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On Some Degenerate Elliptic Equations Arising in Geometric Problems

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Abstract

We consider some fully nonlinear degenerate elliptic operators and we investigate the validity of certain properties related to the maximum principle. In particular, we establish the equivalence between the sign propagation property and the strict positivity of a suitably defined generalized principal eigenvalue. Furthermore, we show that even in the degenerate case considered in the present paper, the well-known condition introduced by Keller–Osserman on the zero-order term is necessary and sufficient for the existence of entire weak subsolutions.

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References

  1. L. Ambrosio and H.M. Soner, “Level set approach to mean curvature flow in arbitrary codimension,” J. Differ. Geom., 43, 693–737 (1996).

    Article  MathSciNet  MATH  Google Scholar 

  2. M. E. Amendola, G. Galise, and A. Vitolo, “Riesz capacity, maximum principle and removable sets of fully nonlinear second-order elliptic operators,” Differ. Integr. Equ., 26, 845–866 (2013).

    MathSciNet  MATH  Google Scholar 

  3. M. E. Amendola, G. Galise, and A. Vitolo, “On the uniqueness of blow-up solutions of fully nonlinear elliptic equations,” In: Dynamical Systems and Differential Equations, DCDS Suppl., 771–780 (2013).

  4. S. N. Armstrong, “Principal eigenvalues and an anti-maximum principle for homogeneous fully nonlinear elliptic equations,” J. Differ. Equ., 246, No. 7, 2958–2987 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  5. J. Bao and X. Ji, “Necessary and sufficient conditions on solvability for Hessian inequalities,” Proc. Am. Math. Soc., 138, 175–188 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  6. J. Bao and X. Ji, “Existence and nonexistence theorem for entire subsolutions of k-Yamabe type equations,” J. Differ. Equ., 253, 2140–2160 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  7. G. Barles and J. Burdeau, “The Dirichlet problem for semilinear second-order degenerate elliptic equations and applications to stochastic exit time control problems,” Commun. Part. Differ. Equ., 20, No. 1-2, 129–178 (1995).

    Article  MathSciNet  MATH  Google Scholar 

  8. H. Berestycki, I. Capuzzo Dolcetta, A. Porretta, and L. Rossi, “Maximum Principle and generalized principal eigenvalue for degenerate elliptic operators,” J. Math. Pures Appl., https://doi.org/10.1016/j.matpur.2014.10.012 (2014).

  9. H. Berestycki, L. Nirenberg, and S. R. S. Varadhan, “The principal eigenvalue and maximum principle for second-order elliptic operators in general domains,” Commun. Pure Appl. Math., 47, No. 1, 47–92 (1994).

    Article  MathSciNet  MATH  Google Scholar 

  10. H. Berestycki and L. Rossi, “Generalizations and properties of the principal eigenvalue of elliptic operators in unbounded domains,” Commun. Pure Appl. Math., 68, No. 6, 1014–1065 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  11. I. Birindelli and F. Demengel, “First eigenvalue and Maximum Principle for fully nonlinear singular operators,” Adv. Differ. Equ., 11, No. 1, 91–119 (2006).

    MathSciNet  MATH  Google Scholar 

  12. I. Birindelli and F. Demengel, “Eigenvalue, maximum principle and regularity for fully nonlinear homogeneous operators,” Commun. Pure Appl. Anal., 6, No. 2, 335–366 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  13. L. Boccardo, T. Gallouet, and J. L. Vazquez, “Nonlinear elliptic equations in ℝN without growth restriction on the data,” J. Differ. Equ., 105 (2), 334–363 (1993).

    Article  MathSciNet  MATH  Google Scholar 

  14. L. Boccardo, T. Gallouet, and J. L. Vazquez, “Solutions of nonlinear parabolic equations without growth restrictions on the data,” Electron. J. Differ. Equ., 2001 (60), 1–20 (2001).

  15. H. Brezis, “Semilinear equations in ℝn without conditions at infinity,” Appl. Math. Optim., 12, 271–282 (1984).

    Article  MathSciNet  MATH  Google Scholar 

  16. L. A. Caffarelli and X. Cabré, Fully Nonlinear Elliptic Equations, Am. Math. Soc., Providence (1995).

  17. L. A. Caffarelli, Y. Y. Li, and L. Nirenberg, “Some remarks on singular solutions of nonlinear elliptic equations. III: Viscosity solutions, including parabolic operators,” Commun. Pure Appl. Math., 66, 109—143 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  18. P. Cannarsa, G. Da Prato, and H. Frankowska, “Invariant measures associated to degenerate elliptic operators,” Indiana Univ. Math. J., 59, No. 1, 53–78 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  19. I. Capuzzo Dolcetta, F. Leoni, and A. Vitolo, “Entire subsolutions of fully nonlinear degenerate elliptic equations,” Bull. Inst. Math. Acad. Sin. (N.S.), 9, 147–161 (2014).

  20. I. Capuzzo Dolcetta, F. Leoni, and A. Vitolo, “On the inequality F(x, D 2 u) ≥ f(u) + g(u) + g(u)|Du|q ,” Math. Ann., 365, No. 1-2, 423–448 (2016).

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

  22. L. D’Ambrosio and E. Mitidieri, “A priori estimates, positivity results, and nonexistence theorems for quasilinear degenerate elliptic inequalities,” Adv. Math., 224, 967–1020 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  23. G. Diaz, “A note on the Liouville method applied to elliptic eventually degenerate fully nonlinear equations governed by the Pucci operators and the Keller–Osserman condition,” Math. Ann., 353, 145–159 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  24. M. J. Esteban, P. L. Felmer, and A. Quaas, “Superlinear elliptic equations for fully nonlinear operators without growth restrictions for the data,” Proc. Edinb. Math. Soc. (2), 53, No. 1, 125–141 (2010).

