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We are interested in the lower part of the spectrum of the Dirichlet Laplacian $A^\varepsilon$ in a thin waveguide $\Pi^\varepsilon$ obtained by repeating periodically a pattern, itself constructed by scaling an inner field geometry $\Omega$ by a small factor $\varepsilon>0$. The Floquet-Bloch theory ensures that the spectrum of $A^\varepsilon$ has a band-gap structure. Due to the Dirichlet boundary conditions, these bands all move to $+\infty$ as $O(\varepsilon^{-2})$ when $\varepsilon\to0^+$. Concerning their widths, applying techniques of dimension reduction, we show that the results depend on the dimension of the so-called space of almost standing waves in $\Omega$ that we denote by $\mathrm{X}_\dagger$. Generically, i.e. for most $\Omega$, there holds $\mathrm{X}_\dagger=\{0\}$ and the lower part of the spectrum of $A^\varepsilon$ is very sparse, made of bands of length at most $O(\varepsilon)$ as $\varepsilon\to0^+$. For certain $\Omega$ however, we have $\mathrm{dim}\,\mathrm{X}_\dagger=1$ and then there are bands of length $O(1)$ which allow for wave propagation in $\Pi^\varepsilon$. The main originality of this work lies in the study of the behaviour of the spectral bands when perturbing $\Omega$ around a particular $\Omega_\star$ where $\mathrm{dim}\,\mathrm{X}_\dagger=1$. We show a breathing phenomenon for the spectrum of $A^\varepsilon$: when inflating $\Omega$ around $\Omega_\star$, the spectral bands rapidly expand before shrinking. In the process, a band dives below the normalized threshold $\pi^2/\varepsilon^2$, stops breathing and becomes extremely short as $\Omega$ continues to inflate.

We investigate the spectrum of a Laplace operator with mixed boundary conditions in an unbounded chamfered quarter of layer. This problem arises in the study of the spectrum of the Dirichlet Laplacian in thick polyhedral domains having some symmetries such as the so-called Fichera layer. The geometry we consider depends on two parameters gathered in some vector $\kappa=(\kappa_1,\kappa_2)$ which characterizes the domain at the edges. By exchanging the axes and/or modifying their orientations if necessary, it is sufficient to restrict the analysis to the cases $\kappa_1\ge0$ and $\kappa_2\in[-\kappa_1,\kappa_1]$. We identify the essential spectrum and establish different results concerning the discrete spectrum with respect to $\kappa$. In particular, we show that for a given $\kappa_1>0$, there is some $h(\kappa_1)>0$ such that discrete spectrum exists for $\kappa_2\in[-\kappa_1,0)\cup(h(\kappa_1),\kappa_1]$ whereas it is empty for $\kappa_2\in[0,h(\kappa_1)]$. The proofs rely on classical arguments of spectral theory such as the max-min principle. The main originality lies rather in the delicate use of the features of the geometry.

In this paper, we provide uniform bounds for convergence rates of the low frequencies of a parametric family of problems for the Laplace operator posed on a rectangular perforated domain of the plane of height $H$. The perforations are periodically placed along the ordinate axis at a distance $O(\epsilon)$ between them, where $\epsilon$ is a parameter that converges towards zero. Another parameter $\eta$, the Floquet-parameter, ranges in the interval $[-\pi, \pi]$. The boundary conditions are quasi-periodicity conditions on the lateral sides of the rectangle and Neumann over the rest. We obtain precise bounds for convergence rates which are uniform on both parameters $\epsilon$ and $\eta$ and strongly depend on $H$. As a model problem associated with a waveguide, one of the main difficulties in our analysis comes near the nodes of the limit dispersion curves.

We give a description of the lower part of the spectrum of the Dirichlet Laplacian in an unbounded 3D periodic lattice made of thin bars (of width $\varepsilon\ll1$) which have a square cross section. This spectrum coincides with the union of segments which all go to $+\infty$ as $\varepsilon$ tends to zero due to the Dirichlet boundary condition. We show that the first spectral segment is extremely tight, of length $O(e^{-\delta/\varepsilon})$, $\delta>0$, while the length of the next spectral segments is $O(\varepsilon)$. To establish these results, we need to study in detail the properties of the Dirichlet Laplacian $A^{\Omega}$ in the geometry $\Omega$ obtained by zooming at the junction regions of the initial periodic lattice. This problem has its own interest and playing with symmetries together with max-min arguments as well as a well-chosen Friedrichs inequality, we prove that $A^{\Omega}$ has a unique eigenvalue in its discrete spectrum, which generates the first spectral segment. Additionally we show that there is no threshold resonance for $A^{\Omega}$, that is no non trivial bounded solution at the threshold frequency for $A^{\Omega}$. This implies that the correct 1D model of the lattice for the next spectral segments is a graph with Dirichlet conditions at the vertices. We also present numerics to complement the analysis.

