Carbone, Luciano; Corbo Esposito, Antonio; De Arcangelis, Riccardo Homogenization of Neumann problems for unbounded integral functionals. (English) Zbl 0940.49015 Boll. Unione Mat. Ital., Sez. B, Artic. Ric. Mat. (8) 2, No. 2, 463-491 (1999). The paper deals with the homogenization problems for integral functionals of Calculus of Variations which are defined on some classes of functions belonging to Sobolev spaces \(W^{1,p}(\Omega)\) (or to \(W^{1,\infty}(\Omega)\)) whose gradients are subject to pointwise oscillating constraints. Using the \(\Gamma\)-convergence techniques the authors obtain some integral representations of the homogenized functionals. Then they apply the results to the minimization of the Neumann type problems: \[ \begin{split} j_{\varepsilon}^p(\Omega,\beta,\lambda) = \min \Biggl\{ \int_{\Omega} \phi (\tfrac x\varepsilon,Du) dx + \int_{\Omega} \beta u dx + \lambda \int_{\Omega}|u|dx:\\ u \in W^{1,p}(\Omega), |Du(x)|\leq m(\tfrac x\varepsilon) \text{ a.e. }x \in \Omega \Biggr\}. \end{split} \] Above \(\Omega\) denotes an open and bounded subset in \({\mathbb R}^n\) and the functions \(\phi(\cdot,z), m(\cdot)\) are \(Y\)-periodic (\(Y = ]0,1[^n\)). Due to the general \(\Gamma\)-convergence theory they get the convergence (as \(\varepsilon\) tends to \(0\)) of the minima \(j_{\varepsilon}^p(\Omega,\beta,\lambda)\) and the \(\varepsilon\)-minimizers, respectively, to the minimum value \(j_{\text{hom}}^p(\Omega,\beta,\lambda)\) and to the minimizers of the limit (homogenized) problem which has similar form as above with the integrand \(\phi\) replaced by \(\phi_{\text{hom}}(Du)\). The last (homogenized) integrand can be calculated by solving the auxiliary minimization problem. Reviewer: Z.Denkowski (Kraków) Cited in 2 Documents MSC: 49J45 Methods involving semicontinuity and convergence; relaxation 35B27 Homogenization in context of PDEs; PDEs in media with periodic structure Keywords:unbounded integral functionals; homogenization; Neumann minimum problems; gradient constrained problems; \(\Gamma\)-convergence PDFBibTeX XMLCite \textit{L. Carbone} et al., Boll. Unione Mat. Ital., Sez. B, Artic. Ric. Mat. (8) 2, No. 2, 463--491 (1999; Zbl 0940.49015) Full Text: EuDML