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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 ε ≪ 1) which have a square cross section. This spectrum coincides with the union of segments which all go to +∞ as ε tends to zero due to the Dirichlet boundary condition. We show that the first spectral segment is extremely tight, of length O(e−δ/ε), δ > 0, while the length of the next spectral segments is O(ε). To establish these results, we need to study in detail the properties of the Dirichlet Laplacian AΩ in the geometry Ω 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 Poincaré–Friedrichs inequality, we prove that AΩ has a unique eigenvalue in its discrete spectrum, which generates the first spectral segment. Additionally we show that there is no threshold resonance for AΩ, that is no non trivial bounded solution at the threshold frequency for AΩ. This implies that the correct 1D model of the lattice for the next spectral segments is a system of ordinary differential equations set on the limit graph with Dirichlet conditions at the vertices. We also present numerics to complement the analysis. [ABSTRACT FROM AUTHOR]

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(ε)$O(\varepsilon)$ between them, where ε is a parameter that converges toward zero. Another parameter η, the Floquet‐parameter, ranges in the interval [−π,π]$[-\pi ,\pi ]$. 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 ε and η 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. [ABSTRACT FROM AUTHOR]

Curved channels with constant cross sections are constructed which support a trapped surface wave. Since corresponding eigenvalues are embedded in the continuous spectrum of the water wave problem and therefore possess the natural instability, the construction procedure requires "fine-tuning" of several parameters in the (small) curvature of the channel as well as geometrical restrictions on the cross section. In particular, the mirror symmetry of the cross section with respect to the vertical axis disrupts the procedure, but examples of suitable non-symmetric cross sections are provided. [ABSTRACT FROM AUTHOR]

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 ε > 0. The periodicity cell itself contains a string of holes at a distance O (ε) 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 ε → 0. We demonstrate that, for ε 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. [ABSTRACT FROM AUTHOR]

We consider the propagation of time-harmonic acoustic waves in a device made of three unbounded channels connected by thin slits. The wavenumber is chosen such that only one mode can propagate. The main goal of this work is to present a device 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, adjusting slightly 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. [ABSTRACT FROM AUTHOR]

We consider a Dirichlet spectral problem for a second order differential operator, with piecewise constant coefficients, in a domain Ω ε in the plane R 2. Here Ω ε is Ω ∪ ω ε ∪ Γ , where Ω is a fixed bounded domain with boundary Γ, ω ε is a curvilinear band of width O (ε) , and Γ = Ω ‾ ∩ ω ‾ ε. The density and stiffness constants are of order ε − m − t and ε − t respectively in this band, while they are of order 1 in Ω; t ≥ 1 , m > 2 , and ε is a small positive parameter. We address the asymptotic behavior, as ε → 0 , for the eigenvalues and the corresponding eigenfunctions. In particular, we show certain localization effects for eigenfunctions associated with low frequencies. This is deeply involved with the extrema of the curvature of Γ. [ABSTRACT FROM AUTHOR]

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 by 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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We study the Steklov spectral problem for the Laplace operator in a bounded domain Ω ⊂ R d , d ≥ 2 , with a cusp such that the continuous spectrum of the problem is non-empty, and also in the family of bounded domains Ω ε ⊂ Ω , ε > 0 , obtained from Ω by blunting the cusp at the distance of ε from the cusp tip. While the spectrum in the blunted domain Ω ε consists for a fixed ε of an unbounded positive sequence { λ j ε } j = 1 ∞ of eigenvalues, we single out different types of behavior of some eigenvalues as ε → + 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 Ω, when ε → + 0. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We develop and investigate radiation conditions at infinity for composite piezo-elastic waveguides. The approach is based on the Mandelstam radiation principle according to which the energy flux at infinity is directed away from the source and which implies constraints on the (sign of the) group velocities. On the other side, the Sommerfeld radiation condition implies limitations on the wave phase velocity and is, in fact, not applicable in the context of piezo-elastic wave guides. We analyze the passage to the limit when the piezo-electric moduli tend to zero in certain regions yielding purely elastic inclusions there. We provide a number of examples, e.g. elastic and acoustic waveguides as well as purely elastic insulating and conducting inclusions. [ABSTRACT FROM AUTHOR]

Abstract We study the Laplacian in a 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 a 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 its spectrum by the method of matched asymptotic expansions. A part of the spectrum acquires a strange behavior when the small perturbation parameter ε > 0 tends to zero, namely it becomes almost periodic in the logarithmic scale | ln ⁡ ε | , and in this way “wanders” along the real axis at a speed O ( ε − 1 ). Résumé Nous étudions le laplacien dans un domaine borné, avec une condition à la frontière de type Robin, variable et singulière en un point. La forme quadratique associée n'est pas bornée inférieurement, et le laplacien correspondant n'est pas self-adjoint ; son spectre résiduel couvre entièrement le plan complexe. Nous décrivons ses extensions self-adjointes et nous en montrons une anti-symétrique, pertinente en physique. Nous approchons la condition de frontière à l'aide d'une famille d'opérateurs self-adjoints et nous décrivons son spectre par la méthode d'appariement des développements asymptotiques. Une partie du spectre adopte un comportement étrange quand le paramètre ε > 0 de petite perturbation tend vers zéro ; précisément, il devient presque périodique en échelle logarithmique | log ⁡ ( ε ) | , et ainsi « erre » le long de l'axe réel à une vitesse O ( ε − 1 ). [ABSTRACT FROM AUTHOR]

