The Monge-Ampère Equation pp 193-209 | Cite as

# Interior Hölder Estimates for Second Derivatives

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## 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 *W*^{2, 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## Bibliography

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