Fine properties of minimizers of mechanical Lagrangians with Sobolev potentials

Alessio Figalli (1) , Vito Mandorino (2)

1 Department of Mathematics and Statistics [Texas Tech]
2 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision

In this paper we study the properties of curves minimizing mechanical Lagrangian where the potential is Sobolev. Since a Sobolev function is only defined almost everywhere, no pointwise results can be obtained in this framework, and our point of view is shifted from single curves to measures in the space of paths. This study is motived by the goal of understanding the properties of variational solutions to the incompressible Euler equations.

Mots clés: value function.; Non-smooth Lagrangians; Euler-Lagrange equations; action-minimizing measures; value function

Domaines: Optimisation et contrôle [math.OC]; Equations aux dérivées partielles [math.AP]; Systèmes dynamiques [math.DS]

Alessio Figalli, Vito Mandorino. Fine properties of minimizers of mechanical Lagrangians with Sobolev potentials. Discrete and Continuous Dynamical Systems - Series A, 2011, 31 (4), pp.1325 -- 1346. ⟨10.3934/dcds.2011.31.1325⟩. ⟨hal-00636941⟩