Interior C^1,1 regularity of solutions to degenerate Monge-Ampère type equations

2018/06/05 by Feida Jiang, Jiang, Feida, Juhua Shi +3
Mathematics · #35J60 #35J70 #35J96 #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1806.01720

openalex publication_date 2018/06/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we study the interior C^1,1 regularity of viscosity solutions for a degenerate Monge-Ampère type equation det[D²u-A(x, u, Du)]=B(x, u, Du) when B ≥ 0 and B^(1)/(n-1)∈ C^1,1(Ω×ℝ× ℝⁿ). We prove that u∈ C^1,1(Ω) under the A3 condition and A3w⁺ condition respectively. In the former case, we construct a suitable auxiliary function to obtain uniform \it a priori estimates directly. In the latter case, the main argument is to establish the Pogorelov type estimates, which are interesting independently.

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