R, compare Ch. 4. A key point in applying direct methods is compactness, compactness of sublevel sets, or compactness of all minimizing sequences.

The vectors ei yield a basis of the tangent plane to the graph of the map x Gx in , Rn+N. Aen) is then associated to the graph of Gx and, we recall, it is called the tangent n-vector to the graph of G. It orients the graph of the map x -+ Cx in terms of the orientation e1 A ... Aen of R1 and does not depend explicitely on the chosen bases. Recall also that, conversely, if f is a unit simple n-vector with positive first component t:0° > 0, then represents an n-plane with no vertical vector, and therefore there exists a linear map L : R' -> R' such that t; = M(L) (L) .

3) such that the maps vj(x) := tQ(i) (x) if x E Q(j) converge strongly in L1 to u. Passing to subsequences and using Fatou's lemma we have jy ff(u)dx < liminfJ f(vj)dx. (9) n S? i) 00 (10) < 1im inf k--oo lQ2i) f f (uQ=,)) dx < I 1im } f (uk(x)) dx oo i=1 QJ(i) f f (uk) dx < lim inf f f (uk) dx , x=1Q(j) and the result follows from (9) and (10). k--boo fl 11 For our future applications Theorem 5 is not sufficient, yet. We need a semicontinuity result for the integral in (1) under the convergence of Uk to u in the sense of measures.

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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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