C^1,α regularity of variational problems with a convexity constraint

2024/03/07 by Ling Wang, Bin Zhou, Wang, Ling +1
Computer Science · Mathematics · #Analysis of PDEs (math.AP) #Contact Mechanics and Variational Inequalities #FOS: Mathematics #Nonlinear Partial Differential Equations #Optimization and Variational Analysis

paper · pdf · doi:10.48550/arxiv.2403.04235

openalex publication_date 2024/03/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we establish the interior C^1,α regularity of minimizers of a class of functionals with a convexity constraint, which includes the principal-agent problems studied by Figalli-Kim-McCann (J. Econom. Theory 146 (2011), no. 2, 454-478). The C^1,1 regularity was previously proved by Caffarelli-Lions in an unpublished note when the cost is quadratic, and recently extended to the case where the cost is uniformly convex with respect to a general preference function by McCann-Rankin-Zhang(arXiv:2303.04937v3). Our main result does not require the uniform convexity assumption on the cost function. In particular, we show that the solutions to the principal-agent problems with q-power cost are C^1,(1)/(q-1) when q > 2 and C^1,1 when 1

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