We consider the propagation of acoustic waves in a waveguide containing a penetrable dissipative inclusion. We prove that as soon as the dissipation, characterized by some coefficient $\eta$, is non zero, the scattering solutions are uniquely defined. Additionally, we give an asymptotic expansion of the corresponding scattering matrix when $\eta\to0^+$ (small dissipation) and when $\eta\to+\infty$ (large dissipation). Surprisingly, at the limit $\eta\to+\infty$, we show that no energy is absorbed by the inclusion. This is due to the so-called skin-effect phenomenon and can be explained by the fact that the field no longer penetrates into the highly dissipative inclusion. These results guarantee that in monomode regime, the amplitude of the reflection coefficient has a global minimum with respect to $\eta$. The situation where this minimum is zero, that is when the device acts as a perfect absorber, is particularly interesting for certain applications. However it does not happen in general. In this work, we show how to perturb the geometry of the waveguide to create 2D perfect absorbers in monomode regime. Asymptotic expansions are justified by error estimates and theoretical results are supported by numerical illustrations.

We examine the band-gap structure of the spectrum of the Neumann problem for the Laplace operator in a strip with periodic dense transversal perforation by identical holes of a small diameter $\varepsilon>0$. The periodicity cell itself contains a string of holes at a distance $O(\varepsilon)$ between them. Under assumptions on the symmetry of the holes, we derive and justify asymptotic formulas for the endpoints of the spectral bands in the low-frequency range of the spectrum as $\varepsilon \to 0$. We demonstrate that, for $\varepsilon$ small enough, some spectral gaps are open. The position and size of the opened gaps depend on the strip width, the perforation period, and certain integral characteristics of the holes. The asymptotic behavior of the dispersion curves near the band edges is described by means of a `fast Floquet variable' and involves boundary layers in the vicinity of the perforation string of holes. The dependence on the Floquet parameter of the model problem in the periodicity cell requires a serious modification of the standard justification scheme in homogenization of spectral problems. Some open questions and possible generalizations are listed.

We consider the propagation of acoustic waves in a 2D waveguide unbounded in one direction and containing a compact obstacle. The wavenumber is fixed so that only one mode can propagate. The goal of this work is to propose a method to cloak the obstacle. More precisely, we add to the geometry thin outer resonators of width $\varepsilon$ and we explain how to choose their positions as well as their lengths to get a transmission coefficient approximately equal to one as if there were no obstacle. In the process we also investigate several related problems. In particular, we explain how to get zero transmission and how to design phase shifters. The approach is based on asymptotic analysis in presence of thin resonators. An essential point is that we work around resonance lengths of the resonators. This allows us to obtain effects of order one with geometrical perturbations of width $\varepsilon$. Various numerical experiments illustrate the theory.

The goal of this work is to design an acoustic mode converter. More precisely, the wave number is chosen so that two modes can propagate. We explain how to construct geometries such that the energy of the modes is completely transmitted and additionally the mode 1 is converted into the mode 2 and conversely. To proceed, we work in a symmetric waveguide made of two branches connected by two thin ligaments whose lengths and positions are carefully tuned. The approach is based on asymptotic analysis for thin ligaments around resonance lengths. We also provide numerical results to illustrate the theory., Comment: arXiv admin note: text overlap with arXiv:2011.06810

We consider the propagation of time harmonic acoustic waves in a device with three channels. The wave number is chosen such that only the piston mode can propagate. The main goal of this work is to present a geometry which can serve as an energy distributor. More precisely, the geometry is first designed so that for an incident wave coming from one channel, the energy is almost completely transmitted in the two other channels. Additionally, tuning a bit two geometrical parameters, we can control the ratio of energy transmitted in the two channels. The approach is based on asymptotic analysis for thin slits around resonance lengths. We also provide numerical results to illustrate the theory.

