A Partial Differential Equation for the Convex Envelope - Adam Oberman Math

Introduction

While convex functions and the convex envelope have been the subject of study for many years, it was not observed until recently that the convex envelope satisfies a partial differential equation in the form of a nonlinear obstacle problem.

The equation for the convex envelope, $u$ , of the function $g:R^n \to R$ , is

$\max \left\{ u(x) - g(x), -\lambda_1[u](x) \right\} = 0.$

Here $λ 1 [u](x)$ is the smallest eigenvalue of the Hessian $D 2 u (x)$ .

The convex envelope satisfies a PDE obstacle problem

Related work

The equation is derived in

The convex envelope is the solution of a nonlinear obstacle problem, Proc. Amer. Math. Soc. 135 (2007), no. 6, 1689--1694

A numerical scheme for the solution of the equation can be found in

Computing the convex envelope using a nonlinear partial differential equation, Mathematical Models and Methods in Applied Sciences (M3AS), Vol. 18. No 5 (2008) 759-780

The analysis of the Dirichlet problem for the convex envelope is performed in

(with Luis Silvestre) The Dirichlet Problem for the Convex Envelope,

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