Skip to main content
SpringerLink
Log in

New strong maximum and comparison principles for fully nonlinear degenerate elliptic PDEs

  • Published:
Calculus of Variations and Partial Differential Equations Aims and scope Submit manuscript

Abstract

We introduce a notion of subunit vector field for fully nonlinear degenerate elliptic equations. We prove that an interior maximum of a viscosity subsolution of such an equation propagates along the trajectories of subunit vector fields. This implies strong maximum and minimum principles when the operator has a family of subunit vector fields satisfying the Hörmander condition. In particular these results hold for a large class of nonlinear subelliptic PDEs in Carnot groups. We prove also a strong comparison principle for degenerate elliptic equations that can be written in Hamilton–Jacobi–Bellman form, such as those involving the Pucci’s extremal operators over Hörmander vector fields.

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. Bardi, M., Capuzzo-Dolcetta, I.: Optimal Control and Viscosity Solutions of Hamilton–Jacobi–Bellman Equations. Birkhäuser Boston Inc, Boston, MA (1997)

    Book  Google Scholar 

  2. Bardi, M., Cesaroni, A.: Liouville properties and critical value of fully nonlinear elliptic operators. J. Differ. Equ. 261(7), 3775–3799 (2016)

    Article  MathSciNet  Google Scholar 

  3. Bardi, M., Da Lio, F.: On the strong maximum principle for fully nonlinear degenerate elliptic equations. Arch. Math. (Basel) 73(4), 276–285 (1999)

    Article  MathSciNet  Google Scholar 

  4. Bardi, M., Da Lio, F.: Propagation of maxima and strong maximum principle for viscosity solutions of degenerate elliptic equations. I. Convex operators. Nonlinear Anal. 44(8, Ser. A: Theory Methods), 991–1006 (2001)

    Article  MathSciNet  Google Scholar 

  5. Bardi, M., Da Lio, F.: Propagation of maxima and strong maximum principle for viscosity solutions of degenerate elliptic equations. II. Concave operators. Indiana Univ. Math. J. 52(3), 607–627 (2003)

    Article  MathSciNet  Google Scholar 

  6. Bardi, M., Mannucci, P.: On the Dirichlet problem for non-totally degenerate fully nonlinear elliptic equations. Commun. Pure Appl. Anal. 5(4), 709–731 (2006)

    Article  MathSciNet  Google Scholar 

  7. Barles, G., Busca, J.: Existence and comparison results for fully nonlinear degenerate elliptic equations without zeroth-order term. Commun. Partial Differ. Equ. 26(11–12), 2323–2337 (2001)

    Article  MathSciNet  Google Scholar 

  8. Beatrous, F.H., Bieske, T.J., Manfredi, J.J.: The maximum principle for vector fields. In: The p-Harmonic Equation and Recent Advances in Analysis, volume 370 of Contemp. Math., pp. 1–9. Amer. Math. Soc., Providence, RI (2005)

  9. Bieske, T.: On \(\infty \)-harmonic functions on the Heisenberg group. Commun. Partial Differ. Equ. 27(3–4), 727–761 (2002)

    Article  MathSciNet  Google Scholar 

  10. Bieske, T.: A sub-Riemannian maximum principle and its application to the \(p\)-Laplacian in Carnot groups. Ann. Acad. Sci. Fenn. Math. 37(1), 119–134 (2012)

    Article  MathSciNet  Google Scholar 

  11. Bieske, T., Capogna, L.: The Aronsson-Euler equation for absolutely minimizing Lipschitz extensions with respect to Carnot–Carathéodory metrics. Trans. Am. Math. Soc. 357(2), 795–823 (2005)

    Article  Google Scholar 

  12. Bieske, T., Martin, E.: The parabolic infinite-Laplace equation in Carnot groups. Michigan Math. J. 65(3), 489–509 (2016)

    Article  MathSciNet  Google Scholar 

  13. Bonfiglioli, A., Lanconelli, E., Uguzzoni, F.: Stratified Lie Groups and Potential Theory for Their Sub-Laplacians. Springer, Berlin (2007)

    MATH  Google Scholar 

  14. Bony, J.-M.: Principe du maximum, inégalite de Harnack et unicité du problème de Cauchy pour les opérateurs elliptiques dégénérés. Ann. Inst. Fourier (Grenoble) 19(fasc. 1), 277–304 xii (1969)

    Article  MathSciNet  Google Scholar 

  15. Caffarelli, L.A., Cabré, X.: Fully Nonlinear Elliptic Equations. American Mathematical Society, Providence, RI (1995)

