Games for eigenvalues of the Hessian and concave/convex envelopes

TL;DR

This paper explores the eigenvalues of the Hessian matrix in PDEs, providing geometric interpretations, domain conditions for solutions, and introducing a game-theoretic approach with mean value characterizations.

Contribution

It offers a new geometric perspective on viscosity solutions, establishes domain criteria for existence of solutions, and develops a game-based approximation method.

Findings

Geometric interpretation of viscosity solutions via convex/concave envelopes

Necessary and sufficient domain conditions for continuous solutions

A two-player game approximates solutions to the PDE

Abstract

We study the PDE $\lambda_j(D^2 u) = 0$, in $\Omega$, with $u=g$, on $\partial \Omega$. Here $\lambda_1(D^2 u) \leq ... \leq \lambda_N (D^2 u)$ 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.