\(\varGamma \)-convergence of quadratic functionals with oscillating linear perturbations
In the present paper the authors take into account the \(\Gamma\)-convergence for functionals \[ I_{\varepsilon} (u) = \int_{\Omega} [(A_{\varepsilon}(x) \cdot \nabla u(x), \nabla u (x)) + b_{\varepsilon}(x) \cdot \nabla u(x) ] dx \] where \(A_{\varepsilon}(x) = A(x, \langle \frac{x}{l_1(\varepsilon)} \rangle)\) and \(b_{\varepsilon}(x) = b(x, \langle \frac{x}{l_2(\varepsilon)} \rangle)\) (\(\langle y \rangle\) stands for the fractional part of \(y \in {\mathbb R}\)) and \(l_j(\varepsilon) \searrow 0\) when \(\varepsilon \searrow 0\), \(j=1,2\), possibly \(l_1 = l_2\). The goal is to investigate which is the interaction between \(A_{\varepsilon}\) and \(b_{\varepsilon}\) in the limit. In the case \(n=1\) they study the general case (not only the periodic one). For \(n \geq 2\) they study the periodic case in detail and in the last section they also give some examples to try to answer the question in the general case.
scientific article; zbMATH DE number 5551962