We consider the linear water-wave problem in a periodic channel $\Pi^h \subset \mathbb{R}^3$, which is shallow except for a periodic array of deep potholes in it. Motivated by applications to surface wave propagation phenomena, we study the band-gap structure of the essential spectrum in the linear water-wave system, which includes the spectral Steklov boundary condition posed on the free water surface. We apply methods of asymptotic analysis, where the most involved step is the construction and analysis of an appropriate boundary layer in a neighborhood of the joint of the potholes with the thin part of the channel. Consequently, the existence of a spectral gap for small enough $h$ is proven.

We consider the propagation of the piston mode in an acoustic waveguide obstructed by two screens with small holes. In general, due to the features of the geometry, almost no energy of the incident wave is transmitted through the structure. The goal of this article is to show that tuning carefully the distance between the two screens, which form a resonator, one can get almost complete transmission. We obtain an explicit criterion, not so obvious to intuit, for this phenomenon to happen. Numerical experiments illustrate the analysis.

We consider the propagation of surface water waves in a straight planar channel perturbed at the bottom by several thin curved tunnels and wells. We propose a method to construct non reflecting underwater topographies of this type at an arbitrary prescribed wave number. To proceed, we compute asymptotic expansions of the diffraction solutions with respect to the small parameter of the geometry taking into account the existence of boundary layer phenomena. We establish error estimates to validate the expansions using advances techniques of weighted spaces with detached asymptotics. In the process, we show the absence of trapped surface waves for perturbations small enough. This analysis furnishes asymptotic formulas for the scattering matrix and we use them to determine underwater topographies which are non-reflecting. Theoretical and numerical examples are given.

We investigate a time-harmonic wave problem in a waveguide. We work at low frequency so that only one mode can propagate. It is known that the scattering matrix exhibits a rapid variation for real frequencies in a vicinity of a complex resonance located close to the real axis. This is the so-called Fano resonance phenomenon. And when the geometry presents certain properties of symmetry, there are two different real frequencies such that we have either $R=0$ or $T=0$, where $R$ and $T$ denote the reflection and transmission coefficients. In this work, we prove that without the assumption of symmetry of the geometry, quite surprisingly, there is always one real frequency for which we have $T=0$. In this situation, all the energy sent in the waveguide is backscattered. However in general, we do not have $R=0$ in the process. We provide numerical results to illustrate our theorems.

We consider the Robin Laplacian in the domains $\Omega$ and $\Omega^\varepsilon$, $\varepsilon >0$, with sharp and blunted cusps, respectively. Assuming that the Robin coefficient $a$ is large enough, the spectrum of the problem in $\Omega$ is known to be residual and to cover the whole complex plane, but on the contrary, the spectrum in the Lipschitz domain $\Omega^\varepsilon$ is discrete. However, our results reveal the strange behavior of the discrete spectrum as the blunting parameter $\varepsilon$ tends to 0: we construct asymptotic forms of the eigenvalues and detect families of "hardly movable" and "plummeting" ones. The first type of the eigenvalues do not leave a small neighborhood of a point for any small $\varepsilon > 0$ while the second ones move at a high rate $O(|\ln \varepsilon|)$ downwards along the real axis $\mathbb{R}$ to $ -\infty$. At the same time, any point $\lambda \in \mathbb{R}$ is a "blinking eigenvalue", i.e., it belongs to the spectrum of the problem in $\Omega^\varepsilon$ almost periodically in the $|\ln \varepsilon|$-scale. Besides standard spectral theory, we use the techniques of dimension reduction and self-adjoint extensions to obtain these results.

We study the Steklov spectral problem for the Laplace operator in a bounded domain $\Omega \subset \mathbb{R}^d$, $d \geq 2$, with a cusp such that the continuous spectrum of the problem is non-empty, and also in the family of bounded domains $\Omega^\varepsilon \subset \Omega$, $\varepsilon > 0$, obtained from $\Omega$ by blunting the cusp at the distance of $\varepsilon$ from the cusp tip. While the spectrum in the blunted domain $\Omega^\varepsilon$ consists for a fixed $\varepsilon$ of an unbounded positive sequence $\{ \lambda_j^\varepsilon \}_{j=1}^\infty$ of eigenvalues, we single out different types of behavior of some eigenvalues as $\varepsilon \to +0$: in particular, stable, blinking, and gliding families of eigenvalues are found. We also describe a mechanism which transforms the family of the eigenvalue sequences into the continuous spectrum of the problem in $\Omega$, when $\varepsilon \to +0$.

