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A homogenization result for unbounded variational functionals. (English) Zbl 0949.49013

In the paper the problem of homogenization of some integral functionals defined on functions subject to oscillating constraints on their gradients is considered. The motivation comes from the theory of elastic-plastic torsion. In abstract setting the problem concerns the asymptotic behaviour (for every open and bounded \(\Omega \subset {\mathbb R}^n\) and \(\beta \in L^1(\Omega)\), as \(h \rightarrow + \infty\)) of solutions to variational problems: \[ m_h(\Omega, \beta) = \min \biggl\{\int_{\Omega}f(hx,u,Du) dx + \int_{\Omega}\beta u dx \biggr\}, \] where \(\min\) is taken over all Lipschitz continuous functions \(u\) such that \(u = 0\) on \(\partial \Omega\), \(|Du(x)|\leq \varphi (x)\) for a.e. \(x \in \Omega\). Above, \(\varphi (\cdot)\) and \(f(\cdot,s,z)\) are measurable and \({]0,1[}^n\)-periodic functions satisfying some additional regularity assumptions. Using \(\Gamma\) convergence theory the authors find the integral representation of the \(\Gamma\) limit (in \(C_0^0(\Omega)\)) and prove also the convergence of minimal values of the problems to the minimal value of the limit problem.

MSC:

49J45 Methods involving semicontinuity and convergence; relaxation
74C05 Small-strain, rate-independent theories of plasticity (including rigid-plastic and elasto-plastic materials)
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