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

By Mariano Giaquinta

This monograph (in volumes) offers with non scalar variational difficulties coming up in geometry, as harmonic mappings among Riemannian manifolds and minimum graphs, and in physics, as solid equilibrium configuations in nonlinear elasticity or for liquid crystals. The presentation is selfcontained and available to non experts. themes are handled so far as attainable in an ordinary means, illustrating effects with easy examples; in precept, chapters or even sections are readable independently of the final context, in order that components might be simply used for graduate classes. Open questions are usually pointed out and the ultimate portion of each one bankruptcy discusses references to the literature and infrequently supplementary effects. eventually, a close 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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For o < dist (Sl',852) set as usual cpa(x) = v-Thcp(x/o). Since u E L1(Sl RN), u * cpa -+ u strongly in L' ((2', R') and passing to a subsequence by Fatou's lemma ff(u(x))dx (5) r < lim onf J f (u * cpo) dx . On the other hand for fixed a, Uk * CPa --i u * co as k - oo, uniformly in S2', hence again by Fatou's lemma (6) f f(u*cpa)dx < lim f k-oo S2, f f(uk*cpa)dx. n, From Jensen's inequality we readily get (7) ff(uk*)dx < J dl f(uk)dx S2, for all k and all a. The conclusion then follows from (5), (6) and (7) and letting O D' T Q.

C. l f (u) dµ < liminf f f (uk) dIL k--oo . 0 Proof. First we observe that f is the pointwise supremum of an increasing sequence of functions fj which are convex, Lipschitz, and piecewise linear. Actually each fj is the pointwise supremum of a finite number of affine functions fj(p) = max{2i(p) £i : 1R" -, IR affine, i = 1, ... , N(j)} , fj(p) > 0 Thus it suffices to prove the result for such fj that from now on we denote again by f. e. in [2 and strongly in Ll (12, IRN) which are constant on suitable dyadic cubes QM(').

The first one is based on Jensen inequality Proposition 2 (Jensen inequality). c. function. Let µ be a Radon measure on (, with µ(,f2) < oc. Then for any u E Ll (fl, IRN; A) we have f u(x) dµ/ n f (u (x)) dlv Proof. , a E RN and b E RN such that t < a + (b, ) and there exists an affine hyperplane in a + (b, y) < f (p) d p E ]RN . It follows a + (b, u(x)) < f (u(x)) V x E S2, and, integrating over Sl, t < a + (b, f f (u(x)) dp, n for any t < f This concludes the proof. Third proof of Theorem 1. Fix S2' CC Sl and let io E C'°(B(0,1)), f cp = 1.

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