We investigate a time-harmonic wave problem in a waveguide. By means of asymptotic analysis techniques, we justify the so-called Fano resonance phenomenon. More precisely, we show that the scattering matrix considered as a function of a geometrical parameter $\varepsilon$ and of the frequency $\lambda$ is in general not continuous at a point $(\varepsilon,\lambda)=(0,\lambda^0)$ where trapped modes exist. In particular, we prove that for a given $\varepsilon\ne0$ small, the scattering matrix exhibits a rapid change for frequencies varying in a neighbourhood of $\lambda^0$. We use this property to construct examples of waveguides such that the energy of an incident wave propagating through the structure is perfectly transmitted (non reflection) or perfectly reflected in monomode regime. We provide numerical results to illustrate our theorems.

We study the Laplacian in a smooth bounded domain, with a varying Robin boundary condition singular at one point. The associated quadratic form is not semi-bounded from below, and the corresponding Laplacian is not self-adjoint, it has the residual spectrum covering the whole complex plane. We describe its self-adjoint extensions and exhibit a physically relevant skew-symmetric one. We approximate the boundary condition, giving rise to a family of self-adjoint operators, and we describe their eigenvalues by the method of matched asymptotic expansions. These eigenvalues acquire a strange behaviour when the small perturbation parameter $\varepsilon>0$ tends to zero, namely they become almost periodic in the logarithmic scale $|\ln \epsilon|$ and, in this way, "wander" along the real axis at a speed $O(\eps^{-1})$.

In the junction $\Omega$ of several semi-infinite cylindrical waveguides we consider the Dirichlet Laplacian whose continuous spectrum is the ray $[\lambda_\dagger, +\infty)$ with a positive cut-off value $\lambda_\dagger$. We give two different criteria for the threshold resonance generated by nontrivial bounded solutions to the Dirichlet problem for the Helmholtz equation $-\Delta u=\lambda_\dagger u$ in $\Omega$. The first criterion is quite simple and is convenient to disprove the existence of bounded solutions. The second criterion is rather involved but can help to detect concrete shapes supporting the resonance. Moreover, the latter distinguishes in a natural way between stabilizing, i.e., bounded but non-descending solutions and trapped modes with exponential decay at infinity., Comment: 25 pages, 3 figures

We consider a time-harmonic wave problem, appearing for example in water-waves and in acoustics, in a setting such that the analysis reduces to the study of a 2D waveguide problem with a Neumann boundary condition. The geometry is symmetric with respect to an axis orthogonal to the direction of propagation of waves. Moreover, the waveguide contains one branch of finite length. We analyse the behaviour of the complex scattering coefficients $\mathcal{R}$, $\mathcal{T}$ as the length of the branch increases and we show how to design geometries where non reflectivity ($\mathcal{R}=0$, $|\mathcal{T}|=1$), perfect reflectivity ($|\mathcal{R}|=1$, $\mathcal{T}=0$) or perfect invisibility ($\mathcal{R}=0$, $\mathcal{T}=1$) hold. Numerical experiments illustrate the different results.

We are interested in a time harmonic acoustic problem in a waveguide with locally perturbed sound hard walls. We consider a setting where an observer generates incident plane waves at $-\infty$ and probes the resulting scattered field at $-\infty$ and $+\infty$. Practically, this is equivalent to measure the reflection and transmission coefficients respectively denoted $R$ and $T$. In [9], a technique has been proposed to construct waveguides with smooth walls such that $R=0$ and $|T|=1$ (non reflection). However the approach fails to ensure $T=1$ (perfect transmission without phase shift). In this work, first we establish a result explaining this observation. More precisely, we prove that for wavenumbers smaller than a given bound $k_{\star}$ depending on the geometry, we cannot have $T=1$ so that the observer can detect the presence of the defect if he/she is able to measure the phase at $+\infty$. In particular, if the perturbation is smooth and small (in amplitude and in width), $k_{\star}$ is very close to the threshold wavenumber. Then, in a second step, we change the point of view and, for a given wavenumber, working with singular perturbations of the domain, we show how to obtain $T=1$. In this case, the scattered field is exponentially decaying both at $-\infty$ and $+\infty$. We implement numerically the method to provide examples of such undetectable defects., Comment: Journal de Math\'ematiques Pures et Appliqu\'ees, 12/08/2017

