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Homogenization of Neumann problems for unbounded integral functionals. (English) Zbl 0940.49015

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.

MSC:

49J45 Methods involving semicontinuity and convergence; relaxation
35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
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