By Mariano Giaquinta

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 solid equilibrium configuations in nonlinear elasticity or for liquid crystals. The presentation is selfcontained and obtainable to non experts. issues are handled so far as attainable in an uncomplicated approach, 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 part of every one bankruptcy discusses references to the literature and occasionally supplementary effects. eventually, an in depth desk of Contents and an in depth Index are of support to refer to this monograph

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**Sample text**

Then Cf (x) +oo rCf (x) = lim inf C f (y) y_x In particular rCf = Cf on x E int (co (D f)) x co (D f ) x E rco (D f) V \ aco (D f) . Proposition 4 can be slightly generalized as follows. Suppose that the effective domain E )f of f is contained in an affine subspace W of V, Df C W C V, and denote by awD f the boundary of D f in W. Then we have Proposition 5. Let f : V -> RU {+oo}. Suppose that f has an affine minorant and D f C W, W being an of rie subspace of V. Then rCf = Cf on V\8wDf . Moreover rCf = C f = +oc on V \ W.

1. f V iff f () = f (G£) V l; E El, if Equalities in (2) follow as we take into account the homogeneity of f . 1 Mass and comass.. c. envelop is also the result of a duality procedure which is very similar to the duality mass and comass introduced by Whitney. 2 Parametric Polyconvex Envelops of Integrands 39 For any k-vector E and any k-covector w respectively in AkR' and AkI[8n we denote the Euclidean norms of and w by Iwl = (Ew2)1/2 ICI = (ES2)1/2 As we have seen in Chapter 2 for 2 < k < n - 2, the set of all simple k-vectors E C AkR' is strictly contained in AkR'.

This means that (a+ao,b+bo) E UnT(x) for any x E A. c.. Remark 1. It is worth to notice that the assumption (i) in Theorem 1 can be replaced by (i)' There exists uo E C°(Q) such that f(x,uo(x)) E C°(Sl) In fact in this case we can apply Theorem 1 to g(x, u) := f (x, u+uo) - f (x, uo), and find again f(x,u) =sup{ah(x) + (bh(x),u) I ah E C°(S2), bh E C'o(0, h )} . ) and find f (x, u) = sup{ah(x) + (bh(x), u) I ah E L- (Q), bh E Co(Sl, RN)} . h We now state Theorem 2. Let f Sl x RN --f A+ be any Borel function.