Strict convexity and C1 regularity of solutions to generated Jacobian equations in dimension two

We present a proof of strict g -convexity in 2D for solutions of generated Jacobian equations with a g -Monge–Ampère measure bounded away from 0. Subsequently this implies C 1 differentiability in the case of a g -Monge–Ampère measure bounded from above. Our proof follows one given by Trudinger and...

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Bibliographic Details
Published in:Calculus of variations and partial differential equations 2021, Vol.60 (6)
Main Author: Rankin, Cale
Format: Article
Language:English
Subjects:
Analysis
Calculus of Variations and Optimal Control
Optimization
Control
Convexity
Domains
Mathematical analysis
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Systems Theory
Theoretical
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Summary:We present a proof of strict g -convexity in 2D for solutions of generated Jacobian equations with a g -Monge–Ampère measure bounded away from 0. Subsequently this implies C 1 differentiability in the case of a g -Monge–Ampère measure bounded from above. Our proof follows one given by Trudinger and Wang in the Monge–Ampère case. Thus, like theirs, our argument is local and yields a quantitative estimate on the g -convexity. As a result our differentiability result is new even in the optimal transport case: we weaken previously required domain convexity conditions. Moreover in the optimal transport case and the Monge–Ampère case our key assumptions, namely A3w and domain convexity, are necessary.
ISSN:0944-2669
1432-0835
DOI:10.1007/s00526-021-02093-4