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2 defined by C f (x) := sup{g(x) I g : V --* Il8, g convex, g < f } . 28 1. Regular Variational Integrals (a) (b) Fig. 1. c. envelop. Since the pointwise supremum of convex functions is clearly convex, Cf is convex. Moreover Theorem 2 reads as Theorem 4.

Regular and smooth integrands in the Calculus of Variations. Let f (x, u, p) be a non-negative, smooth function from Sl x RN x RnN into R, which is convex with respect to the last argument p. 2 Some Classical Lower Semicontinuity Theorems 15 Theorem 6 (Tonelli-Morrey). c. with respect to the weak convergence in W1,1 Theorem 6 is an immediate consequence of the following Theorem 7. Let f be as previously. f2 x RnN)weak. Proof. Let uk , u strongly in L' and Pk p weakly in L'. F(u,p) < oo). Since f is convex in p we have n N f (x, uk, P) + E E af (x, uk, P) (P,k - Pa ) f(x,uk,Pk) U=1 i=1 n N f(x,uk,P) + pa (x,u,P)(Pak - Pa) EEf aPa Pa Q=1 i=1 N n a=1i=1 r [a af (x,uk,P) - aP f (x,u,P)j (Pak -Pa) 11 Since pk - p in L1 and fp(x, u, p) is bounded in K, we get k-moo f fp (x, u, P) (Pk - P) dx = 0 .

F2 x R' -> R+ be any Borel function. c. in L' (0, RN; µ) with respect to the weak* convergence and that F is not identically +oo. e. f2 and all u. E RN. n defined on some subclass of the class of smooth maps u : 1? C Rn - I[8N Definition 1. ) Lxn(gu,rl) for all admissible smooth functions u. Correspondingly, we say that the integrand f (x, u, G) is regular if there exists a positive constant v such that (3) f (x, u, G) > v jM(G) j for all admissible N x n-matrices G. 24 1. F-energies are well identified as Cartesian currents.

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