The Monge-Ampère Equation

Abstract

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General Schwarz Lemmata and their applications

“…It is also reminiscent of Pogorelov's Lemma [22] (cf. Lemma 4.1.1 of [7]) for Monge-Ampère equation, since the maximum eigenvalue of ∇ 2 u is the · 0 for the normal map ∇u for any smooth u. A consequence of Theorem 1.2 asserts that the equivalence of the negative amplitude of the holomorphic sectional curvature implies the equivalence of the metrics.…”

A proof of the Krylov–Safonov theorem without localization

“…We conclude that V = ∂w(F ), where w := 1 2 |x| 2 + 1 a v. Since det D 2 w ≥ det D 2 (|x| 2 /2) = 1 in the Alexandrov sense (see e.g. [5] for the definition), the result follows.…”

Nonsmooth solutions for a class of fully nonlinear PDE’s on Lie groups

“…The proof of this theorem uses Pogorelov's counterexamples, see [18] or [8,Section 5.5], and its extensions developed by Urbas in [20] and by Gutierrez, Lanconelli and Montanari in [9] to show existence of viscosity non classical solutions to real curvature equations and to Gauss-Levi curvature equations, respectively. If condition ( H ℓ ) holds true for some ℓ ≥ k − 1, then u ̸ ∈ C 1,β X for any β > 1 − 2 k .…”

“…A viscosity solution for (8) is either a viscosity sub-solution and a viscosity super-solution for (8).…”