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**Additional resources for Aubry-Mather theory**

**Example text**

T ∈ (0, T ). 16) Moreover, there exists a function h ∈ L1 (0, T ) such that t ψ t, u(t) ≤ ψ s, u(s) + h(σ) dσ for all 0 ≤ s ≤ t ≤ T. 9 (D. T˘ ataru, [T˘ ataru]). 18) + A(t)x 2 H + A(s)y 2 H for all t, s ∈ [0, T ], x ∈ D A(t) , and y ∈ D A(s) , where M is a positive constant and g is a function of bounded variation on [0, T ]. a. t ∈ (0, T ). 9. 9. 7 still holds with L = L( x H ), where L(·) is a nondecreasing function (actually, this is Kato’s original assumption). 9 holds under more general conditions so that Kato’s original result can again be derived as a special case (see [T˘ataru]).

Since V1 is embedded compactly ∗ into H, then also Vk+1 ⊂ Vk and Vk∗ ⊂ Vk+1 compactly, so that (iii) holds. Moreover, Vk ⊂ H compactly. The duality mapping Jk : Vk → Vk∗ is given by Jk (u)(v) = (u, v)k for all u, v ∈ Vk . 16) Clearly, J1 = BE . We easily see that en = λ(1+k)/2 en for all n, k ∈ IN∗ . Jk λ(1−k)/2 n n Hence (ii) is also satisfied. a. 1) where B: D(B) ⊂ X → X is a linear unbounded operator, f : IR+ → X, and X is a Banach space. In this section we briefly recall some definitions and results from the theory of semigroups of bounded linear operators.

Since H is dense in HE , g = 0. 1 is proved. For our regularity considerations we shall need some subspaces of HE , which we define by powers of the operator A. 4) and let λ1 , λ2 , . . and e1 , e2 , . . 1, and γ ≥ 0. 13) n=1 and u ∈ D(Aγ ) whenever this series converges. Then Aγ is selfadjoint and strongly monotone. PROOF Since H is separable, Aγ is selfadjoint, by [Zeidler, p. 294]. Since {λn } is increasing,Aγ ∞ λγk u, (Aγ u, u)H = ∞ √ λk ek k=1 2 H λγ1 u, ≥ √ λk ek 2 H = λγ1 u 2 H. k=1 Thus Aγ is strongly monotone.

### Aubry-Mather theory

by George

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