Homogenization for conductive thin layers of pre-fractal type
The authors prove a convergence result for an energy problem associated to an elliptic problem posed in 2D domain with a thin layer of pre-fractal type. Let \(\Omega \) be the rectangle \((0,1]\times [ -1,1]\) and \(K_{0}\) be the segment joining \((0,0)\) to \((1,0)\).\ Given \(\alpha \in (2,4]\) and \( \theta =\arcsin \sqrt{\alpha -\alpha ^{2}/4}\), the authors introduce the four contractive similarities \(\varphi _{1}\), \(\varphi _{2}\), \(\varphi _{3}\) and \(\varphi _{4}\) associated to these parameters \(\alpha \) and \(\theta \). Applying a combination of \(n\) such similarities to \(K_{0}\) and taking the union of these images, the authors obtain a pre-fractal set \(K_{\varepsilon }^{n}\). In a quite similar way, they build a \(\varepsilon \)-neighborhood \( \Sigma _{\varepsilon }^{n}\) of \(K^{n}\) which can be decomposed in two subsets. The authors then introduce a weight \(w_{\varepsilon }^{n}\) which takes the value 1 outside \(\Sigma _{\varepsilon }^{n}\) and two other different values in the two subsets of \(\Sigma _{\varepsilon }^{n}\). These two values involve some constant \(c_{0}\). The energy functional under consideration is \(F_{\varepsilon }^{n}(u)=\int_{\Omega }| \nabla u| ^{2}w_{\varepsilon }^{n}\,d\xi\, d\eta \), defined on the appropriate weighted Sobolev space \(H_{0}^{1}(\Omega ;w_{\varepsilon }^{n})\), equipped with the appropriate weighted norm. The main convergence result of the paper asserts that \((F_{\varepsilon }^{n})_{\varepsilon }\) Mosco-converges in \( L^{2}(\Omega )\) to the functional \(F^{n}\) defined as \(F^{n}(u)=\int_{\Omega }| \nabla u| ^{2}\,d\xi \,d\eta +c_{0}\int_{K^{n}}| \nabla _{\tau }u| ^{2}\,ds\). Here \( \nabla _{\tau }u\) means the tangential gradient of \(u\) on each side of \(K^{n} \). The Mosco-convergence has been defined in [\textit{U. Mosco}, Adv. Math. 3, 510--585 (1969; Zbl 0192.49101)]. The authors first prove a uniform (with respect to \( \varepsilon \) and \(n\)) Poincaré inequality in \(H_{0}^{1}(\Omega ;w_{\varepsilon }^{n})\).
scientific article; zbMATH DE number 5312284
scientific article