Archive for Rational Mechanics and Analysis

, Volume 228, Issue 3, pp 923–967 | Cite as

Regularity for Fully Nonlinear Elliptic Equations with Oblique Boundary Conditions

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Abstract

In this paper, we obtain a series of regularity results for viscosity solutions of fully nonlinear elliptic equations with oblique derivative boundary conditions. In particular, we derive the pointwise C α, C 1,α and C 2,α regularity. As byproducts, we also prove the A–B–P maximum principle, Harnack inequality, uniqueness and solvability of the equations.

Mathematics Subject Classification

35J25 35B65 35J60 35D40 

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References

  1. 1.
    Caffarelli, L.A.; Cabré, X.: Fully Nonlinear Elliptic Equations. American Mathematical Society, Providence (1995)CrossRefMATHGoogle Scholar
  2. 2.
    Crandall, M.G.; Ishii, H.; Lions, P.-L.: User’s guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc. 27, 1–67 (1992)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Gilbarg, D.; Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Springer, Berlin (2001)MATHGoogle Scholar
  4. 4.
    Ishii, H.: Fully nonlinear oblique derivative problems for nonlinear second-order elliptic PDEs. Duke Math. J. 62, 633–661 (1991)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Kazdan, J.L.: Prescribing the Curvature of a Riemannian Manifold. American Mathematical Society, Providence (1985)CrossRefMATHGoogle Scholar
  6. 6.
    Krylov, N.V.: Boundedly inhomogeneous elliptic and parabolic equations in a domain. (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 47, 75–108, 1983Google Scholar
  7. 7.
    Lieberman, G.M.: Solvability of quasilinear elliptic equations with nonlinear boundary conditions. Trans. Am. Math. Soc. 273, 753–765 (1982)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Lieberman, G.M.: Oblique derivative problems in Lipschitz domains. I. Continuous boundary data. Boll. Un. Mat. Ital. B (7) 1, 1185–1210, 1987Google Scholar
  9. 9.
    Lieberman, G.M.: Oblique Derivative Problems for Elliptic Equations. World Scientific Publishing Co. Pte. Ltd., Hackensack, 2013Google Scholar
  10. 10.
    Lieberman, G.M.; Trudinger, N.S.: Nonlinear oblique boundary value problems for nonlinear elliptic equations. Trans. Am. Math. Soc. 295, 509–546 (1986)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Lions, P.-L.: Neumann type boundary conditions for Hamilton-Jacobi equations. Duke Math. J. 52, 793–820 (1985)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Milakis, E.; Silvestre, L.E.: Regularity for fully nonlinear elliptic equations with Neumann boundary data. Commun. Partial Differ. Equ. 31, 1227–1252 (2006)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Safonov, M.V.: On the oblique derivative problem for second order elliptic equations. Commun. Partial Differ. Equ. 20, 1349–1367 (1995)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Safonov, M.V.: On the Boundary Value Problems for Fully Nonlinear Elliptic Equations of Second Order. http://www-users.math.umn.edu/~safon002/NOTES/BVP_94/BVP.pdf 2015. Accessed 13 Apr 2017
  15. 15.
    Wang, L.H.: A maximum principle for elliptic and parabolic equations with oblique derivative boundary problems. J. Partial Differ. Equ. 5, 23–27 (1992)MathSciNetMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsXi’an Jiaotong UniversityXi’anChina
  2. 2.Department of Applied MathematicsNorthwestern Polytechnical UniversityXi’anChina

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