Oscillations and Concentrations in Sequences of Gradients up to the Boundary
Stefan Krömer, Martin Kruÿz ́ õk · Journal of convex analysis · 2013
Oscillations and concentrations in sequences of gradients \{ abla u_k\} { ∇ u k } , bounded in L^p(\Omega; \mathbb{R}^{M\times N}) L p ( Ω ; R M × N ) if p>1 p > 1 and \Omega\subset\mathbb{R}^n Ω ⊂ R n is a bounded domain with the extension property in W^{1,p} W 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. Kałamajska and M. Kružík [“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.