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

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 The C^1,1 regularity was previously proved by Caffarelli-Lions in an unpublished note when the cost is quadratic, and recently extended to the case where the cost is uniformly convex with respect to a general preference function by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更多