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

For any $$\theta <\frac{1}{3}$$ θ < 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...

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Bibliographic Details
Published inCalculus of variations and partial differential equations Vol. 64; no. 5
Main Authors Cao, Wentao, Hirsch, Jonas, Inauen, Dominik
Format Journal Article
LanguageEnglish
Published 01.06.2025
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ISSN0944-2669
1432-0835
DOI10.1007/s00526-025-03019-0

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Summary:For any $$\theta <\frac{1}{3}$$ θ < 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.
ISSN:0944-2669
1432-0835
DOI:10.1007/s00526-025-03019-0