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 strong equilibrium configuations in nonlinear elasticity or for liquid crystals. The presentation is selfcontained and available to non experts. themes are taken care of so far as attainable in an effortless manner, illustrating effects with basic examples; in precept, chapters or even sections are readable independently of the final context, in order that components could 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 occasionally supplementary effects. eventually, an in depth desk of Contents and an intensive Index are of support 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 By Proposition 3 for j = 1, 2, ... we can find a partitioning of 1? 3) such that the maps vj(x) := tQ(i) (x) if x E Q(j) converge strongly in L1 to u. Passing to subsequences and using Fatou's lemma we have jy ff(u)dx < liminfJ f(vj)dx. (9) n S? i) 00 (10) < 1im inf k--oo lQ2i) f f (uQ=,)) dx < I 1im } f (uk(x)) dx oo i=1 QJ(i) f f (uk) dx < lim inf f f (uk) dx , x=1Q(j) and the result follows from (9) and (10). k--boo fl 11 For our future applications Theorem 5 is not sufficient, yet. We need a semicontinuity result for the integral in (1) under the convergence of Uk to u in the sense of measures.

C. c. with respect to the weak topology of V. In particular we infer Theorem 5. c. with respect to the second variable. c. with respect to the weak convergence in L1(S1, RN; µ) Next proof is more analytic or less functional Second proof of Theorem 1. First suppose that f (p) is of class C1, convex, and with gradient bounded in RN. Assume Uk u weakly in L1, f f (u(x)) dy < oo, and fix e > 0. By Lusin's theorem (µ(S2) < oo) we can find a compact set K C Sl such that µ(S1\K)

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. We would like to present some of them. Recall the following well-known results on Banach spaces Theorem 2 (Banach-Saks). Let V be a Banach space and let {uk} be a sequence in V which is weakly converging to u. 7k>0 3= j=1 which converge strongly to u. Theorem 3. Let C be a convex set of a Banach space V. Then C is strongly closed if and only if C is weakly closed. First proof of Theorem 1.

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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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