By Jean Mawhin
FACHGEB the decade has visible a massive improvement in serious element idea in endless dimensional areas and its software to nonlinear boundary price difficulties. particularly, amazing effects have been received within the classical challenge of periodic strategies of Hamiltonian platforms. This ebook presents a scientific presentation of the main easy instruments of severe aspect thought: minimization, convex services and Fenchel rework, twin least motion precept, Ekeland variational precept, minimax equipment, Lusternik- Schirelmann conception for Z2 and S1 symmetries, Morse concept for almost certainly degenerate serious issues and non-degenerate severe manifolds. each one strategy is illustrated through purposes to the dialogue of the lifestyles, multiplicity, and bifurcation of the periodic strategies of Hamiltonian structures. one of the handled questions are the periodic ideas with mounted interval or fastened strength of self sustaining platforms, the lifestyles of subharmonics within the non-autonomous case, the asymptotically linear Hamiltonian structures, unfastened and compelled superlinear difficulties. software of these effects to the equations of mechanical pendulum, to Josephson platforms of sturdy kingdom physics and to questions from celestial mechanics are given. the purpose of the ebook is to introduce a reader accepted to extra classical thoughts of standard differential equations to the robust procedure of recent serious aspect idea. the fashion of the exposition has been tailored to this target. the recent topological instruments are brought in a revolutionary yet targeted manner and instantly utilized to differential equation difficulties. The summary instruments is usually utilized to partial differential equations and the reader also will locate the fundamental references during this path within the bibliography of greater than 500 goods which concludes the booklet. ERSCHEIN
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Extra info for Critical Point Theory and Hamiltonian Systems
0 JO Then, for a > a2, JTF(t,a)dt IJ T LF(t, a2) + 0 fo I0 1 f (t, (1 - s)a2 + sa)(a - a2) dsJ dt F(t,a2)dt+c2(a-a2) and fo F(t, a) dt -* +0o if a -+ +oo. 7. 2. 9 reduces to the usual Fredholm necessary and sufficient condition rT f(t)dt=0 Jo for the solvability of (25). 9 implies existence if and only if h = (11T) fo h(t) dt belongs to the range of g. ) acting on T-periodic functions will be open (resp. closed) if the range of g is open (resp. closed) in R. Historical and Bibliographical Notes The direct method of the calculus of variations has its origin in the Dirichlet principle, which consists in connecting the existence of a solution of the Dirichlet problem Du(x) = 0, u(x) = h(x), x E Q (27) z E c 9Q where S2 C RN is an open bounded set, 0 = E 1 D; is the Laplacian, h 9Q -r R is a given function, to the existence of a minimum of the : Dirichlet integral D(u) = L IDu(x)12dx, (28) Historical and Bibliographical Notes 23 over a class of sufficiently regular real functions on SI which are equal to h on 852.
Wirtinger's inequality implies that l1uCII < c3. ) and (10) we obtain H 2 =H < < ()i)) < H(t, u(t)) + H(t, uE(t)) 2 (V H(t, ue(t)), ue(t)) + 2 H(t, 0) + 4 Iii (t) 12 + 1 (-Jiz (t), uE(t)) - 2IUe(t)I2 + 4 IuE(t)I2 +7(t) 2 3. 2, we have J T H(t, uE)dt < 0 T < J T (JuE(t), uE(t)) dt + IucIL2 + yo T 4 Iuc1Ls+ 4IuEILa+ryo <_ 4 C2+ 4C3+-to = C4- By assumption (A3), IueI < c5. Finally, 11U,11 <- 11i i,11 + IIuEII < C3 + NFTC5 = C6. c) Existence of a solution for the original problem. Since IIuEII < c6, there is a sequence (c,,) in ]0, co] tending to 0 and some u E HT such that uEn converge weakly to u in HT.
9, see Mawhin [Maw,]. Further historical references on the Dirichlet principle are given in [Maw2], as well in [Bert], [Boll], [Coui], [Fun,] (which contains Hamel's treatment of the forced pendulum equation), [Had,], [Ler,], [McS1,2], [San1], [Vo11,2], [VoP1]. For recent contributions to convexity and semi-continuity questions in the calculus of variations, see [Ces,], [C1a3,4,5], and see [Bre,], [EkT1], [ScLi] for elegant treatments of the foundations of convex analysis. 6 to a more general situation which covers the forced double pendulum in [CFSI].