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.