The Dirichlet problem for the convex envelope

TL;DR

This paper investigates the Dirichlet problem for the PDE characterizing the convex envelope, establishing optimal regularity results and linking the convex envelope to stochastic control and nonlinear elliptic PDEs.

Contribution

It provides the first regularity results for the Dirichlet problem of the convex envelope PDE and characterizes the convex envelope via stochastic control and elliptic PDEs.

Findings

C^{1,α} regularity of solutions with boundary data differentiability

Convex envelope as a stochastic control value function

Characterization as an underestimator for nonlinear elliptic PDEs

Abstract

The Convex Envelope of a given function was recently characterized as the solution of a fully nonlinear Partial Differential Equation (PDE). In this article we study a modified problem: the Dirichlet problem for the underlying PDE. The main result is an optimal regularity result. Differentiability ($C^{1,\alpha}$ regularity) of the boundary data implies the corresponding result for the solution in the interior, despite the fact that the solution need not be continuous up to the boundary. Secondary results are the characterization of the convex envelope as: (i) the value function of a stochastic control problem, and (ii) the optimal underestimator for a class of nonlinear elliptic PDEs.