Localization effect for a spectral problem in a perforated domain with Fourier boundary conditions (Q2843450)

The paper describes the asymptotic behaviour of the eigenpairs for elliptic problems posed in periodically perforated domains and with Fourier boundary conditions on the boundary of the perforations. Let \(\Omega \) be a bounded domain of \(\mathbb{R}^{d}\), \(d\geq 2\), with Lipschitz boundary \(\partial \Omega \), \(K=[0,1)^{d}\), \(E\subset \mathbb{R}^{d}\) be a \(K\)-periodic, open and connected set with Lipschitz boundary \(\Sigma \), and \(B=\mathbb{R} ^{d}\setminus \Sigma \). \(Y=K\cap E\) is supposed to be connected and \(K\cap B\Subset K\). For every \(i\in \mathbb{Z}^{d}\), let \(Y_{\varepsilon }^{i}=\varepsilon (i+Y)\), \(\Sigma _{\varepsilon }^{i}=\varepsilon \Sigma \cap Y_{\varepsilon }^{i}\), \(B_{\varepsilon }^{i}=\varepsilon B\cap Y_{\varepsilon }^{i}\) and \(\Omega _{\varepsilon }=\Omega \setminus \bigcup _{i\in I_{\varepsilon }}B_{\varepsilon }^{i}\), where \(I_{\varepsilon }=\{i\in \mathbb{Z}^{d}:Y_{\varepsilon }^{i}\subset \Omega \}\). The authors consider the spectral problem \(-\mathrm{div}(a^{\varepsilon }(x)\nabla u^{\varepsilon }(x))=\lambda ^{\varepsilon }u^{\varepsilon }(x)\), posed in \(\Omega _{\varepsilon }\) with the Fourier boundary condition \(a^{\varepsilon }(x)\nabla u^{\varepsilon }(x)\cdot n=-q(x)u^{\varepsilon }(x)\) and the homogeneous Dirichlet boundary conditions \(u^{\varepsilon }(x)=0\) on \( \partial \Omega \).NEWLINENEWLINEIn the first problem under consideration \(a^{\varepsilon }(x)=a(x/\varepsilon )\), where \(a\) is a \(d\times d\) matrix of \(Y\)-periodic and bounded coefficients which satisfies a uniform coercivity condition. \( q\in C^{3}(\mathbb{R}^{d})\) is a positive function. It is supposed that \(q\) has a unique global minimum at \(0\in \Omega \) and that in the vicinity of \(0\) it holds \(q(x)=q(0)+\frac{1}{2}x^{T}H(q)x+o(x^{2})\) for a positive definite Hessian matrix \(H(q)\). The authors introduce the eigenpairs \((\lambda _{j}^{\varepsilon },u_{j}^{\varepsilon })\) for this problem. The first main result proves the asymptotic behaviour of the eigenpairs \((\lambda _{j}^{\varepsilon },u_{j}^{\varepsilon })\) in terms of the eigenpairs of a rescaled problem. The second main result proves quite similar results but now considering \(a^{\varepsilon }(x)=a(x,x/\varepsilon )\) and \( q(x)=q^{\varepsilon }(x)=q(x,x/\varepsilon )\), where \(a\) and \(q\) satisfy quite similar hypotheses than the preceding ones. The third (resp. fourth) main result is devoted to the study of the case obtained when \( a^{\varepsilon }(x)=a(x/\varepsilon )\) and \(q(x)=\varepsilon q(x)\) (resp. \( a^{\varepsilon }(x)=a(x/\varepsilon )\) and \(q(x)=q(x)/\varepsilon \)). For the proofs of these results, the authors introduce some rescaled problem and they study its solution using the formal asymptotic expansion series tool. They also study the corresponding effective problem.