Games for eigenvalues of the Hessian and concave/convex envelopes

Abstract

We study the PDE λj(D2u)=0, in Ω, with u=g, on ∂Ω. Here λ1(D2u)≤...≤λN(D2u) are the ordered eigenvalues of the Hessian D2u. 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.