A generalization of Aubry–Mather theory to partial differential equations and pseudo-differential equations
Rafael de la Llave, Enrico Valdinoci · Annales de l Institut Henri Poincaré C Analyse Non Linéaire · 2008
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 Ψ DEs 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.