C^1,(1)/(3)- very weak solutions to the two dimensional Monge-Ampére equation

2023/10/10 by Wentao Cao, Cao, Wentao, Jonas Hirsch +3 · 1 citation
Mathematics · Physics and Astronomy · #35J96 #74B2 #Analysis of PDEs (math.AP) #Black Holes and Theoretical Physics #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.2310.06693

openalex publication_date 2023/10/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For any θ<(1)/(3), we show that very weak solutions to the two-dimensional Monge-Ampère equation with regularity C^1,θ are dense in the space of continuous functions. This result is shown by a convex integration scheme involving a subtle decomposition of the defect at each stage. The decomposition diagonalizes the defect and, in addition, incorporates some of the leading-order error terms of the first perturbation, effectively reducing the required amount of perturbations to one.

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