Nonexistence of solutions in nonconvex multidimensional variational problems

Tomáš Roubíček, Vladimir Sverak

Research output: Contribution to journalArticlepeer-review

4 Scopus citations

Abstract

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 ∫Ω W(x, ∇u) dx with a Carathéodory (if n ≥ 2) or even continuous (if n ≥ 3) integrand W.

Original languageEnglish (US)
Pages (from-to)427-435
Number of pages9
JournalJournal of Convex Analysis
Volume7
Issue number2
StatePublished - Dec 1 2000

Keywords

  • Bauer principle
  • Cantor sets
  • Extreme points
  • Gradient Young measures
  • Integration factors
  • Nonattainment

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