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

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 easy method, illustrating effects with basic examples; in precept, chapters or even sections are readable independently of the final context, in order that elements could be simply used for graduate classes. Open questions are usually pointed out and the ultimate element of each one bankruptcy discusses references to the literature and occasionally supplementary effects. eventually, a close desk of Contents and an in depth Index are of support to refer to this monograph

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**Example text**

E. point of A. This in fact depends upon the rectifiability of A, compare Theorem 5 in Sec. 4 Remark 1. Consider the measures 1-tk and 7{1, k _< n, in R n. Although they can be very different, it is easily seen that 7dk (A) = 0 if and only if Hk 00 (A) = 0. e. x E E, and one can show that, if Q, Qv v = 1, 2.... are compact se and if every open set A D Q contains Q for v sufficiently large, then 7{k (Q) > lim sup 7{1 (Qv) . v-oo Notice that the last inequality is in general false for the Hausdorff measure 7-Ck.

E. in X 1. e. x E X and we may take Z = {x E X I Dµv(x) = +oo} . (iv) of Theorem 3 hold if X = Rn. The proof of Theorem 3 is based on the following lemma whose proof uses in turn Vitali's property. In view of Theorem 2 (ii), the claims (i) Lemma 3. Let A be any subset of X. (i) If Dµv(x) < a in A, then v(A) < ay(A) (ii) If D,,v(x) > a in A, then v(A) > ap(A) A simple consequence of Theorem 3 is the following theorem to which we shall return in Sec. 1. Theorem 4 (Lebesgue-Besicovitch's differentiation theorem).

3 Hausdorff Measures rn/2 13 n=1,2,... (n/2)I'(n/2) where F(t) is Euler's gamma function 00 j xt-1 exp(-x) dx , we may and do take as ws WS :_ (s/2)F(s/2) . Fig. 1. k (E) - 2, f' (E) - 4. Definition 1. Let X be a metric space. i is then defined by 7-ls(A) (2) Observe that 71 bola(A), AC X > 7-1o2 for 61 < b2i thus 7-1 is well defined and 'H'(A) = supW (A) . 5>0 Notice that it is essential to use the approximating measures 7-lb in order to define H'. The naive definition ECUCk} 7ls(E) k k 1. General Measure Theory 14 in fact would have very unpleasant features.