The interior C2 estimate for the Monge–Ampère equation in dimension n = 2

Abstract

Abstract. In this paper, we introduce a new auxiliary function, and establish the interior C 2 estimate for Monge-Ampère equation in dimension n = 2, which was firstly proved by Heinz [5] by a geometric method.

Interior C2 regularity of convex solutions to prescribing scalar curvature equations

“…where constant C depending only on n, u C 1 (B 1 ), f C 2 (B 1 ) and 1 f L ∞ (B 1 ). In the case of n = 2, the above results were proved by Heinz [7] (see also [3]). One observes that equations (3) and (2) are Monge-Ampère type equation, all admissible solutions are automatically convex.…”

“…The above result is related to a longstanding problem in fully nonlinear partial differential equations: the interior C 2 estimate for solutions of the following prescribing scalar curvature equation and σ 2 -Hessian equation, (2) σ 2 (κ 1 (x), · · · , κ n (x)) = f (X, ν(x)) > 0, X ∈ B r × R ⊂ R n+1 and (3) σ 2 (∇ 2 u(x)) = f (x, u(x), ∇u(x)) > 0, x ∈ B r ⊂ R n where κ 1 , · · · , κ n are the principal curvatures and ν the normal of the given hypersurface as a gragh over a ball B r ⊂ R n respectively, σ k the k-th elementary symmetric function, 1 ≤ k ≤ n. Equations (2) and (3) are special cases of σ k -Hessian and curvature equations developed by Caffarealli-Nirenberg-Spruck in [1], as an integrated part of fully nonlinear PDE. A C 2 function u is called an admissible solution to equation (3) if u satisfies the equation and ∆u > 0. Similarly, if a graph (x, u(x)) is called an admissible solution to equation (2) if u satisfies the equation with positive mean curvature.…”

The interior gradient estimate of prescribed Hessian quotient curvature equations

“…In this paper, our proof, which is based on a suitable choice of auxiliary functions, is elementary and avoids geometric computations on the graph of solutions. This technique is from [2]. Remark 1.3.…”

“…in higher dimensions is a longstanding problem, where σ 2 (∇ 2 u) = σ 2 (λ(∇ 2 u)) = 1≤i 1 <i 2 ≤n λ i 1 λ i 2 , λ(∇ 2 u) = (λ 1 , · · · , λ n ) are the eigenvalues of ∇ 2 u, and f > 0. For n = 2, (1.2) is the Monge-Ampère equation, and Heinz [5] obtained the estimate by the convex hypersurface geometry method (see [2] for an elementary analytic proof). For Monge-Ampère equations with dimension n ≥ 3, Pogorelov [7] constructed his famous counterexample, namely irregular solutions to Monge-Ampère equations.…”

On interior <inline-formula><tex-math id="M1">\begin{document} $C^2$ \end{document}</tex-math></inline-formula>-estimates for the Monge-Ampère equation

“…We briefly refer the reader to the books [4,Chapter 3], [7,Section 17.6], and [8,Chapter 4], the survey articles [3,Section 2.3] and [13,Section 3.4], and the references therein for various formulations of such estimates. Fairly recent additions to the literature include a new proof of Pogorelov's classical estimates in dimension two in [2] and extensions of Pogorelov's estimates to Monge-Ampère-type equations arising from optimal transport in [10].…”