Journal of Convex Analysis 27 (2020), No. 3, 811--832
Copyright Heldermann Verlag 2020
Differentiability of the Argmin Function and a Minimum Principle for Semiconcave Subsolutions
Julius Ross
Dept. of Mathematics, Statistics and Computer Science, University of Illinois, Chicago, IL 60607, U.S.A.
julius@math.uic.edu
David Witt Nyström
Dept. of Mathematical Sciences, University of Gothenburg, 41296 Göteborg, Sweden
david.witt.nystrom@gu.se
[Abstract-pdf]
Suppose $f(x,y) + \frac{\kappa}{2} \|x\|^2 - \frac{\sigma}{2}\|y\|^2$ is convex where $\kappa\ge 0, \sigma>0$, 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.
Keywords: Argmin function, differentiability, minimum principle, semiconcave subsolutions.
MSC: 28B20, 58C06.
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