  25. G. Fichera, “Sulle equazioni differenziali lineari ellittico-paraboliche del secondo ordine,” Atti Accad. Naz. Lincei. Mem. Cl. Sci. Fis. Mat. Nat. Sez. I. (8), 5, 1–30 (1956).

  26. B. Franchi and E. Lanconelli, “Une métrique associée à une classe d’opérateurs elliptiques dégénérés,” Rend. Semin. Mat. Univ. Politec. Torino, Proc. Conf. on Linear partial and pseudodifferential operators (Torino, 1982), 105–114 (1983).

  27. A. Friedman and M.A. Pinsky, “Asymptotic stability and spiraling properties for solutions of stochastic equations,” Trans. Am. Math. Soc., 186, 331–358 (1973).

    Article  MathSciNet  MATH  Google Scholar 

  28. G. Galise and A. Vitolo, “Viscosity solutions of uniformly elliptic equations without boundary and growth conditions at infinity,” Int. J. Differ. Equ., 2011, 453727 (2011).

  29. Y. Giga, Surface Evolution Equations. A Level Set Approach, Birkhäuser, Basel (2006).

  30. F. R. Harvey and H.B. Lawson Jr., “Existence, uniqueness and removable singularities for nonlinear partial differential equations in geometry,” In: Surveys in Differential Geometry, 18, 102–156, International Press, Somerville (2013).

  31. F.R. Harvey and H.B. Lawson Jr., “Removable singularities for nonlinear subequations,” Indiana Univ. Math. J., 63, 1525–1552 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  32. F.R. Harvey and H.B. Lawson Jr., “Characterizing the strong maximum principle,” Preprint, http://arxiv-web3.library.cornell.edu/abs/1309.1738 (2014).

  33. N.K. Hayman and P.B. Kennedy, Subharmonic Functions. Vol. I, Academic Press, London (1976).

  34. N. Ikoma and H. Ishii, “Eigenvalue problem for fully nonlinear second-order elliptic PDE on balls,” Ann. Inst. H. Poincar´e Anal. Non Linéaire, 29, 783–812 (2012).

  35. H. Ishii, “Perron’s method for Hamilton–Jacobi equations,” Duke Math. J., 55, No. 2, 369–384 (1987).

    Article  MathSciNet  MATH  Google Scholar 

  36. Q. Jin, Y. Y. Li, and H. Xu, “Nonexistence of positive solutions for some fully nonlinear elliptic equations,” Methods Appl. Anal., 12, 441–449 (2005).

    MathSciNet  MATH  Google Scholar 

  37. J. B. Keller, “On solutions of Δu = f(u),” Commun. Pure Appl. Math., 10, 503–510 (1957).

  38. N. S. Landkof, Foundations of Modern Potential Theory, Springer-Verlag, Heidelberg–New York (1972).

  39. F. Leoni, “Nonlinear elliptic equations in ℝN with absorbing zero order terms,” Adv. Differ. Equ., 5, 681–722 (2000).

    MathSciNet  MATH  Google Scholar 

  40. F. Leoni and B. Pellacci, “Local estimates and global existence for strongly nonlinear parabolic equations with locally integrable data,” J. Evol. Equ., 6, 113–144 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  41. P. L. Lions, “Bifurcation and optimal stochastic control,” Nonlinear Anal., 7, No. 2, 177–207 (1983).

    Article  MathSciNet  Google Scholar 

  42. A. Oberman and L. Silvestre, “The Dirichlet problem for the convex envelope,” Trans. Am. Math. Soc., 363 (11), 5871–5886 (2011).

  43. O.A. Oleĭnik and E.V. Radkevič, Second Order Equations with Nonnegative Characteristic Form, Plenum Press, New York (1973).

  44. R. Osserman, “On the inequality Δu ≥ f(u),” Pacific J. Math., 7, 1141–1147 (1957).

    Article  Google Scholar 

  45. M. H. Protter and H. F.Weinberger, Maximum Principles in Differential Equations, Prentice-Hall, Englewood Cliffs (1967).

  46. A. Quaas and B. Sirakov, “Principal eigenvalues and the Dirichlet problem for fully nonlinear elliptic operators,” Adv. Math., 218 (2008).

  47. J.-P. Sha, “p-convex Riemannian manifolds,” Invent. Math., 83, 437–447 (1986).

    Article  MathSciNet  MATH  Google Scholar 

  48. K. Suzuki, “The first boundary value and eigenvalue problems for degenerate elliptic equations,” Publ. Res. Inst. Math. Sci. Kyoto Univ. Ser. A, 4 (1), 179–200 (1968).

    Article  MathSciNet  MATH  Google Scholar 

  49. H. Wu, “Manifolds of partially positive curvature,” Indiana Univ. Math. J., 36, 525–548 (1987).

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Dipartimento di Matematica, Sapienza Università di Roma, Roma, Italy

    I. Capuzzo Dolcetta & F. Leoni

  2. Dipartimento di Matematica, Università di Salerno, Salerno, Italy

    A. Vitolo

Authors
  1. I. Capuzzo Dolcetta
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  2. F. Leoni
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  3. A. Vitolo
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Correspondence to I. Capuzzo Dolcetta.

Additional information

Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 58, Proceedings of the Seventh International Conference on Differential and Functional Differential Equations and InternationalWorkshop “Spatio-Temporal Dynamical Systems” (Moscow, Russia, 22–29 August, 2014). Part 1, 2015.

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Capuzzo Dolcetta, I., Leoni, F. & Vitolo, A. On Some Degenerate Elliptic Equations Arising in Geometric Problems. J Math Sci 233, 446–461 (2018). https://doi.org/10.1007/s10958-018-3937-3

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