Game-Theoretic Approach to Hölder Regularity for PDEs Involving Eigenvalues of the Hessian

We prove a local Hölder estimate for any exponent 0<δ <1/2 for solutions of the dynamic programming principle [ u^ε (x) = ∑_j=1^nα_j(S)=jinf|v|=1v∈ Ssupu^ε (x + ε v) + u^ε (x - ε v)/2 ] with α 1 , α n > 0 and α 2 ,⋯ , α n − 1 ≥ 0. The proof is based on a new coupling idea from game theory. As an application, we get the same regularity estimate for viscosity solutions of the PDE ∑_i=1^nα_iλ_i(D^2u)=0, where λ 1 ( D 2 u ) ≤⋯ ≤ λ n ( D 2 u ) are the eigenvalues of the Hessian.