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A survey of partial differential equations methods in weak KAM theory. (English) Zbl 1040.37046

Author’s introduction: “This paper is an expanded version of the Courant lectures I gave at NYU in March 2002 and provides an idiosyncratic overview of some partial differential equation methods recently developed for weak KAM theory. I am pushing here the viewpoint that useful information can be extracted from (i) examining two coupled PDEs, a generalized eikonal equation and a related continuity equation, and (ii) exploiting elementary convexity arguments. I adopt throughout an expository, heuristic style, with the particular aim of emphasizing these two principles. Consult the original papers for precise assertions and full proofs. I discuss mostly my own work (much joint with D. Gomes) [Arch. Ration. Mech. Anal. 161, 271–305 (2002; Zbl 1100.37039); ibid. 157, 1–33 (2001; Zbl 0986.37056)]; and [the author, Calc. Var. Partial Differ. Equ. 17, 159–177 (2003; Zbl 1032.37048) and Commun. Math. Phys. 244, 311–334 (2004; Zbl 1073.81037)] and leave out details about the discoveries of Mather, Fathi, and many others: this omission is entirely due to my lack of expertise. While it is currently not so certain that really new dynamical information can be extracted from these arguments, this PDE approach seems to me interesting, at the very least pouring old wine into new bottles.”
After introducting the Hamilton-Jacobi equations and J. N. Mather’s generalized action [Math. Z. 207, 169–207 (1991; Zbl 0696.58027)], Section 2 concerns the associated infinite-dimensional linear programming, Section 3 the derived effective Hamiltonian, and Section 4 partial regularity with respect to a certain measure. Section 5 investigates a new variational principle reminiscent of “Aronsson’s variational principle” for the effective Hamiltonian, and finally Section 6 comments on relations to quantum mechanics.

MSC:

37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion
37J50 Action-minimizing orbits and measures (MSC2010)
35F20 Nonlinear first-order PDEs
49J45 Methods involving semicontinuity and convergence; relaxation
37-02 Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory
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