Potential Analysis

, Volume 48, Issue 3, pp 325–335 | Cite as

Nonlinear Elliptic Equations with Mixed Singularities

Article

Abstract

We study non-variational degenerate elliptic equations with mixed singular structures, both at the set of critical points and on the ground touching points. No boundary data are imposed and singularities occur along an a priori unknown interior region. We prove that positive solutions have a universal modulus of continuity that does not depend on their infimum value. We further obtain sharp, quantitative regularity estimates for non-negative limiting solutions.

Keywords

Singular PDEs Regularity theory 

Mathematics Subject Classification (2010)

35B65 35J60 

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References

  1. 1.
    Alt, H., Caffarelli, L.: Existence and regularity for a minimum problem with free boundary. J. Reine Angew. Math. 325, 105–144 (1981)MathSciNetMATHGoogle Scholar
  2. 2.
    Alt, H., Phillips, D.: A free boundary problem for semilinear elliptic equations. J Reine Angew. Math. 368, 63–107 (1986)MathSciNetMATHGoogle Scholar
  3. 3.
    Araújo, D., Ricarte, G., Teixeira, E.: Geometric gradient estimates for solutions to degenerate elliptic equations. Calc. Var Partial Differential Equations 53 (3–4), 605–625 (2015)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Araújo, D., Teixeira, E.: Geometric approach to singular elliptic equations. Arch. Ration. Mech. Anal. 3, 1019–1054 (2013)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Barles, G., Chasseigne, E., Imbert, C.: hölder continuity of solutions of second-order non-linear elliptic integro-differential equations. J. Eur. Math. Soc. (JEMS) 13(1), 1–26 (2011)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    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)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Caffarelli, L.: The regularity of free boundaries in higher dimensions. Acta Math. 139(3–4), 155–184 (1977)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Caffarelli, L., Kinderlehrer, D.: Potential methods in variational inequalities. J. Analyse Math. 37, 285–295 (1980)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Crandall, M., Ishii, H., Lions, P.-L.: User’s guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. (N.S.) 27(1), 1–67 (1992)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Crandall, M.G., Rabinowitz, P.H., Tartar, L.: On a Dirichlet problem with a singular nonlinearity. Comm. Partial Differential Equations 2(2), 193–222 (1977)MathSciNetCrossRefMATHGoogle Scholar
  11. 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(19–20), 1165–1168 (2009)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    dos Prazeres, D., Teixeira, E.: Cavity problems in discontinuous media. Calc. Var. Partial Differential Equations 55(1), 55:10 (2016)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Imbert, C., Silvestre, S.: C 1,a regularity of solutions of some degenerate fully non-linear elliptic equations. Adv. Math. 233, 196–206 (2013)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Imbert, C., Silvestre, S.: Estimates on elliptic equations that hold only where the gradient is large. To appear in J. Eur. Math. Soc. (JEMS)Google Scholar
  15. 15.
    Ishii, H., Lions, P.L.: Viscosity solutions of fully nonlinear second-order elliptic partial differential equations. J. Differ. Equ. 83(1), 26–78 (1990)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Ricarte, G., Teixeira, E.: Fully nonlinear singularly perturbed equations and asymptotic free boundaries. J. Funct. Anal. 261(6), 1624–1673 (2011)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Teixeira, E.: Universal moduli of continuity for solutions to fully nonlinear elliptic equations. Arch. Ration. Mech. Anal. 211(3), 911–927 (2014)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Teixeira, E.: Hessian continuity at degenerate points in nonvariational elliptic problems. Int. Math. Res. Not. 2015, 6893–6906 (2015)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Teixeira, E.: Regularity for the fully nonlinear dead-core problem. Math Ann. 364(3–4), 1121–1134 (2016)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2017

Authors and Affiliations

  1. 1.University of Central FloridaOrlandoUSA

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