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Some new PDE methods for weak KAM theory. (English) Zbl 1032.37048

The author continues earlier work [L. C. Evans and D. Gomes, Arch. Ration. Mech. Anal. 157, 1-33 (2001; Zbl 0986.37056) and Arch. Ration. Mech. Anal. 161, 271-305 (2002; Zbl 1100.37039)] concerning systems on \({\mathbb R}^n\) that are governed by a Hamiltonian \(H=H(p, x)\) which is \(1\)-periodic in each \(x_j\), \(1\leq j\leq n\). It is furthermore assumed that \(H(\cdot, x)\) is uniformly convex and that basically \(H(p, x)\) behaves as \(|p|^2\). Under these hypotheses it is first shown that for every fixed \(P\in {\mathbb R}^n\) and \(k\in {\mathbb N}\) there exists a unique minimizer \(v^k\in C^1({\mathbb T}^n)\) (with \({\mathbb T}^n\) denoting the flat torus) to the functional \(I_k(v)=\int_{{\mathbb T}^n}e^{kH(P+\nabla v(x), x)} dx\) such that \(\int_{{\mathbb T}^n}v^k dx=0\). Thereafter the existence of the limits \(v^k\to v\) uniformly and \(d\sigma^k(x)=(I_k(v^k))^{-1}e^{kH(\nabla u^k(x), x)} \to d\sigma(x)\) as measures is established (along subsequences), where \(u^k(P, x)=P\cdot x+v^k(P, x)\); note that \(v^k\) depends on \(P\). Among other things the main results assert that \[ \lim_{k\to\infty}\int_{{\mathbb T}^n}H(\nabla u^k(x), x) d\sigma^k(x) =\lim_{k\to\infty} (1/k)\ln\bigg(\int_{{\mathbb T}^n} e^{kH(\nabla u^k(x), x)} dx\bigg)=\overline{H}(P), \] with \(\overline{H}=\overline{H}(P)\) the effective Hamiltonian related to the “cell problem” and characterized by \(\overline{H}(P)=\inf_{v\in C^1({\mathbb T}^n)}\max_{x\in {\mathbb T}^n} H(P+\nabla v(x), x)\). Moreover, \(H(\nabla u(x), x)=\overline{H}(P)\) for \(d\sigma\)-a.e. \(x\in {\mathbb T}^n\) and also \(\text{ div}(\sigma \nabla_p H)=0\) in \({\mathbb T}^n\) is satisfied.

MSC:

37J50 Action-minimizing orbits and measures (MSC2010)
35F20 Nonlinear first-order PDEs
37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion
35A15 Variational methods applied to PDEs
49R50 Variational methods for eigenvalues of operators (MSC2000)
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