Download Cartesian Currents in the Calculus of Variations II: by Mariano Giaquinta PDF

By Mariano Giaquinta

ISBN-10: 354064010X

ISBN-13: 9783540640103

This monograph (in volumes) bargains with non scalar variational difficulties bobbing up in geometry, as harmonic mappings among Riemannian manifolds and minimum graphs, and in physics, as sturdy equilibrium configuations in nonlinear elasticity or for liquid crystals. The presentation is selfcontained and available to non experts. subject matters are taken care of so far as attainable in an basic means, illustrating effects with basic examples; in precept, chapters or even sections are readable independently of the overall context, in order that elements should be simply used for graduate classes. Open questions are usually pointed out and the ultimate element of every one bankruptcy discusses references to the literature and infrequently supplementary effects. eventually, an in depth desk of Contents and an intensive Index are of aid to refer to this monograph

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Additional info for Cartesian Currents in the Calculus of Variations II: Variational Integrals (Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge A Series of Modern Surveys in Mathematics)

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Q Dividing by ti(Q) and letting the size of Q tending to zero, we then obtain (4) tf (x, P) + (1 - t)f (x, q) > f (x, tp + (1 - t)q) for all x except on a p-null set which however depends in principle on t, p, q. e. c. with respect to the weak* convergence in L°°(0, IRN; µ). The necessary condition in Proposition 1 is actually sufficient, too. In fact more generally we have Theorem 1. c. and convex. , IRN; p), i. g. F(uk,12) k-+oo There are several different proofs of Theorem 1: they use the convexity of f at different levels.

Y E A2R4 of the form wry = dx l Adx2 + 7dx3Adx4 we have 11will =1 (11) if 171<1. Since 11 wo II = 1 and -y - lI wry 11 is convex, it suffices to prove that 11 wti II <_ 1 if 171 <_ 1. Consider any simple vector C = v1Av2 E A2IR4. y, C> = Al + 7a3b4 and by Schwartz inequality 1I2 < [(a')2 +72(a3)21[(bl)2 + (b4)2] :5 IV, 1I2 + Iv212 = IC12 < 1 which proves (11). 2 Parametric Polyconvex Envelops of Integrands 41 Va, b . II ae1Ae2 + be3Ae4 II = la + Ib1 This proves (10). Finally, we notice that dual claims hold for k-covectors, for instance dW E AkRn T-1/21WI C IIW 11 s IWI if and only if u) is simple IIW11 =WI and again I W is a norm on Ak We have seen that for any non-negative f : M(N, n) F(C) = f ** (e) we have V 6 E lln,l[Sn+N where f(e) { I +00 6 E L+ otherwise.

28 1. Regular Variational Integrals (a) (b) Fig. 1. c. envelop. Since the pointwise supremum of convex functions is clearly convex, Cf is convex. Moreover Theorem 2 reads as Theorem 4. Let f : V -+ IR U {+oo} and n = dim V. Then we have n+1 n+1 Cf (x) = inf { \if (xi) i=1 I l Ai > 0, E Ai = 1, xi E V } i=1 . c.. In this case the epigraph of f epif :_ {(x,y)EVxRjy? f(x)} is convex and closed. Therefore Hahn-Banach's theorem yields Proposition 2. c.. Then f is never -oo if and only if there exists an affine map P : V -+ 1R such that $(x) < f (x) V x E V.

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Cartesian Currents in the Calculus of Variations II: Variational Integrals (Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge A Series of Modern Surveys in Mathematics) by Mariano Giaquinta


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