The norm resolvent convergence for elliptic operators in multi-dimensional domains with small holes
Let \(\Omega\subseteq \mathbb{R}^n\) be an open \(C^2\)-domain, let \(M\in \Omega\) and consider \(\omega\subseteq \mathbb{R}^n\) bounded and smooth. For \(\varepsilon>0\) put \(\omega_\varepsilon = M+\epsilon \omega\) as well as \(\Omega_\varepsilon = \Omega\setminus \omega_\varepsilon\). Furthermore, let \(A=(A_{i,j})_{i,j}\) be a symmetric-real-matrix valued \(C^1\)-function on \(\overline{\Omega}\), being bounded and strictly positive definite. Moreover, let \(V\in C(\overline{\Omega})\) and consider the operators \[ \mathcal{H}_\varepsilon = -\text{div } A \text{ grad} + V \] in \(L^2(\Omega_\varepsilon)\) subject to homogeneous Dirichlet boundary conditions on \(\partial\Omega\) and Dirichlet, Neumann or Robin type boundary conditions on \(\partial \omega_\varepsilon\). The operator \(\mathcal{H}_0\) is defined with the same differential expression as \(\mathcal{H}_\varepsilon\) with homogeneous Dirichlet boundary conditions on \(\partial\Omega\) acting as an operator on \(L^2(\Omega)\). The main result of the present paper is the following. For Neumann (and certain Robin-type) boundary conditions at \(\partial \omega_\varepsilon\) and \(\lambda \in \mathbb{C}\setminus\mathbb{R}\) there exists \(C\geq 0\) schapteruch that for \(\varepsilon\in (0,1)\) we have \[ \| \big(\mathcal{H}_\varepsilon -\lambda\big)^{-1} - \big(\mathcal{H}_0 -\lambda\big)^{-1} \|_{L^2(\Omega)\to H^1(\Omega_\varepsilon)}\leq C\varepsilon. \] For Dirichlet type boundary conditions the right-hand side of this estimate reads \(C\sqrt{\varepsilon}\) \((n=3)\), \(C\varepsilon\sqrt{|\ln\varepsilon|}\) \((n=4)\), and \(C\varepsilon\) \((n\geq 5)\). One reason for this dimension-dependent behaviour is that the estimates of certain averages over the wholes \(\omega_\varepsilon\) for solutions \(u_0\) to the problem on \(\Omega\), behave in a dimension-dependent way on the right-hand side of the resolvent problem.
dimension-dependent convergence estimates
scientific article; zbMATH DE number 6919768
scientific article