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We study the PDE $\lambda_j(D^2 u) = 0$, in $\Omega$, with $u=g$, on$\partial \Omega$. Here $\lambda_1(D^2 u) \leq ... \leq \lambda_N (D^2 u)$ arethe ordered eigenvalues of the Hessian $D^2 u$. First, we show a geometricinterpretation of the viscosity solutions to the problem in terms ofconvex/concave envelopes over affine spaces of dimension $j$. In one of ourmain results, we give necessary and sufficient conditions on the domain so thatthe problem has a continuous solution for every continuous datum $g$. Next, weintroduce a two-player zero-sum game whose values approximate solutions to thisPDE problem. In addition, we show an asymptotic mean value characterization forthe solution the the PDE.

Games for eigenvalues of the Hessian and concave/convex envelopes