DOI: 10.68381/jca20041 ISSN: 0944-6532
Oscillations and Concentrations in Sequences of Gradients up to the Boundary
Stefan Krömer, Martin Kružík
Oscillations and concentrations in sequences of gradients
\{\nabla 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.