Abstract

In the scalar n-dimensional situation, the extreme points in the set of certain gradient LpL^p-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 ∫ΩW(x,∇u) dx\int_\Omega W(x,\nabla u)\,\mathrm{d}x with a Caratheodory (if n ≥ 2) or even continuous (if n ≥ 3) integrand W.

Author information

Contact details are reproduced from the original publication and may be historical.

Tomáš Roubíček

Mathematical Institute, Charles University, Sokolovská 83, 18675 Praha 8, Czech Republic
and: Institute of Information Theory and Automation, Academy of Sciences, Pod vodárenskou věží 4, CZ-182 08 Praha 8, Czech Republic

Vladimír Šverák

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

Keywords

  • Gradient Young measures
  • extreme points
  • Cantor sets
  • integration factors
  • Bauer principle
  • nonattainment

Mathematics Subject Classification

49J99

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Suggested citation

T. Roubíček, V. Šverák. “Nonexistence of Solutions in Nonconvex Multidimensional Variational Problems.” Journal of Convex Analysis 7 (2000), No. 2, 427–435. https://doi.org/10.68381/jca07022

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Published by Heldermann Verlag, 2000. Rights now held by Banach Press.