Abstract

Oscillations and concentrations in sequences of gradients {∇uk}\{\nabla u_k\}, bounded in Lp(Ω;RM×N)L^p(\Omega; \mathbb{R}^{M\times N}) if p>1p>1 and Ω⊂Rn\Omega\subset\mathbb{R}^n is a bounded domain with the extension property in W1,pW^{1,p}, and their interaction with local integral functionals can be described by a generalization of Young measures due to DiPerna and Majda. We characterize such DiPerna-Majda measures, thereby extending a result by A. Ka{ł}amajska and M. Kruž{í}k [``Oscillations and concentrations in sequences of gradients'', ESAIM, Control Optim. Calc. Var. 14(1) (2008) 71–104], where the full characterization was possible only for sequences subject to a fixed Dirichlet boundary condition. As an application we state a relaxation result for noncoercive multiple-integral functionals.

Author information

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

Martin Kružík

Institute of Information Theory and Automation, Academy of Sciences of the Czech Republic, Pod vodárenskou věží 4, 182 08 Praha 8, Czech Republic
and: Faculty of Civil Engineering, Czech Technical University, Thákurova 7, 166 29 Praha 6, Czech Republic

kruzik@utia.cas.cz

Keywords

  • Sequences of gradients
  • concentrations
  • oscillations
  • quasiconvexity

Mathematics Subject Classification

49J45, 35B05

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Suggested citation

S. Krömer, M. Kružík. “Oscillations and Concentrations in Sequences of Gradients up to the Boundary.” Journal of Convex Analysis 20 (2013), No. 3, 723–752. https://doi.org/10.68381/jca20041

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Published by Heldermann Verlag, 2013. Rights now held by Banach Press LLC.