We propose a PDE approach to the Aubry-Mather theory using viscosity solutions. This allows to treat Hamiltonians (on the flat torus T-N) just coercive. continuous and quasiconvex, for which a Hamiltonian flow cannot necessarily be defined. The analysis is focused on the family of Hamilton-Jacobi equations H(x, Du) = a with a real parameter, and in particular on the unique equation of the family. corresponding to the so-called critical value a = c, for which there is a viscosity solution on T-N. We define generalized projected Aubry and Mather sets and recover several properties of these sets holding for regular Hamiltonians.
PDE aspects of Aubry-Mather theory for quasiconvex Hamiltonians / Albert, Fathi; Siconolfi, Antonio. - In: CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS. - ISSN 0944-2669. - 22:2(2005), pp. 185-228. [10.1007/s00526-004-0271-z]
PDE aspects of Aubry-Mather theory for quasiconvex Hamiltonians
SICONOLFI, Antonio
2005
Abstract
We propose a PDE approach to the Aubry-Mather theory using viscosity solutions. This allows to treat Hamiltonians (on the flat torus T-N) just coercive. continuous and quasiconvex, for which a Hamiltonian flow cannot necessarily be defined. The analysis is focused on the family of Hamilton-Jacobi equations H(x, Du) = a with a real parameter, and in particular on the unique equation of the family. corresponding to the so-called critical value a = c, for which there is a viscosity solution on T-N. We define generalized projected Aubry and Mather sets and recover several properties of these sets holding for regular Hamiltonians.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.