Norm-resolvent convergence for elliptic operators in domain with perforation along curve
Let \(\Omega\equiv\{x=(x_1,x_2)\in \mathbb{R}^2 : 0 <x_2<d\}\) be an infinite strip and let \(\gamma\) be a curve in \(\Omega\) with positive distance from \(\partial \Omega\). For \(\epsilon\) small and positive, let \(\theta^\epsilon_0\), \(\theta^\epsilon_1\) be the unions of two families of small holes along the curve \(\gamma\), such that the size of the perforations tends to \(0\) as \(\epsilon\to 0^+\). In the perforated domain \(\Omega^\epsilon \equiv \Omega \setminus (\theta^\epsilon_0 \cup \theta^\epsilon_1)\) the authors consider the operator \(\mathcal{H}^\epsilon\) in \(L_2(\Omega^\epsilon)\) introduced by the differential expression NEWLINE\[NEWLINE-\sum_{i,j=1}^2 \frac{\partial}{\partial x_i}A_{ij}\frac{\partial}{\partial x_j}+\sum_{j=1}^2\bigg(A_j \frac{\partial}{\partial x_j}-\frac{\partial }{\partial x_j}\overline{A_j}\bigg)+A_0NEWLINE\]NEWLINE in \(\Omega^\epsilon\), with Dirichlet condition on \(\partial \Omega \cup \partial \theta^\epsilon_0\) and Robin condition on \(\partial \theta^\epsilon_1\). The coefficients \(A_{ij}\) satisfy a standard uniform ellipticity condition in \(\Omega\), \(A_{ij}, A_i \in W^1_\infty (\Omega)\), \(A_{ij}=A_{ji}\) for \(i,j =1,2\), \(A_0 \in L_\infty (\Omega)\), and \(A_{ij}, A_0\) are real valued. Then the authors investigate the norm-resolvent convergence of the perturbed operator to a homogenized one, together with some estimates on the rate of convergence. The problem is motivated by waveguide theory and the perforations can be seen as micro-defects in the media.
scientific article; zbMATH DE number 6361366
scientific article