Two approaches to minimax formula of the additive eigenvalue for quasiconvex Hamiltonians

TL;DR

This paper presents two novel proofs for a minimax formula of additive eigenvalues in quasiconvex Hamilton-Jacobi equations, expanding understanding beyond convex cases.

Contribution

It introduces two distinct proofs for the minimax formula applicable to quasiconvex Hamiltonians, using Jensen-like inequalities and elementary calculations.

Findings

Established a Jensen-like inequality for quasiconvex functions

Provided a new elementary proof based on mollified functions

Linked methods to viscosity solution approximation

Abstract

Two different proofs for an inf-sup type representation formula (minimax formula) of the additive eigenvalues corresponding to first-order Hamilton-Jacobi equations are given for quasiconvex (level-set convex) Hamiltonians not necessarily convex. The first proof, which is similar to known proofs for convex Hamiltonians, invokes a Jensen-like inequality for quasiconvex functions instead of the standard Jensen's inequality. The second proof is completely different with elementary calculations. It is based on convergence of derivatives of mollified Lipschitz continuous functions whose proof is also given. These methods also relate to an approximation problem of viscosity solutions.