Regularity of optimal transport maps and partial differential inclusions

In this paper we deal with the regularity of planar optimal transport maps, in a “critical” case not covered presently in the literature. Following a suggestion by J.Maly, and the ideas in [1], we relate the regularity problem to a “rigidity” problem for partial differential inclusions which might be interesting in its own right. Let us start with the first problem. We give a formulation in terms of subdifferentials (i.e. cyclically monotone operators), deferring the relations with optimal transport theory to Section 6. Problem 1. Let u : R → P (R) be a subdifferential, A ⊂ Dom(u) open, and let us assume that 1 L L 2 A ≤ Ju A ≤ LL 2 A.