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

  • Martino BardiEmail author
  • Alessandro Goffi


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.

Mathematics Subject Classification

Primary: 35B50 35J70 35J60 Secondary: 49L25 35H20 



  1. 1.
    Bardi, M., Capuzzo-Dolcetta, I.: Optimal Control and Viscosity Solutions of Hamilton–Jacobi–Bellman Equations. Birkhäuser Boston Inc, Boston, MA (1997)CrossRefGoogle Scholar
  2. 2.
    Bardi, M., Cesaroni, A.: Liouville properties and critical value of fully nonlinear elliptic operators. J. Differ. Equ. 261(7), 3775–3799 (2016)MathSciNetCrossRefGoogle Scholar
  3. 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)MathSciNetCrossRefGoogle Scholar
  4. 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)MathSciNetCrossRefGoogle Scholar
  5. 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)MathSciNetCrossRefGoogle Scholar
  6. 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)MathSciNetCrossRefGoogle Scholar
  7. 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)MathSciNetCrossRefGoogle Scholar
  8. 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)Google Scholar
  9. 9.
    Bieske, T.: On \(\infty \)-harmonic functions on the Heisenberg group. Commun. Partial Differ. Equ. 27(3–4), 727–761 (2002)MathSciNetCrossRefGoogle Scholar
  10. 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)MathSciNetCrossRefGoogle Scholar
  11. 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)CrossRefGoogle Scholar
  12. 12.
    Bieske, T., Martin, E.: The parabolic infinite-Laplace equation in Carnot groups. Michigan Math. J. 65(3), 489–509 (2016)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Bonfiglioli, A., Lanconelli, E., Uguzzoni, F.: Stratified Lie Groups and Potential Theory for Their Sub-Laplacians. Springer, Berlin (2007)zbMATHGoogle Scholar
  14. 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)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Caffarelli, L.A., Cabré, X.: Fully Nonlinear Elliptic Equations. American Mathematical Society, Providence, RI (1995)CrossRefGoogle Scholar
  16. 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)MathSciNetCrossRefGoogle Scholar
  17. 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)MathSciNetCrossRefGoogle Scholar
  18. 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)Google Scholar
  19. 19.
    Feleqi, E., Rampazzo, F.: Iterated Lie brackets for nonsmooth vector fields. NoDEA Nonlinear Differ. Equ. Appl. 24(6), 61 (2017)MathSciNetCrossRefGoogle Scholar
  20. 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)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order, 2nd edn. Springer, Berlin (1983)CrossRefGoogle Scholar
  22. 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)Google Scholar
  23. 23.
    Jensen, R.: Uniqueness criteria for viscosity solutions of fully nonlinear elliptic partial differential equations. Indiana Univ. Math. J. 38(3), 629–667 (1989)MathSciNetCrossRefGoogle Scholar
  24. 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)MathSciNetCrossRefGoogle Scholar
  25. 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)MathSciNetCrossRefGoogle Scholar
  26. 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)MathSciNetzbMATHGoogle Scholar
  27. 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. (2003)
  28. 28.
    Montanari, A., Lanconelli, E.: Pseudoconvex fully nonlinear partial differential operators: strong comparison theorems. J. Differ. Equ. 202(2), 306–331 (2004)MathSciNetCrossRefGoogle Scholar
  29. 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)Google Scholar
  30. 30.
    Taira, K.: Diffusion Processes and Partial Differential Equations. Academic Press Inc, Boston, MA (1988)zbMATHGoogle Scholar
  31. 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)MathSciNetCrossRefGoogle Scholar
  32. 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)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mathematics “T. Levi-Civita”University of PadovaPadovaItaly
  2. 2.Gran Sasso Science InstituteL’AquilaItaly

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