Journal of Convex Analysis, Vol. 07, No. 2, pp. 427-435 (2000)

Nonexistence of Solutions in Nonconvex Multidimensional Variational Problems

Roubícek, Tomás; Sverak, Vladimír

Mathematical Institute Charles University Sokolovská 83 18675 Praha 8 Czech Republic
School of Mathematics Vincent Hall University of Minnesota Minneapolis, MN 55455 U.S.A.

Abstract: In the scalar n-dimensional situation, the extreme points in the set of certain gradient L<sup>p</sup>-Young measures are studied. For n = 1, such Young measures must be composed from Diracs, while for n ≥ 2 there are non-Dirac extreme points among them, for n ≥ 3, some are even weakly* continuous. This is used to construct nontrivial examples of nonexistence of solutions of the minimization-type variational problem Integral<sub>0</sub> W(x, nabla u) dx with a Caratheodory (if n ≥ 2) or even continuous (if n ≥ 3) integrand W.

Classification (MSC2000): 49J99

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