Anisotropic singular perturbations – the vectorial case. (English) Zbl 0804.49013
Summary: We obtain the \(\Gamma (L^ 1 (\Omega))\)-limit of the sequence \(J_ \varepsilon (u)= {1\over \varepsilon} E_ \varepsilon (u)\), where \(E_ \varepsilon\) is the family of anisotropic perturbations
\[
E_ \varepsilon (u):= \int_ \Omega W(u(x)) dx+ \varepsilon^ 2 \int_ \Omega h^ 2(x, \nabla u(x))dx
\]
of the nonconvex functional of vector- valued functions \(E_ 0(u)= \int_ \Omega W(u(x))dx\). The proof relies on the blow-up argument introduced by Fonseca and Müller.
MSC:
| 49J45 | Methods involving semicontinuity and convergence; relaxation |
Keywords:
anisotropic perturbationsReferences:
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