By Mariano Giaquinta

ISBN-10: 354064010X

ISBN-13: 9783540640103

This monograph (in volumes) offers with non scalar variational difficulties bobbing up in geometry, as harmonic mappings among Riemannian manifolds and minimum graphs, and in physics, as reliable equilibrium configuations in nonlinear elasticity or for liquid crystals. The presentation is selfcontained and available to non experts. issues are handled so far as attainable in an user-friendly method, illustrating effects with basic examples; in precept, chapters or even sections are readable independently of the overall context, in order that elements will be simply used for graduate classes. Open questions are usually pointed out and the ultimate portion of every one bankruptcy discusses references to the literature and infrequently supplementary effects. ultimately, a close desk of Contents and an intensive Index are of support to refer to this monograph

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**Extra resources 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)**

**Example text**

Such a theorem needs some more work and will be treated in the next subsection. Here we consider the autonomous case which is simpler 1. Regular Variational Integrals 18 Theorem S. c.. 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, ...

A contradiction. ) <<- 1, 0(77) >_ 0} . 42 1. Regular Variational Integrals Observing that for any 0 E Ano+N the function 0(77) - 1(,q) is positively homogeneous of degree one we see at once that f*(O) := Sup{95(x) - f(77) I qE AnRn+N} must be +oo or 0. e. (c) and (d) of (iii) are equivalent. (ii) If f 0 for some E E+, l An]Rn+N, I (C) # 0. We set 77 = / f f () 0, then clearly I10 I If = +00. Let e E so that 1(77) = 1. Then 0(6) = 0(77) <- 11 0 If and 0 II ff V E A,,Rn+N sup {g5(M(G))/ f (G) I G E M(N, n)} <- 11011f.

Moreover I'C f is proper if f is not identically +oo and has an affine minorant. Also, Hahn-Banach's theorem yields Theorem 5. Suppose that TC f : V -+ lk has an of fine minorant. Then I'C f (x) = sup{2(x) I 1 : V -> R, 2 affine, P < f } sup{g(x) I g : V -+ 1R, g convex, g < f } . 1 Polyconvexity and Polyconvex Envelops 29 Notice that in the second equality of Theorem 5 the convex functions g take values in R, not in R U {+oo}. c. minorant of a function f : V -+ Ilk will be denoted by r f . Then we have Proposition 3.

### 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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