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Extra resources for Aubry-Mather theory
17e). 17e) immediately implies that |ˆ ν (N 1 , s) − νˆ(N 2 , s)| ≤ g (|N 1 − N 2 |) , which implies that νˆ is continuous and gives a modulus of continuity for it. 9). We thus obtain a quasilinear system of partial diﬀerential equations for the components of r. 11). 14). 4) by specifying the history of r up to time 0. It proves mathematically convenient to recast these initial-boundaryvalue problems in an entirely diﬀerent form, called the weak form of the equations by mathematicians and the Principle of Virtual Power (or the Principle of Virtual Work) by physicists and engineers.
11) νˆ G(ξ) + K, ξ dξ. 12) Φ(K) := νˆ G(ξ) + K, ξ dξ = L. 0 Note that since G is the indeﬁnite integral of the integrable function g, it is absolutely continuous. It follows that the function Φ(·), just like νˆ(·, s), strictly increases from 0 to ∞ as its argument increases from −∞ to ∞. 12) has a unique solution K (depending on G and L). 11). 10). Let us now suppose that g is continuous. Then G is continuously diﬀerentiable. 10) implies that the solution z is twice continuously diﬀerentiable. 13) d ˆ ds N z (s), s + g(s) = 0.
1) for its integrals to make sense as Lebesgue integrals and for our boundary and initial conditions to have consistent generalizations. These generalizations are the highlights of the ensuing development, the details of which can be omitted by the reader unfamiliar with real analysis. 2) ∀ s ∈ [0, 1]. 4) lim t 0 (ρA)(s)[r(s, t) − u(s)] ds = o ∀ [a, b] ⊂ [0, 1], a and that v is integrable on [0, 1]. 4) are consistent with the local integrability of rs and rt (see Adams (1975), Neˇcas (1967)). 1), as the presence there of v suggests.