Calculus of Variations and Geometric Measure Theory

The Aronsson-Euler Equation for 1-dimensional Hamiltonians

Albert Fathi ((ENS Lyon)

created by alberti on 06 Feb 2005

9 feb 2005

Abstract.

Seminari di Calcolo delle Variazioni e Teoria Geometrica della Misura

when: Wednesday, February 9, at 5.30 PM

where: Departiment of Mathematics, Pisa (Sala Riunioni)

speaker: Albert Fathi (ENS Lyon)

title: The Aronsson-Euler Equation for 1-dimensional Hamiltonians

abstract: In relation with the Hamilton-Jacobi equation $H(x, D_x u)=c$, Craig Evans introduced the Aronsson-Euler equation $$ -Dx2 ( {\partial H\over\partial p}(xDx u), {\partial H\over\partial p}(xDx u) ) - < {\partial H\over\partial x}(xDx u); {\partial H\over\partial x}(xDx u) > =0 \ . $$ He showed relations with weak KAM theory. Yifeng Yu has extended the work of Evans. In this joint work with Antonio Siconolfi, we study the 1-dimensional case. We show by examples that the Aubry set cannot be the uniqueness set for this Aronsson-Euler equation. We describe a uniqueness set.