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Asymptotic Methods in Regularity Theory for Nonlinear Elliptic Equations: A Survey

  • Edgard A. Pimentel
  • Makson S. Santos
Chapter
Part of the Springer INdAM Series book series (SINDAMS, volume 28)

Abstract

We survey recent asymptotic methods introduced in regularity theory for fully nonlinear elliptic equations. Our presentation focuses mainly on the recession function. We detail the role of this class of techniques through examples and results. Our applications include regularity in Sobolev and Hölder spaces. In addition, we produce a density result and examine ellipticity-invariant quantities, such as the Escauriaza’s exponent.

Keywords

Fully nonlinear elliptic equations Regularity Theory Asymptotic Methods Recession Operator 

Mathematics Subject Classification (2010)

35J60 35B65 

References

  1. 1.
    Amaral, M., Teixeira, E.: Free transmission problems. Commun. Math. Phys. 337(3), 1465–1489 (2015)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Caffarelli, L.: Interior a priori estimates for solutions of fully nonlinear equations. Ann. Math. 130(1), 189–213 (1989)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Caffarelli, L., Cabré, X.: Fully Nonlinear Elliptic Equations. American Mathematical Society Colloquium Publications, vol. 43. American Mathematical Society, Providence (1995)Google Scholar
  4. 4.
    Castillo, R., Pimentel, E.: Interior Sobolev regularity for fully nonlinear parabolic equations. Calc. Var. Partial Differ. Equ. 56, 127 (2017)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Escauriaza, L.: W 2, n a priori estimates for solutions to fully nonlinear equations. Indiana Univ. Math. J. 42(2), 413–423 (1993)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Evans, L.C.: Classical solutions of fully nonlinear, convex, second-order elliptic equations. Commun. Pure Appl. Math. 35(3), 333–363 (1982)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Fabes, E., Stroock, D.: The L p-integrability of Green’s functions and fundamental solutions for elliptic and parabolic equations. Duke Math. J. 51(4), 997–1016 (1984)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Fleming, W., Souganidis, P.: On the existence of value functions of two-player, zero-sum stochastic differential games. Indiana Univ. Math. J. 38(2), 293–314 (1989)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Krylov, N.: Estimates for derivatives of the solutions of nonlinear parabolic equations. Dokl. Akad. Nauk SSSR 274(1), 23–26 (1984)MathSciNetGoogle Scholar
  10. 10.
    Krylov, N., Safonov, M.: An estimate for the probability of a diffusion process hitting a set of positive measure. Dokl. Akad. Nauk SSSR 245(1), 18–20 (1979)MathSciNetGoogle Scholar
  11. 11.
    Krylov, N., Safonov, M.: A property of the solutions of parabolic equations with measurable coefficients. Izv. Akad. Nauk SSSR Ser. Mat. 44(1), 161–175, 239 (1980)Google Scholar
  12. 12.
    Li, D., Zhang, K.: W 2, p interior estimates of fully nonlinear elliptic equations. Bull. Lond. Math. Soc. 47(2), 301–314 (2015)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Lin, F.: Second derivative L p-estimates for elliptic equations of nondivergent type. Proc. Am. Math. Soc. 96(3), 447–451 (1986)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Nadirashvili, N., Vladut, S.: Nonclassical solutions of fully nonlinear elliptic equations. Geom. Funct. Anal. 17(4), 1283–1296 (2007)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Nadirashvili, N., Vladut, S.: Singular viscosity solutions to fully nonlinear elliptic equations. J. Math. Pures Appl. (9) 89(2), 107–113 (2008)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Nadirashvili, N., Vladut, S.: Singular solutions of Hessian fully nonlinear elliptic equations. Adv. Math. 228(3), 1718–1741 (2011)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Pimentel, E.: Regularity theory for the Isaacs equation through approximation methods. Ann. Inst. H. Poincaré C Anal. Non Linéaire (2018, to appear)Google Scholar
  18. 18.
    Pimentel, E., Teixeira, E.: Sharp Hessian integrability estimates for nonlinear elliptic equations: an asymptotic approach. J. Math. Pures Appl. 106(4), 744–767 (2016)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Silvestre, L., Teixeira, E.: Regularity estimates for fully non linear elliptic equations which are asymptotically convex. In: Contributions to Nonlinear Elliptic Equations and Systems, pp. 425–438. Springer, Berlin (2015)CrossRefGoogle Scholar
  20. 20.
    Świȩch, A.: Another approach to the existence of value functions of stochastic differential games. J. Math. Anal. Appl. 204(3), 884–897 (1996)Google Scholar
  21. 21.
    Świȩch, A.: W 1, p-interior estimates for solutions of fully nonlinear, uniformly elliptic equations. Adv. Diff. Equ. 2(6), 1005–1027 (1997)Google Scholar
  22. 22.
    Teixeira, E.: Universal moduli of continuity for solutions to fully nonlinear elliptic equations. Arch. Ration. Mech. Anal. 211(3), 911–927 (2014)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Winter, N.: W 2, p and W 1, p-estimates at the boundary for solutions of fully nonlinear, uniformly elliptic equations. Z. Anal. Anwend. 28(2), 129–164 (2009)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Department of MathematicsPontifical Catholic University of Rio de Janeiro – PUC-RioRio de JaneiroBrazil

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