We investigate the eigenvalue problem $-\text{div}(\sigma \nabla u) = \lambda u\ (\mathscr{P})$ in a 2D domain $\Omega$ divided into two regions $\Omega_{\pm}$. We are interested in situations where $\sigma$ takes positive values on $\Omega_{+}$ and negative ones on $\Omega_{-}$. Such problems appear in time harmonic electromagnetics in the modeling of plasmonic technologies. In a recent work [15], we highlighted an unusual instability phenomenon for the source term problem associated with $(\mathscr{P})$: for certain configurations, when the interface between the subdomains $\Omega_{\pm}$ presents a rounded corner, the solution may depend critically on the value of the rounding parameter. In the present article, we explain this property studying the eigenvalue problem $(\mathscr{P})$. We provide an asymptotic expansion of the eigenvalues and prove error estimates. We establish an oscillatory behaviour of the eigenvalues as the rounding parameter of the corner tends to zero. We end the paper illustrating this phenomenon with numerical experiments., Comment: Mathematical Modelling and Numerical Analysis (ESAIM: M2AN), 09/12/2016. arXiv admin note: text overlap with arXiv:1304.4788

We consider blood flow in a vessel with an attached capillary system. The latter is modeled with the help of a corresponding fractal graph whose edges are supplied with ordinary differential equations obtained by the dimension-reduction procedure from a three-dimensional model of blood flow in thin vessels. The Kirchhoff transmission conditions must be satisfied at each interior vertex. The geometry and physical parameters of this system are described by a finite number of scaling factors which allow the system to have self-reproducing solutions. Namely, these solutions are determined by the factors' values on a certain fragment of the fractal graph and are extended to its rest part by virtue of these scaling factors. The main result is the existence and uniqueness of self-reproducing solutions, whose dependence on the scaling factors of the fractal graph is also studied. As a corollary we obtain a relation between the pressure and flux at the junction, where the capillary system is attached to the blood vessel. This relation leads to the Robin boundary condition at the junction and this condition allows us to solve the problem for the flow in the blood vessel without solving it for the attached capillary system., Comment: 33 pages, 7 figures

We are interested in a time harmonic acoustic problem in a waveguide containing flies. The flies are modelled by small sound soft obstacles. We explain how they should arrange to become invisible to an observer sending waves from $-\infty$ and measuring the resulting scattered field at the same position. We assume that the flies can control their position and/or their size. Both monomodal and multimodal regimes are considered. On the other hand, we show that any sound soft obstacle (non necessarily small) embedded in the waveguide always produces some non exponentially decaying scattered field at $+\infty$ for wavenumbers smaller than a constant that we explicit. As a consequence, for such wavenumbers, the flies cannot be made completely invisible to an observer equipped with a measurement device located at $+\infty$.

We investigate a time harmonic acoustic scattering problem by a penetrable inclusion with compact support embedded in the free space. We consider cases where an observer can produce incident plane waves and measure the far field pattern of the resulting scattered field only in a finite set of directions. In this context, we say that a wavenumber is a non-scattering wavenumber if the associated relative scattering matrix has a non trivial kernel. Under certain assumptions on the physical coefficients of the inclusion, we show that the non-scattering wavenumbers form a (possibly empty) discrete set. Then, in a second step, for a given real wavenumber and a given domain D, we present a constructive technique to prove that there exist inclusions supported in D for which the corresponding relative scattering matrix is null. These inclusions have the important property to be impossible to detect from far field measurements. The approach leads to a numerical algorithm which is described at the end of the paper and which allows to provide examples of (approximated) invisible inclusions., Comment: 20 pages, 7 figures

We study a spectral problem $(\mathscr{P}^{\delta})$ for a diffusion like equation in a 3D domain $\Omega$. The main originality lies in the presence of a parameter $\sigma^{\delta}$, whose sign changes on $\Omega$, in the principal part of the operator we consider. More precisely, $\sigma^{\delta}$ is positive on $\Omega$ except in a small inclusion of size $\delta>0$. Because of the sign-change of $\sigma^{\delta}$, for all $\delta>0$ the spectrum of $(\mathscr{P}^{\delta})$ consists of two sequences converging to $\pm\infty$. However, at the limit $\delta=0$, the small inclusion vanishes so that there should only remain positive spectrum for $(\mathscr{P}^{\delta})$. What happens to the negative spectrum? In this paper, we prove that the positive spectrum of $(\mathscr{P}^{\delta})$ tends to the spectrum of the problem without the small inclusion. On the other hand, we establish that each negative eigenvalue of $(\mathscr{P}^{\delta})$ behaves like $\delta^{-2}\mu$ for some constant $\mu<0$. We also show that the eigenfunctions associated with the negative eigenvalues are localized around the small inclusion. We end the article providing 2D numerical experiments illustrating these results.