The largest eigenvalue of a convex function, duality, and a theorem of Slodkowski

TL;DR

This paper explores Slodkowski's theorem on the largest eigenvalue of convex functions, providing a duality perspective and an alternative proof, which advances understanding in nonlinear elliptic PDEs.

Contribution

It offers a dual interpretation of the largest eigenvalue and presents an alternative proof, enhancing the theoretical framework for convex analysis and PDEs.

Findings

Extension of a.e. lower bounds to pointwise bounds for convex functions

Dual interpretation of the largest eigenvalue via Legendre-Fenchel transform

Alternative proof of Slodkowski's theorem using duality

Abstract

First, we provide an exposition of a theorem due to Slodkowski regarding the largest "eigenvalue" of a convex function. In his work on the Dirichlet problem, Slodkowski introduces a generalized second-order derivative which for $C^2$ functions corresponds to the largest eigenvalue of the Hessian. The theorem allows one to extend an a.e lower bound on this largest "eigenvalue" to a bound holding everywhere. Via the Dirichlet duality theory of Harvey and Lawson, this result has been key to recent progress on the fully non-linear, elliptic Dirchlet problem. Second, we give a dual interpretation of this largest eigenvalue using the Legendre-Fenchel transform, and use this dual perspective to provide an alternative proof to an important step in the proof of the theorem.