Singular Set of a Convex Potential in Two Dimensions
Yifeng Yu · Communications in Partial Differential Equations · 2007
Let u be a convex potential of the optimal transfer map from a convex open set X to a nonconvex open set Y in the plane. If u only has singularities whose sets of supports are one dimension, then under some mild assumptions on Y, we show that the singular set of u are disjoint union of countably many C 1 curves. Or we can say that the singular set is a C 1 manifold.