Periodic Riemannian manifold with preassigned gaps in spectrum of Laplace-Beltrami operator

It is known (E.L. Green (1997), O. Post (2003)) that for an arbitrary $m\in\mathbb{N}$ one can construct a periodic non-compact Riemannian manifold $M$ with at least $m$ gaps in the spectrum of the corresponding Laplace-Beltrami operator $-\Delta_M$. In this work we want not only to produce a new type of periodic manifolds with spectral gaps but also to control the edges of these gaps. The main result of the paper is as follows: for arbitrary pairwise disjoint intervals (\a_j,\b_j)\subset[0,\infty), $j=1,...,m$ ($m\in\mathbb{N}$), for an arbitrarily small $\delta>0$ and for an arbitrarily large $L>0$ we construct a periodic non-compact Riemannian manifold $M$ with at least $m$ gaps in the spectrum of the operator $-\Delta_{M}$, moreover the edges of the first $m$ gaps belong to $\delta$-neighbourhoods of the edges of the intervals (\a_j,\b_j), while the remaining gaps (if any) are located outside the interval $[0,L]$.Comment: 30 pages, 2 Figures; Journal of Differential Equations (2012

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

Spectral properties of elliptic operator with double-contrast coefficients near a hyperplane

In this paper we study the asymptotic behaviour as $\varepsilon\to 0$ of the spectrum of the elliptic operator $\mathcal{A}^\varepsilon=-{1\over b^\varepsilon}\mathrm{div}(a^\varepsilon\nabla)$ posed in a bounded domain $\Omega\subset\mathbb{R}^n$ $(n \geq 2)$ subject to Dirichlet boundary conditions on $\partial\Omega$. When $\varepsilon\to 0$ both coefficients $a^\varepsilon$ and $b^\varepsilon$ become high contrast in a small neighborhood of a hyperplane $\Gamma$ intersecting $\Omega$. We prove that the spectrum of $\mathcal{A}^\varepsilon$ converges to the spectrum of an operator acting in $L^2(\Omega)\oplus L^2(\Gamma)$ and generated by the operation $-\Delta$ in $\Omega\setminus\Gamma$, the Dirichlet boundary conditions on $\partial\Omega$ and certain interface conditions on $\Gamma$ containing the spectral parameter in a nonlinear manner. The eigenvalues of this operator may accumulate at a finite point. Then we study the same problem, when $\Omega$ is an infinite straight strip ("waveguide") and $\Gamma$ is parallel to its boundary. We show that $\mathcal{A}^\varepsilon$ has at least one gap in the spectrum when $\varepsilon$ is small enough and describe the asymptotic behaviour of this gap as $\varepsilon\to 0$. The proofs are based on methods of homogenization theory.Comment: In the second version we added the case r=0, also the presentation is essentially improved. The manuscript is submitted to a journa

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