Operator estimates for the crushed ice problem

Let $\Delta_{\Omega_\varepsilon}$ be the Dirichlet Laplacian in the domain $\Omega_\varepsilon:=\Omega\setminus\left(\cup_i D_{i \varepsilon}\right)$. Here $\Omega\subset\mathbb{R}^n$ and $\{D_{i \varepsilon}\}_{i}$ is a family of tiny identical holes ("ice pieces") distributed periodically in $\mathbb{R}^n$ with period $\varepsilon$. We denote by $\mathrm{cap}(D_{i \varepsilon})$ the capacity of a single hole. It was known for a long time that $-\Delta_{\Omega_\varepsilon}$ converges to the operator $-\Delta_{\Omega}+q$ in strong resolvent sense provided the limit $q:=\lim_{\varepsilon\to 0} \mathrm{cap}(D_{i\varepsilon}) \varepsilon^{-n}$ exists and is finite. In the current contribution we improve this result deriving estimates for the rate of convergence in terms of operator norms. As an application, we establish the uniform convergence of the corresponding semi-groups and (for bounded $\Omega$) an estimate for the difference of the $k$-th eigenvalue of $-\Delta_{\Omega_\varepsilon}$ and $-\Delta_{\Omega_\varepsilon}+q$. Our proofs relies on an abstract scheme for studying the convergence of operators in varying Hilbert spaces developed previously by the second author.Comment: now 24 pages, 3 figures; some typos fixed and references adde

Inverse homogenization problem for Poisson equation and relation between potential and capacity of holes

We consider Poisson problems $-\Delta u^\varepsilon=f$ on perforated domains, and characterize the limit of $u^\varepsilon$ as the solution to $(-\Delta+\mu)u=f$ on domain $\Omega$ with some potential $\mu\in W^{-1,\infty}(\Omega).$ It is known that $\mu$ is related to the capacity of holes when $\mu\in L^\infty(\Omega).$ In this paper, we characterize $\mu$ as the limit of the density of the capacity of holes also for some $\mu\notin L^\infty(\Omega).$ We apply the result for the inverse homogenization problem.Comment: 10 pages, 3 figure

NORM CONVERGENCE OF THE RESOLVENT FOR WILD PERTURBATIONS (presentation)

International audienceWe present here recent progress in the convergence of the resolvent of Laplace operators under wild perturbations. In particular, we show convergence in norm in a generalised sense. We focus here on the excision of many small balls in a complete Riemannian manifold with bounded geometry

Wildly perturbed manifolds: norm resolvent and spectral convergence

International audienceThe publication of the important work of Rauch and Taylor [RT75] started a hole branch of research on wild perturbations of the Laplace-Beltrami operator. Here, we extend certain results and show norm convergence of the resolvent. We consider a (not necessarily compact) manifold with many small balls removed, the number of balls can increase as the radius is shrinking, the number of balls can also be infinite. If the distance of the balls shrinks less fast than the radius, then we show that the Neumann Laplacian converges to the unperturbed Laplacian, i.e., the obstacles vanish. In the Dirichlet case, we consider two cases here: if the balls are too sparse, the limit operator is again the unperturbed one, while if the balls concentrate at a certain region (they become "solid" there), the limit operator is the Dirichlet Laplacian on the complement of the solid region. Norm resolvent convergence in the limit case of ho-mogenisation is treated elsewhere, see [KP18] and references therein. Our work is based on a norm convergence result for operators acting in varying Hilbert spaces described in the book [P12] by the second author

Operator estimates for Neumann sieve problem

Let $\Omega$ be a domain in $\mathbb{R}^n$, $\Gamma$ be a hyperplane intersecting $\Omega$, $\varepsilon>0$ be a small parameter, and $D_{k,\varepsilon}$, $k=1,2,3\dots$ be a family of small "holes" in $\Gamma\cap\Omega$; when $\varepsilon \to 0$, the number of holes tends to infinity, while their diameters tends to zero. Let $\mathscr{A}_\varepsilon$ be the Neumann Laplacian in the perforated domain $\Omega_\varepsilon=\Omega\setminus\Gamma_\varepsilon$, where $\Gamma_\varepsilon=\Gamma\setminus (\cup_k D_{k,\varepsilon})$ ("sieve"). It is well-known that if the sizes of holes are carefully chosen, $\mathscr{A}_\varepsilon$ converges in the strong resolvent sense to the Laplacian on $\Omega\setminus\Gamma$ subject to the so-called $\delta'$-conditions on $\Gamma$. In the current work we improve this result: under rather general assumptions on the shapes and locations of the holes we derive estimates on the rate of convergence in terms of $L^2\to L^2$ and $L^2\to H^1$ operator norms; in the latter case a special corrector is required.Comment: 33 pages, 3 figure