Skip to main content
SpringerLink
Log in

Geometric gradient estimates for solutions to degenerate elliptic equations

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

Abstract

We establish new, optimal geometric gradient estimates for solutions to a class of 2nd order partial differential equations, \(\mathcal {L}(X, \nabla u, D^2 u) = f\), whose diffusion properties (ellipticity) degenerate along the a priori unknown set of critical points of an existing solution, \(\mathcal {S}(u) := \{ X : \nabla u(X) = 0 \}\). Such a quantitative information plays a decisive role in the analysis of a number of analytic and geometric problems. The results proven in this work are new even for simple equation \(|\nabla u | \cdot \Delta u = 1\). Under natural nondegeneracy condition on the source term \(f\), we further establish the precise asymptotic behavior of solutions at interior gradient vanishing points. These new estimates are then employed in the study of some well known problems in the theory of elliptic PDEs.

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

    Article  MATH  MathSciNet  Google Scholar 

  2. Birindelli, I., Demengel, F.: Eigenvalue, maximum principle and regularity for fully non linear homogeneous operators. Commun. Pure Appl. Anal. 6, 335–366 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  3. Birindelli, I., Demengel, F.: Eigenvalue and Dirichlet problem for fully-nonlinear operators in non-smooth domains. J. Math. Anal. Appl. 352, 822–835 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  4. Birindelli, I., Demengel, F.: \(C^{1,\beta }\) regularity for Dirichlet problems associated to fully nonlinear degenerate elliptic equations. COCV (to appear, 2104)

  5. Caffarelli, L.A.: Interior a priori estimates for solutions of fully nonlinear equations. Ann. Math. (2) 130(1), 189–213 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  6. Caffarelli, L.A., Cabré, X.: Fully nonlinear elliptic equations. American Mathematical Society Colloquium Publications, vol. 43. American Mathematical Society, Providence (1995)

  7. Caffarelli, L., Salazar, J.: Solutions of fully nonlinear elliptic equations with patches of zero gradient: existence, regularity and convexity of level curves. Trans. Am. Math. Soc. 354(8), 3095–3115 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  8. Caffarelli, L., Salazar, J., Shahgholian, H.: Free-boundary regularity for a problem arising in superconductivity. Arch. Rational. Mech. Anal. 171(1), 115–128 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  9. Chapman, S.J.: A mean-field model of superconducting vortices in three dimensions. SIAM J. App. Math. 55, 1259–1274 (1995)

    Article  MATH  Google Scholar 

  10. 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  MATH  MathSciNet  Google Scholar 

  11. 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, 1165–1168 (2009)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  Google Scholar 

  13. Evans, L.C., Savin, O.: \(C^{1, \alpha }\) regularity for infinity harmonic functions in two dimensions. Calc. Var. Partial Differ. Equ. 32, 325–347 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  14. Evans, L.C., Smart, C.K.: Everywhere differentiability of infinity harmonic functions. Calc. Var. Partial Differ. Equ. 42, 289–299 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  15. Guisti, E.: Direct Methods in the Calculus of Variations. World Scientific Publishing, Singapore (2003)

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  18. Ishii, H.: On uniqueness and existence of viscosity solutions of fully nonlinear second-order elliptic PDE’s. Commun. Pure Appl. Math. 42, 14–45 (1989)

    Article  Google Scholar 

  19. Iwaniec, T., Manfredi, J.: Regularity of p-harmonic functions in the plane. Revista Matematica Iberoamericana 5, 1–19 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  20. Juutinen, P., Lindqvist, P., Manfredi, J.J.: On the equivalence of viscosity solutions and weak solutions for a quasi-linear elliptic equation. SIAM J. Math. Anal. 33, 699–717 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  21. Krylov, N.V.: Boundedly nonhomogeneous elliptic and parabolic equations. Izv. Akad. Nak. SSSR Ser. Mat. 46, 487–523 (1982). English transl. in Math. USSR Izv. 20, 459–492 (1983)

  22. Krylov, N.V.: Boundedly nonhomogeneous elliptic and parabolic equations in a domain. Izv. Akad. Nak. SSSR Ser. Mat. 47, 75–108 (1983). English transl. in Math. USSR Izv. 22, 67–97 (1984)

  23. Lee, K., Shahgholian, H.: Hausdorff dimension and stability for the \(p\)-obstacle problem\((2<p<\infty )\). J. Differ. Equ. 195(1), 14–24 (2003)

  24. Nadirashvili, N., Vladut, S.: Nonclassical solutions of fully nonlinear elliptic equations. Geom. Funct. Anal. 17(4), 1283–1296 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  25. Nadirashvili, N., Vladut, S.: Singular viscosity solutions to fully nonlinear elliptic equations. J. Math. Pure Appl. (9) 89(2), 107–113 (2008)

  26. Nadirashvili, N., Vladut, S.: Nonclassical solutions of fully nonlinear elliptic equations II. Hessian equations and octonions. Geom. Funct. Anal. 21, 483–498 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  27. Rossi, J.D., Teixeira, E.V., Urbano, J.: Miguel optimal regularity at the free boundary for the infinity obstacle problem (Preprint)

  28. Savin, O.: \(C^1\) regularity for infinity harmonic functions in two dimensions. Arch. Rational Mech. Anal. 176, 351–361 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  29. Savin, O.: Small perturbation solutions for elliptic equations. Commun. Partial Differ. Equ. 32(4–6), 557–578 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  30. Savin, O.: Regularity of flat level sets in phase transitions. Ann. Math. (2) 169(1), 41–78 (2009)

  31. Teixeira, E.V.: Sharp regularity for general Poisson equations with borderline sources. J. Math. Pure Appl. (9) 99(2), 150–164 (2013)

  32. Teixeira, E.V.: Universal moduli of continuity for solutions to fully nonlinear elliptic equations. Arch. Rational Mech. Anal. 211(3), 911–927 (2014)

  33. Teixeira, E.V.: Regularity for quasilinear equations on degenerate singular sets. Math. Ann. 358(1), 241–256 (2014)

  34. Uraltseva, N.N.: Degenerate quasilinear elliptic systems. Zap. Na. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 7, 184–222 (1968)

Download references

Acknowledgments

The authors would like to thank Luis Silvestre for insightful and stimulating comments and suggestions. This work has been partially supported by CNPq-Brazil.

Author information

Authors and Affiliations

  1. Department of Mathematics, Universidade Federal Ceará, 60455-760 , Fortaleza, CE, Brazil

    Damião J. Araújo, Gleydson Ricarte & Eduardo V. Teixeira

Authors
  1. Damião J. Araújo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Gleydson Ricarte
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Eduardo V. Teixeira
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Eduardo V. Teixeira.

Additional information

Communicated by O. Savin.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Araújo, D.J., Ricarte, G. & Teixeira, E.V. Geometric gradient estimates for solutions to degenerate elliptic equations. Calc. Var. 53, 605–625 (2015). https://doi.org/10.1007/s00526-014-0760-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00526-014-0760-7

Mathematics Subject Classification

Search

Navigation