We consider functions that satisfy the identityHere ε > 0 and α, and β are suitable nonnegative coefficients such that α + β = 1. In particular, we show that these functions are uniquely determined by their boundary values, approximate p-harmonic functions for 2 ≤ p < ∞ (for a choice of p that depends on α and β), and satisfy the strong comparison principle. We also analyze their relation to the theory of tug-of-war games with noise.

We study the PDE λj(D 2 u) = 0, in Ω, with u = g, on ∂Ω. Here λ1(D 2 u) ≤ ... ≤ λN (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.2010 Mathematics Subject Classification. 35D40, 35J25, 26B25.

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