{1, \alpha }\)-regularity for a class of singular/degenerate fully nonlinear elliptic equations. We also derive the existence of a viscosity solution to the Dirichlet problem with the associated operator.

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

Data availability

Data sharing is not applicable to this article as no data sets were generated or analysed.

References

  1. Aikawa, H., Kilpeläinen, T., Shanmugalingam, N., Zhong, X.: Boundary Harnack principle for \(p\)-harmonic functions in smooth Euclidean domains. Potential Anal. 26(3), 281–301 (2007)

    Article  MathSciNet  Google Scholar 

  2. Araújo, D.J., Ricarte, G.C., Teixeira, E.V.: Geometric gradient estimates for solutions to degenerate elliptic equations. Calc. Var. Partial Differ. Equ. 53(3–4), 605–625 (2015)

    Article  MathSciNet  Google Scholar 

  3. Araújo, D.J., Ricarte, G.C., Teixeira, E.V.: Singularly perturbed equations of degenerate type. Ann. Inst. H. Poincaré C Anal. Non Linéaire 34(3), 655–678 (2017)

    Article  MathSciNet  Google Scholar 

  4. Araújo, D.J., Sirakov, B.: Sharp boundary and global regularity for degenerate fully nonlinear elliptic equations. J. Math. Pures Appl. 9(169), 138–154 (2023)

    Article  MathSciNet  Google Scholar 

  5. Baasandorj, S., Byun, S.-S., Lee, K.-A., Lee, S.-C.: \(C^{1, \alpha }\)-regularity for a class of degenerate/singular fully nonlinear elliptic equations. Interfaces Free Bound. arXiv:2209.14581

  6. Bezerra Júnior, E.C., da Silva, J.V., Rampasso, G.C., Ricarte, G.C.: Global regularity for a class of fully nonlinear PDEs with unbalanced variable degeneracy. J. Lond. Math. Soc. (2) 108(2), 622–665 (2023)

    Article  MathSciNet  Google Scholar 

  7. Bezerra Júnior, E.C., da Silva, J.V., Ricarte, G.C.: Fully nonlinear singularly perturbed models with non-homogeneous degeneracy. Rev. Mat. Iberoam. 39(1), 123–164 (2023)

    Article  MathSciNet  Google Scholar 

  8. Birindelli, I., Demengel, F.: Comparison principle and Liouville type results for singular fully nonlinear operators. Ann. Fac. Sci. Toulouse Math. (6) 13(2), 261–287 (2004)

    Article  MathSciNet  Google Scholar 

  9. Birindelli, I., Demengel, F.: Regularity and uniqueness of the first eigenfunction for singular fully nonlinear operators. J. Differ. Equ. 249(5), 1089–1110 (2010)

    Article  MathSciNet  Google Scholar 

  10. Birindelli, I., Demengel, F.: \(C^{1,\beta }\) regularity for Dirichlet problems associated to fully nonlinear degenerate elliptic equations. ESAIM Control Optim. Calc. Var. 20(4), 1009–1024 (2014)

    Article  MathSciNet  Google Scholar 

  11. Bronzi, A.C., Pimentel, E.A., Rampasso, G.C., Teixeira, E.V.: Regularity of solutions to a class of variable-exponent fully nonlinear elliptic equations. J. Funct. Anal. 279(12), 108781 (2020)

    Article  MathSciNet  Google Scholar 

  12. Caffarelli, L.A., Cabré, X.: Fully Nonlinear Elliptic Equations. American Mathematical Society Colloquium Publications, vol. 43. American Mathematical Society, Providence (1995)

  13. Crandall, M.G., Ishii, H., Lions, P.-L.: User’s guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc. (N.S.) 27(1), 1–67 (1992)

    Article  MathSciNet  Google Scholar 

  14. da Silva, J.V., Rampasso, G.C., Ricarte, G.C., Vivas, H.A.: Free boundary regularity for a class of one-phase problems with non-homogeneous degeneracy. Isr. J. Math. 254(1), 155–200 (2023)

    Article  MathSciNet  Google Scholar 

  15. da Silva, J.V., Ricarte, G.C.: Geometric gradient estimates for fully nonlinear models with non-homogeneous degeneracy and applications. Calc. Var. Partial Differ. Equ. 59(5), 33 (2020). (Paper No. 161)

    Article  MathSciNet  Google Scholar 

  16. da Silva, J.V., Vivas, H.: The obstacle problem for a class of degenerate fully nonlinear operators. Rev. Mat. Iberoam. 37(5), 1991–2020 (2021)

    Article  MathSciNet  Google Scholar 

  17. da Silva, J.V., Vivas, H.: Sharp regularity for degenerate obstacle type problems: a geometric approach. Discrete Contin. Dyn. Syst. 41(3), 1359–1385 (2021)

    Article  MathSciNet  Google Scholar 

  18. Dávila, G., Felmer, P., Quaas, A.: Alexandroff–Bakelman–Pucci estimate for singular or degenerate fully nonlinear elliptic equations. C. R. Math. Acad. Sci. Paris 347(19–20), 1165–1168 (2009)

    Article  MathSciNet  Google Scholar 

  19. Dávila, G., Felmer, P., Quaas, A.: Harnack inequality for singular fully nonlinear operators and some existence results. Calc. Var. Partial Differ. Equ. 39(3–4), 557–578 (2010)

