TL;DR
This survey reviews the regularity theory of Alexandrov solutions to the Monge-Ampère equation, including classical estimates and approaches to boundary value problems, highlighting recent and classical methods.
Contribution
It compiles and discusses both classical and recent techniques for analyzing the regularity of solutions to the Monge-Ampère equation, including topics not recently covered.
Findings
Discussion of interior and boundary regularity of solutions
Review of Calabi's interior $C^3$ estimate
Analysis of classical solutions to the Dirichlet problem
Abstract
In this survey article we discuss the interior and boundary regularity of Alexandrov solutions to . We include some topics which it seems were not recently revisited in similar articles, including Calabi's interior estimate, and the approaches of Cheng-Yau and Lions to obtain classical solutions to the Dirichlet problem. The survey grew from two mini-courses given by the author in May 2018. One was for "Advanced Lectures in Nonlinear Analysis" at l'Universit\`{a} degli Studi di Torino, and the other for the Oxford PDE CDT.
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Taxonomy
TopicsGeometric Analysis and Curvature Flows · Geometry and complex manifolds · Nonlinear Partial Differential Equations