Deterministic homogenization of integral functionals with convex integrands
The authors develop a framework adapted to the homogenization of problems involving integral functionals and based on the concept of \(H\)-algebra. The generic problem is written as \(\min \{F_\varepsilon(v):v\in W_0^{1,p}(\Omega ;\mathbb R^n)\}\) where the functional \(F_\varepsilon\) is defined on \(W_{0}^{1,p}(\Omega ;\mathbb R^{n})\) by \(F_{\varepsilon }(v)=\int_{\Omega }f(x,x/\varepsilon ,Dv)\,dx\). Here \(\Omega \) is a bounded and open subset of \(\mathbb R^{N}\), and \(f\) is a function from \(\mathbb R^{N}\times\mathbb R^{N}\times\mathbb R^{nN}\) to \([0,+\infty )\) which satisfies some hypotheses among which are smoothness, measurability and a \(p\) -growth conditions. The main tools of the study are the definitions of homogenization algebras and of a weak \(\Sigma\)-convergence for sequences of functions in \(L^{p}(\Omega )\). The main result of the paper establishes the structure of the limit integral functional which involves an appropriate \(H\)-algebra \(A\) which is supposed to satisfy some ``proper'' further hypothesis. This main result also proves the weak convergence of the sequence \( (u)_{\varepsilon }\) in \(W_{0}^{1,p}(\Omega ;\mathbb R^{n})\) and the weak convergence to \((\frac{\partial u}{\partial x_{i}})_{\varepsilon }\) in \( L^{p}(\Omega )^{n}\). The paper ends with some applications of the general result to some concrete situations. The authors start with a periodic case following with almost periodic cases.
scientific article; zbMATH DE number 5822925
scientific article