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Interior Hölder Estimates for Second Derivatives

  • Cristian E. Gutiérrez
Chapter
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Part of the Progress in Nonlinear Differential Equations and Their Applications book series (PNLDE, volume 89)

Abstract

This chapter is concerned with interior Hölder regularity of second derivatives of solutions to the Monge–Ampère equation where \(\Omega\) is a bounded convex domain in \(\mathbb{R}^{n}\). The main result is Theorem 8.2.4, and its proof is conceptually simpler than the proof of the W2, p estimates shown in Chapter 6 The idea is to adapt the perturbation techniques developed in [Caf89] for fully nonlinear uniformly elliptic equations (see also [CC95]) to Monge–Ampère equation. However, there are some difficulties in carrying out this procedure due to the fact that interior regularity for solutions of the Monge–Ampère equation depends upon smoothness assumptions on Dirichlet boundary data, see Chapter 5 To handle this we will localize the problem by working with sections of solutions and show that they have an appropriate geometric shape.

Keywords

Dirichlet Boundary Data Bounded Convex Domain Interior Regularity Smoothness Assumptions Main Result 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Bibliography

  1. [Caf89]
    L.A. Caffarelli, Interior a priori estimates for solutions of fully nonlinear elliptic equations. Ann. Math. 130, 189–213 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  2. [CC95]
    L.A. Caffarelli, X. Cabré, Fully Nonlinear Elliptic Equations. American Mathematical Society Colloquium Publications, vol. 43 (American Mathematical Society, Providence, 1995)Google Scholar
  3. [GN11]
    C.E. Gutiérrez, T. Nguyen, Interior gradient estimates for solutions to the linearized Monge–Ampère equation. Adv. Math. 228 (4), 2034–2070 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  4. [GT83]
    D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd edn. (Springer, New York, 1983)CrossRefzbMATHGoogle Scholar
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    O. Savin, Pointwise C 2, α- estimates at the boundary for the Monge-Ampère equation. J. Am. Math. Soc. 26, 63–99 (2013)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  • Cristian E. Gutiérrez
    • 1
  1. 1.Department of MathematicsTemple UniversityPhiladelphiaUSA

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