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Journal of Convex Analysis 07 (2000), No. 2, 427--435
Copyright Heldermann Verlag 2000

Nonexistence of Solutions in Nonconvex Multidimensional Variational Problems

Tomás Roubícek
Mathematical Institute, Charles University, Sokolovská 83, 18675 Praha 8, Czech Republic

Vladimír Sverak
School of Mathematics, Vincent Hall, University of Minnesota, Minneapolis, MN 55455, U.S.A.

In the scalar n-dimensional situation, the extreme points in the set of certain gradient Lp-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 Integral0 W(x, nabla u) dx with a Caratheodory (if n ≥ 2) or even continuous (if n ≥ 3) integrand W.

Keywords: Gradient Young measures, extreme points, Cantor sets, integration factors, Bauer principle, nonattainment.

MSC: 49J99

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