Nonlocal Geometric Expansion of Convex Planar Curves

We consider a class of nonlocal geometric equations for expanding curves in the plane, arising in the study of evolutions governed by Monge–Kantorovich mass transfer. We construct convex solutions, given convex initial data. In order to obtain such solutions, we develop a new version of Perronʹs method. We give applications to the problem of characterizing fast/slow diffusion limits.