    Book  Google Scholar 

  16. Capogna, L., Zhou, X.: Strong comparison principle for \(p\)-harmonic functions in Carnot-Caratheodory spaces. Proc. Am. Math. Soc. 146(10), 4265–4274 (2018)

    Article  MathSciNet  Google Scholar 

  17. 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 

  18. Fefferman, C., Phong, D.H.: Subelliptic eigenvalue problems. In: Conference on Harmonic Analysis in Honor of Antoni Zygmund, Vol. I, II, Chicago, Ill., 1981, pp. 590–606. Wadsworth, Belmont, CA (1983)

  19. Feleqi, E., Rampazzo, F.: Iterated Lie brackets for nonsmooth vector fields. NoDEA Nonlinear Differ. Equ. Appl. 24(6), 61 (2017)

    Article  MathSciNet  Google Scholar 

  20. Giga, Y., Ohnuma, M.: On strong comparison principle for semicontinuous viscosity solutions of some nonlinear elliptic equations. Int. J. Pure Appl. Math. 22(2), 165–184 (2005)

    MathSciNet  MATH  Google Scholar 

  21. Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order, 2nd edn. Springer, Berlin (1983)

    Book  Google Scholar 

  22. Goffi, A.: Topics in nonlinear PDEs: from Mean Field Games to problems modeled on Hörmander vector fields. PhD thesis, Gran Sasso Science Institute (2019)

  23. Jensen, R.: Uniqueness criteria for viscosity solutions of fully nonlinear elliptic partial differential equations. Indiana Univ. Math. J. 38(3), 629–667 (1989)

    Article  MathSciNet  Google Scholar 

  24. Kawohl, B., Kutev, N.: Strong maximum principle for semicontinuous viscosity solutions of nonlinear partial differential equations. Arch. Math. (Basel) 70(6), 470–478 (1998)

    Article  MathSciNet  Google Scholar 

  25. 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 

  26. Li, Y., Wang, B.: Strong comparison principles for some nonlinear degenerate elliptic equations. Acta Math. Sci. Ser. B (Engl. Ed.) 38(5), 1583–1590 (2018)

    MathSciNet  MATH  Google Scholar 

  27. Manfredi, J.J.: Nonlinear subelliptic equations on Carnot groups. Notes of a course given at the third school on analysis and geometry in metric spaces. http://www.pitt.edu/~manfredi/papers/fullynonlsubtrentofinal.pdf (2003)

  28. Montanari, A., Lanconelli, E.: Pseudoconvex fully nonlinear partial differential operators: strong comparison theorems. J. Differ. Equ. 202(2), 306–331 (2004)

    Article  MathSciNet  Google Scholar 

  29. Stroock, D.W., Varadhan, S.R.S.: On the support of diffusion processes with applications to the strong maximum principle. In: Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971), Vol. III: Probability theory, pp. 333–359. University of California Press, Berkeley, CA (1972)

  30. Taira, K.: Diffusion Processes and Partial Differential Equations. Academic Press Inc, Boston, MA (1988)

    MATH  Google Scholar 

  31. Trudinger, N.S.: Comparison principles and pointwise estimates for viscosity solutions of nonlinear elliptic equations. Rev. Mat. Iberoam. 4(3–4), 453–468 (1988)

    Article  MathSciNet  Google Scholar 

  32. Wang, C.: The Aronsson equation for absolute minimizers of \(L^\infty \)-functionals associated with vector fields satisfying Hörmander’s condition. Trans. Am. Math. Soc. 359(1), 91–113 (2007)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics “T. Levi-Civita”, University of Padova, Via Trieste 63, 35121, Padova, Italy

    Martino Bardi

  2. Gran Sasso Science Institute, Viale Francesco Crispi 7, 67100, L’Aquila, Italy

    Alessandro Goffi

Authors
  1. Martino Bardi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Alessandro Goffi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Martino Bardi.

Additional information

Communicated by L. Ambrosio.

Publisher's Note

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

The authors are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). The first-named author was partially supported by the research projects “Mean-Field Games and Nonlinear PDEs” of the University of Padova, and “Nonlinear Partial Differential Equations: Asymptotic Problems and Mean-Field Games” of the Fondazione CaRiPaRo. The second-named author wishes to thank the Department of Mathematics of the University of Padova for the hospitality during the preparation of this paper.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bardi, M., Goffi, A. New strong maximum and comparison principles for fully nonlinear degenerate elliptic PDEs. Calc. Var. 58, 184 (2019). https://doi.org/10.1007/s00526-019-1620-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00526-019-1620-2

Mathematics Subject Classification

Search

Navigation