Game-theoretic approach to H\"{o}lder regularity for PDEs involving eigenvalues of the Hessian

TL;DR

This paper establishes a local Hölder regularity estimate for solutions of a game-theoretic dynamic programming principle and applies it to viscosity solutions of a PDE involving eigenvalues of the Hessian.

Contribution

It introduces a novel coupling method from game theory to prove Hölder regularity for PDE solutions involving eigenvalues of the Hessian.

Findings

Proves Hölder regularity with exponent 0<δ<1/2 for the dynamic programming solutions.

Extends the regularity result to viscosity solutions of the eigenvalue PDE.

Introduces a new coupling technique based on game theory.

Abstract

We prove a local H\"{o}lder estimate with an exponent $0<\delta<\frac 12$ for solutions of the dynamic programming principle $$u^\varepsilon (x) =\sum_{j=1}^n \alpha_j\inf_{\dim(S)=j}\sup_{\substack{v\in S\\ |v|=1}}\frac{ u^\varepsilon (x + \varepsilon v) + u^\varepsilon (x - \varepsilon v)}{2}.$$ 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 $$\sum_{i=1}^n \alpha_i\lambda_i(D^2u)=0,$$ where $\lambda_1(D^2 u)\leq\cdots\leq \lambda_n(D^2 u)$ are the eigenvalues of the Hessian.