Interior $C^2$ estimate for Monge-Amp\`ere equation in dimension two
Jiakun Liu
TL;DR
This paper establishes a new local second-derivative estimate for the Monge-Ampère equation in two dimensions using the partial Legendre transform, advancing understanding of regularity in this context.
Contribution
It introduces a novel approach employing the partial Legendre transform to derive genuine local $C^2$ estimates for the Monge-Ampère equation in dimension two.
Findings
Established a local $C^2$ estimate for the Monge-Ampère equation in 2D
Utilized the partial Legendre transform technique
Enhanced regularity results for solutions in two dimensions
Abstract
We obtain a genuine local estimate for the Monge-Amp\`ere equation in dimension two, by using the partial Legendre transform.
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Taxonomy
TopicsGeometry and complex manifolds · Geometric Analysis and Curvature Flows · Nonlinear Partial Differential Equations