$C^{1,{1}/{3}-}$ very weak solutions to the two dimensional Monge–Ampère equation

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2025年5月27日

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Wentao Cao,
Jonas Hirsch,
Dominik Inauen

摘要

For any $\theta <\frac{1}{3}$ $θ < \frac{1}{3}$ , we show that very weak solutions to the two-dimensional Monge–Ampère equation with regularity $C^{1,\theta }$ $C^{1, θ}$ 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.

\(C^{1,{1}/{3}-}\) very weak solutions to the two dimensional Monge–Ampère equation

摘要