Hidden convexity in some nonlinear PDEs from geometry and physics

Yann Brenier (1, 2)

1 JAD - Laboratoire Jean Alexandre Dieudonné
2 IWD - Institut Wolfgang Döblin

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.

Mots clés: Partial differential equations; convexity

Domaines: Equations aux dérivées partielles [math.AP]

Yann Brenier. Hidden convexity in some nonlinear PDEs from geometry and physics. 2009. ⟨hal-00361725⟩