Game-theoretic approach to Hölder regularity for PDEs involving eigenvalues of the Hessian

Abstract

We prove a local Hölder estimate with an exponent $0<\delta <\frac{1}{2}$ for solutions of the dynamic programming principle $$u^{\epsilon }(x)=\sum _{j=1}^n{\alpha }_j\underset{\dim (S)=j}{\inf }\underset{\begin{array}{c}v\in S\\ |v|=1\end{array}}{\sup }\frac{u^{\epsilon }(x+\epsilon v)+u^{\epsilon }(x-\epsilon v)}{2}.$$ 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 $$\sum _{i=1}^n{\alpha }_i{\lambda }_i(D^{2}u)=0,$$ where ${\lambda }_1(D^{2}u)\le \cdots \le {\lambda }_n(D^{2}u)$ are the eigenvalues of the Hessian.

References in corpus (2)

• Tug-of-war games with varying probabilities and the normalized $p(x)$-Laplacian
• The evolution problem associated with eigenvalues of the Hessian