Differentiability of the argmin function and a minimum principle for semiconcave subsolutions

Abstract

Suppose $f(x,y)+\frac{\kappa }{2}\Vert x{\Vert }^2-\frac{\sigma }{2}\Vert y{\Vert }^2$ is convex where $\sigma >0$, and the argmin function $\gamma (x)=\{\gamma :{\inf }_{y}f(x,y)=f(x,\gamma )\}$ exists and is single valued. We will prove $\gamma$ is differentiable almost everywhere. As an application we deduce a minimum principle for certain semiconcave subsolutions.

20 pages, minor changes, accepted version to appear in Journal of Convex Analysis