On boundary-value problems for the Laplacian in bounded domains with micro inhomogeneous structure of the boundaries
The authors consider the bounded 3D domain \(\Omega \) with smooth boundary \( \Gamma \), lying in the half-space \(\{x_{3}<0\}\) and such that the measure of \(\Gamma _{1}=\text{int}(\{x_{3}=0\}\cap \Gamma )\) is non empty. The authors dispose in a \(\varepsilon \)-periodic way on \(\Gamma _{1}\) some \(\delta \) -reproductions of a smooth 2D domain \(\omega \). The small parameter \(\delta \) is supposed to depend on \(\varepsilon \) in such a way that \( \lim_{\varepsilon \rightarrow 0}\varepsilon /\delta =p\in [ 0,+\infty ] \). The authors consider the elliptic problem \(\Delta u_{\varepsilon ,\delta }=f_{\varepsilon }\) in \(\Omega \), with homogeneous Dirichlet boundary conditions on the thin zones and on the part of the boundary of \(\Omega \) not included in \(\{x_{3}=0\}\) and with a Robin type boundary condition \( \partial u_{\varepsilon ,\delta }/\partial x_{3}+qu_{\varepsilon ,\delta }=0\) on the rest of the boundary. Here \(q\) is nonnegative, \(f_{\varepsilon }\) belongs to \(L_{2}(\Omega )\) and converges to some \(f\) in the strong topology of this space. The two main results of the paper describe the asymptotic behaviour of this solution when \(\varepsilon \) goes to 0.\ The authors distinguish between the cases \(p=+\infty \) and \(p<+\infty \). In both cases, the solution \(u_{\varepsilon ,\delta }\) converges in the strong topology of \( H^{1}(\Omega )\) to the solution of an elliptic problem of the same type, but with different kinds of boundary conditions. The two other main results consider a spectral problem, the authors replacing \(f_{\varepsilon }\) by \( \lambda _{\varepsilon ,\delta }u_{\varepsilon ,\delta }\). The proof of these convergence results is based on the use of the variational formulations of these problems, on the derivation of uniform estimates for the solution and on the construction of appropriate test-functions which allow to pass to the limit in the variational formulations. The paper ends with a long list of references dealing with such homogenization problems.
homogeneous Dirichlet boundary conditions
scientific article; zbMATH DE number 5162301
scientific article