
Journal of Convex Analysis 07 (2000), No. 2, 427435 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 ndimensional situation, the extreme points in the set of certain gradient L^{p}Young measures are studied. For n = 1, such Young measures must be composed from Diracs, while for n ≥ 2 there are nonDirac 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 minimizationtype variational problem Integral_{0} 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 [ Fulltextpdf (287 KB)] 