Weak KAM theory for nonregular commuting Hamiltonians

Andrea Davini; Maxime Zavidovique

doi:10.3934/dcdsb.2013.18.57

Article Contents

2013, Volume 18, Issue 1: 57-94. Doi: 10.3934/dcdsb.2013.18.57

This issue Previous Article The basic reproduction number of discrete SIR and SEIS models with periodic parameters Next Article Slow passage through multiple bifurcation points

Weak KAM theory for nonregular commuting Hamiltonians

Andrea Davini^1, and
Maxime Zavidovique^2,

1.
Dip. di Matematica, Università di Roma "La Sapienza", P.le Aldo Moro 2, 00185 Roma
2.
IMJ, Université Pierre et Marie Curie, Case 247, 4 place Jussieu, F-75252 Paris Cédex 05

Received: May 2011

Revised: June 2012

Published: January 2013

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Abstract

In this paper we consider the notion of commutation for a pair of continuous and convex Hamiltonians, given in terms of commutation of their Lax--Oleinik semigroups. This is equivalent to the solvability of an associated multi--time Hamilton--Jacobi equation. We examine the weak KAM theoretic aspects of the commutation property and show that the two Hamiltonians have the same weak KAM solutions and the same Aubry set, thus generalizing a result recently obtained by the second author for Tonelli Hamiltonians. We make a further step by proving that the Hamiltonians admit a common critical subsolution, strict outside their Aubry set. This subsolution can be taken of class $C^{1,1}$ in the Tonelli case. To prove our main results in full generality, it is crucial to establish suitable differentiability properties of the critical subsolutions on the Aubry set. These latter results are new in the purely continuous case and of independent interest.

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Mathematics Subject Classification: Primary: 35F21; Secondary: 49L25, 37J50.

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