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Free boundary problems in optimal transportation

Speaker: Jiakun Liu

Date: Tue, Aug 8, 2023

Location: PIMS, University of British Columbia, Zoom

Conference: Kantorovich Initiative Seminar

Subject: Mathematics

Class: Scientific

CRG: Pacific Interdisciplinary Hub on Optimal Transport

Abstract:

In this talk, we introduce some recent regularity results of free boundary in
optimal transportation. Particularly for higher order regularity, when
densities are Hölder continuous and domains are $C^2$, uniformly convex, we obtain the free boundary is $C^{2,\alpha}$ smooth. We also consider another mode
case that the target consists of two disjoint convex sets, in which
singularities of optimal transport mapping arise. Under similar assumptions,
we show that the singular set of the optimal mapping is an $(n-1)$-dimensional
$C^{2,\alpha}$ regular sub-manifold of $\mathbb{R}^n$. These are based on a
series of joint work with Shibing Chen and Xu-Jia Wang.