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September  2009, 1(3): 271-294. doi: 10.3934/jgm.2009.1.271

Cauchy problems for stationary Hamilton-Jacobi equations under mild regularity assumptions

Olga Bernardi 1, , Franco Cardin 1, and  Antonio Siconolfi 2,

1. 

Dipartimento di Matematica Pura ed Applicata, Via Belzoni 7 - 35131 Padova
2. 

Dipartimento di Matematica, "La Sapienza” Università di Roma, P.le Aldo Moro, 2 - 00185 Roma, Italy

Received  July 2008 Revised  April 2009 Published  November 2009

For a Hamiltonian enjoying rather weak regularity assumptions, we provide necessary and sufficient conditions for the existence of a global viscosity solution to the corresponding stationary Hamilton-Jacobi equation at a fixed level $a$, taking a prescribed value on a given closed subset of the ground space. The analysis also includes the case where $a$ is the Mañé critical value. Our results are based on a metric method extending Maupertuis approach.
   For general underlying spaces, compact or noncompact, we give a global version of the classical characteristic method based on the notion of $a$-characteristic. In the compact case, we propose an inf-sup formula producing the minimal solution of the problem, where the generalized Aubry set is involved.
Keywords: Characteristics method., Stationary Hamilton-Jacobi equation, Weak KAM theory.
Mathematics Subject Classification: Primary: 35F20, 49L25; Secondary: 58F0.
Citation: Olga Bernardi, Franco Cardin, Antonio Siconolfi. Cauchy problems for stationary Hamilton-Jacobi equations under mild regularity assumptions. Journal of Geometric Mechanics, 2009, 1 (3) : 271-294. doi: 10.3934/jgm.2009.1.271
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