Academy of Mathematics and Systems Science, CAS
Colloquia & Seminars

Speaker:	周斌教授，北京大学
Inviter:	邱国寰
Title:	Regularity of variational problems with a convexity constraint
Language:	Chinese
Time & Venue:	2024.03.28 09:30-10:30 思源楼 S813
Abstract:	We establish the interior $C^{1,\alpha}$ regularity of minimizers of a class of functionals with a convexity constraint, which includes the principal-agent problems studied by Figalli-Kim-McCann. The $C^{1,1}$ regularity was previously proved by Caffarelli-Lions in an unpublished note when the cost is quadratic, and recently extended to the case where the cost is uniformly convex with respect to a general preference function by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:2303.04937). 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 $C^{1,\frac{1}{q-1}}$ when $q > 2$ and $C^{1,1}$ when $1<q\leq 2$. Examples can show that this regularity is optimal when $q\geq 2$.