Hidden convexity in some nonlinear PDEs from geometry and physics

TL;DR

This paper demonstrates hidden convex structures in certain nonlinear PDEs from geometry and physics, enabling robust existence and uniqueness results for broad data classes, despite remaining regularity challenges.

Contribution

It reveals hidden convexity in key nonlinear PDEs, providing new existence and uniqueness results in a general framework.

Findings

Convex structure identified in Monge-Ampère equation

Existence and uniqueness results established for Euler and hyperbolic conservation laws

Framework applicable to Born-Infeld system

Abstract

The purpose of the present paper is to show few examples of nonlinear PDEs (mostly with strong geometric features) for which there is a hidden convex structure. This is not only a matter of curiosity. Once the convex structure is unrevealed, robust existence and uniqueness results can be unexpectedly obtained for very general data. Of course, as usual, regularity issues are left over as a hard post-process, but, at least, existence and uniqueness results are obtained in a large framework. The paper will address: THE MONGE-AMPERE EQUATION, THE EULER EQUATION, MULTIDIMENSIONAL HYPERBOLIC SCALAR CONSERVATION LAWS AND THE BORN-INFELD SYSTEM.