2020/11/30 by Cale Rankin · 3 citations
Mathematics · #Applied mathematics #Bounded function #Computer science #Convexity #Differentiable function #Dimension (graph theory) #Direct proof #Domain (mathematical analysis) #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Jacobian matrix and determinant #Mathematical analysis #Mathematics #Measure (data warehouse) #Nonlinear Partial Differential Equations #Pure mathematics #math.AP #msc:35J66 #msc:35J96
paper · pdf · doi:10.1007/s00526-021-02093-4
published in Calculus of Variations and Partial Differential Equations 60(6) (Springer Science+Business Media) · minor modification of domain conditions, one proof moved to appendix, To appear in Calculus of Variations and Partial Differential Equations
openalex publication_date 2021/09/04 · arxiv created 2021/09/22 · arxiv updated 2021/09/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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 C1 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.