D’Apice, Ciro; De Maio, Umberto A homogenization result for unbounded variational functionals. (English) Zbl 0949.49013 Rend. Accad. Naz. Sci. XL, Mem. Mat. Appl. (5) 20, 65-93 (1996). 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. Reviewer: Z.Denkowski (Kraków) MSC: 49J45 Methods involving semicontinuity and convergence; relaxation 74C05 Small-strain, rate-independent theories of plasticity (including rigid-plastic and elasto-plastic materials) Keywords:unbounded integral functionals; homogenization; oscillating constraints on gradients; elastic-plastic torsion PDFBibTeX XMLCite \textit{C. D'Apice} and \textit{U. De Maio}, Rend. Accad. Naz. Sci. XL, Mem. Mat. Appl. (5) 20, 65--93 (1996; Zbl 0949.49013)