Games for eigenvalues of the Hessian and concave/convex envelopes

Abstract

We study the PDE ${\lambda }_j(D^{2}u)=0$, in $\Omega$, with $u=g$, on $\partial \Omega$. Here ${\lambda }_1(D^{2}u)\le ...\le {\lambda }_N(D^{2}u)$ are the ordered eigenvalues of the Hessian $D^{2}u$. First, we show a geometric interpretation of the viscosity solutions to the problem in terms of convex/concave envelopes over affine spaces of dimension $j$. In one of our main results, we give necessary and sufficient conditions on the domain so that the problem has a continuous solution for every continuous datum $g$. Next, we introduce a two-player zero-sum game whose values approximate solutions to this PDE problem. In addition, we show an asymptotic mean value characterization for the solution the the PDE.