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

By Mariano Giaquinta

This monograph (in volumes) bargains with non scalar variational difficulties coming up in geometry, as harmonic mappings among Riemannian manifolds and minimum graphs, and in physics, as good 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 simple method, illustrating effects with easy examples; in precept, chapters or even sections are readable independently of the final context, in order that elements will 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 infrequently supplementary effects. ultimately, a close desk of Contents and an in depth Index are of support to refer to this monograph

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Actually one proves Proposition 1. 7-a and 7{s are measures on X; moreover 1s is Borel-regular for alls>0. If X = RI and A is a Hs-measurable subset of X with 7-ls(A) < oc, then Hs L A is a Radon measure on X. 3 Hausdorff Measures 15 Using the so-called isodiametric inequality Ln (A) <(diarn w2 A ' VA C R, which says that among sets A C ][fin with a given diameter p, the ball with diameter p has the largest Lebesgue measure, one also proves the following important theorem Theorem 1. The Hausdorff measure H-ln in R' coincides with the Lebesgue measure Ln* on IEBn.

We speak of approximate limit of f at x in case aplim f (y) := aplimsup f (y) = apliminf f (y) y-+x 'Y-+x and we say f to be approximately continuous at x if aplim f (y) = f (x) y-x We then have, compare Sec. 4 (i) f is approximately continuous at x if for every open set U C we have R with f (x) E U 0(f-1(u), x) = 1 (ii) f is approximately continuous at x if there exists a measurable set E with x E E such that 0(E,x) = 1 and fj E is continuous at x, (iii) if f E Lioc(R ; µ) and aplimy-x f (y) = f (x), then lim.

E. in X and 1. General Measure Theory 26 UkLlfkI = JLL(Ifk W k) = ULifl The same argument yields the result for v+ and v-. 1 dv+ dv- dlvl dIvl ' then we have v = Ivl L dv dlvl and by definition dv + ( dJvJ ) dv+ du dv- d-lvl (dlvl) dI vI ' If vk, k = 1, ... , m, are signed measures, vk = Ak L fk, defining p ;_ Ek µk we find /k = /LL (fk dd/G ). thus we conclude Proposition 1. If vI, ... , v' are signed measures, then v = (vl, ... , vm) is a vector valued measure in the sense of Definition 2. Therefore µ = (41,..

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