JournalsaihpcVol. 26, No. 4pp. 1309–1344

A generalization of Aubry–Mather theory to partial differential equations and pseudo-differential equations

  • Rafael de la Llave

    University of Texas at Austin, Department of Mathematics, 1 University Station, C1200, Austin, TX 78712-0257, USA

  • Enrico Valdinoci

    Dipartimento di Matematica, Università di Roma Tor Vergata, Via della Ricerca Scientifica, 1, I-00133 Roma, Italy
A generalization of Aubry–Mather theory to partial differential equations and pseudo-differential equations cover
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Abstract

We discuss an Aubry–Mather-type theory for solutions of non-linear, possibly degenerate, elliptic PDEs and other pseudo-differential operators.

We show that for certain PDEs and ΨDEs with periodic coefficients and a variational structure it is possible to find quasi-periodic solutions for all frequencies. This results also hold under a generalized definition of periodicity that makes it possible to consider problems in covers of several manifolds, including manifolds with non-commutative fundamental groups.

An abstract result will be provided, from which an Aubry–Mather-type theory for concrete models will be derived.

Cite this article

Rafael de la Llave, Enrico Valdinoci, A generalization of Aubry–Mather theory to partial differential equations and pseudo-differential equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 26 (2009), no. 4, pp. 1309–1344

DOI 10.1016/J.ANIHPC.2008.11.002