    Article  MathSciNet  Google Scholar 

  20. De Filippis, C.: Regularity for solutions of fully nonlinear elliptic equations with nonhomogeneous degeneracy. Proc. R. Soc. Edinb. Sect. A 151(1), 110–132 (2021)

    Article  MathSciNet  Google Scholar 

  21. Evans, L.C.: Classical solutions of fully nonlinear, convex, second-order elliptic equations. Commun. Pure Appl. Math. 35(3), 333–363 (1982)

    Article  MathSciNet  Google Scholar 

  22. Fang, Y., Rădulescu, V.D., Zhang, C.: Regularity of solutions to degenerate fully nonlinear elliptic equations with variable exponent. Bull. Lond. Math. Soc. 53(6), 1863–1878 (2021)

    Article  MathSciNet  Google Scholar 

  23. Gilbarg, D., Trudinger, N. S.: Elliptic partial differential equations of second order, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, 2nd ed. Springer, Berlin (1983)

  24. Harjulehto, P., Hästö, P.: Orlicz Spaces and Generalized Orlicz Spaces. Lecture Notes in Mathematics, vol. 2236. Springer, Cham (2019)

  25. Imbert, C.: Alexandroff–Bakelman–Pucci estimate and Harnack inequality for degenerate/singular fully non-linear elliptic equations. J. Differ. Equ. 250(3), 1553–1574 (2011)

    Article  MathSciNet  Google Scholar 

  26. Imbert, C., Silvestre, L.: \(C^{1,\alpha }\) regularity of solutions of some degenerate fully non-linear elliptic equations. Adv. Math. 233, 196–206 (2013)

    Article  MathSciNet  Google Scholar 

  27. Junges Miotto, T.: The Aleksandrov–Bakelman–Pucci estimates for singular fully nonlinear operators. Commun. Contemp. Math. 12(4), 607–627 (2010)

    Article  MathSciNet  Google Scholar 

  28. Kawohl, B., Kutev, N.: Comparison principle and Lipschitz regularity for viscosity solutions of some classes of nonlinear partial differential equations. Funkcial. Ekvac. 43(2), 241–253 (2000)

    MathSciNet  Google Scholar 

  29. Kawohl, B., Kutev, N.: Comparison principle for viscosity solutions of fully nonlinear, degenerate elliptic equations. Commun. Partial Differ. Equ. 32(7–9), 1209–1224 (2007)

    Article  MathSciNet  Google Scholar 

  30. Krylov, N.V.: Boundedly inhomogeneous elliptic and parabolic equations. Izv. Akad. Nauk SSSR Ser. Mat. 46(3), 487–523 (1982)

    MathSciNet  Google Scholar 

  31. Krylov, N.V.: Boundedly inhomogeneous elliptic and parabolic equations in a domain. Izv. Akad. Nauk SSSR Ser. Mat. 47(1), 75–108 (1983)

    MathSciNet  Google Scholar 

  32. Krylov, N.V., Safonov, M.V.: An estimate for the probability of a diffusion process hitting a set of positive measure. Dokl. Akad. Nauk SSSR 245(1), 18–20 (1979)

    MathSciNet  Google Scholar 

  33. Krylov, N.V., Safonov, M.V.: A property of the solutions of parabolic equations with measurable coefficients. Izv. Akad. Nauk SSSR Ser. Mat. 44(1), 161–175 (1980)

    MathSciNet  Google Scholar 

  34. Milakis, E., Silvestre, L.E.: Regularity for fully nonlinear elliptic equations with Neumann boundary data. Commun. Partial Differ. Equ. 31(7–9), 1227–1252 (2006)

    Article  MathSciNet  Google Scholar 

  35. Winter, N.: \(W^{2, p}\) and \(W^{1, p}\)-estimates at the boundary for solutions of fully nonlinear, uniformly elliptic equations. Z. Anal. Anwend. 28(2), 129–164 (2009)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors would like to express their sincere gratitude to the anonymous referees who provided valuable comments and suggestions on the earlier version, which improved the clarity and understanding of the manuscript.

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, Seoul National University, Seoul, 08826, South Korea

    Sumiya Baasandorj

  2. Department of Mathematical Sciences, Research Institute of Mathematics, Seoul National University, Seoul, 08826, South Korea

    Sun-Sig Byun & Ki-Ahm Lee

  3. Research Institute of Mathematics, Seoul National University, Seoul, 08826, South Korea

    Se-Chan Lee

Authors
  1. Sumiya Baasandorj
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sun-Sig Byun
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ki-Ahm Lee
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Se-Chan Lee
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Se-Chan Lee.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

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

Sumiya Baasandorj was supported by NRF-2022R1A2C1009312. Sun-Sig Byun and Ki-Ahm Lee were supported by NRF-2021R1A4A1027378. Se-Chan Lee was supported by NRF-2022R1A6A3A01086546.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Baasandorj, S., Byun, SS., Lee, KA. et al. Global regularity results for a class of singular/degenerate fully nonlinear elliptic equations. Math. Z. 306, 1 (2024). https://doi.org/10.1007/s00209-023-03400-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00209-023-03400-9

Keywords

Mathematics Subject Classification

Search

Navigation