We consider a time-harmonic wave problem, appearing, for example, in water-wave theory, in acoustics, or in electromagnetism, 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 analyze the behavior of the complex scattering coefficients R, τas the length of the branch increases, and we show how to design geometries where nonreflectivity R = 0, ∣T ∣ = 1), perfect reflectivity (∣R ∣ = 1,T = 0), or perfect invisibility (R = 0, T = 1) holds. Numerical experiments illustrate the different results. [ABSTRACT FROM AUTHOR]

Abstract We consider the spectral Dirichlet–Laplacian problem on a domain which is formed from a periodic waveguide Π perturbed by non-compact, non-periodic changes of geometry. We show that the domain perturbation causes an addition to the essential spectrum, which consists of isolated points belonging to the discrete spectrum of a model problem. This model problem is posed on a domain, which is just a compact perturbation of Π. We discuss the position of the new spectral components in relation to the essential spectrum of the problem in Π. [ABSTRACT FROM AUTHOR]

We consider blood flow in a vessel with an attached capillary system. The latter is modelled 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. [ABSTRACT FROM AUTHOR]

We address a spectral problem for the Dirichlet-Laplace operator in a waveguide Πε. Πε is obtained from an unbounded two-dimensional strip Π which is periodically perforated by a family of holes, which are also periodically distributed along a line, the so-called "perforation string". We assume that the two periods are different, namely, O(1) and O(ε) respectively, where 0 < ε ≪ 1. We look at the band-gap structure of the spectrum σε as ε → 0. We derive asymptotic formulas for the endpoints of the spectral bands and show that σε has a large number of short bands of length O(ε) which alternate with wide gaps of width O(1). [ABSTRACT FROM AUTHOR]

A problem of scattering by a resonator connecting two 2D waveguides is studied. The incident wave is one of the waveguide modes taken at the spectral parameter close to a threshold of the continuous spectrum. It is shown that in the general case the reflection coefficient for such a mode is close to -1 (this case corresponds to an almost perfect reflection). Also it is shown that in some special cases the reflection coefficient is close to 0, and an almost perfect transmission is observed. The behavior of scattering at near-threshold frequencies is determined by solutions corresponding to the threshold spectral parameter and crucially depends on whether stabilizing solutions do exist or not. Anomalous transmission is observed when there exist only solutions growing at infinity. [ABSTRACT FROM AUTHOR]

A one-dimensional model of an elastic junction that contains hard- and readily-movable thin rods is derived, and asymptotic formulas for eigenvalues with rigorous estimates for remainders are given. In addition to vector functions satisfying classical ordinary differential equations, the model involves algebraic unknowns and algebraic relations corresponding to the longitudinal rigid motion of readily-movable rods. [ABSTRACT FROM AUTHOR]

We consider elastic and piezoelectric waveguides composed from identical beads threaded periodically along a spoke converging at infinity. We show that the essential spectrum constitutes a non-negative monotone unbounded sequence and thus has infinitely many spectral gaps. [ABSTRACT FROM AUTHOR]

Localized potentials in the Dirac equation for the electron dynamics in a zigzag graphene ribbon are constructed to support trapped modes while the corresponding eigenvalues are embedded into the continuous spectrum. [ABSTRACT FROM AUTHOR]

We construct a family of periodic piezoelectric waveguides Π, depending on a small geometrical parameter, with the following property: as ɛ → +0, the number of gaps in the essential spectrum of the piezoelectricity problem on Π grows unboundedly. [ABSTRACT FROM AUTHOR]

For the honeycomb lattice of quantum waveguides, the limit passage is performed when the relative thickness h of ligaments tends to zero and the asymptotic structure of the spectrum of the Dirichlet Laplacian is described. [ABSTRACT FROM AUTHOR]

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 η, is non zero, the scattering solutions are uniquely defined. Additionally, we give an asymptotic expansion of the corresponding scattering matrix when η → 0+ (small dissipation) and when η → +∞ (large dissipation). Surprisingly, at the limit η → +∞, 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 η. 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. [ABSTRACT FROM AUTHOR]

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 ε 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 ε . Various numerical experiments illustrate the theory. [ABSTRACT FROM AUTHOR]

We consider the linear water‐wave problem in a periodic channel Πh⊂R2$\Pi ^h \subset {\mathbb {R}}^2$, which consists of infinitely many identical containers and connecting thin structures. The connecting canals are assumed to be of constant, positive length, but their depth is proportional to a small parameter h. 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 forms a spectral problem where the spectral parameter appears in the Steklov boundary condition posed on the free water surface. We show that for small h there exists a large number of spectral gaps and also find asymptotic formulas for the position of the gaps as h→0${h} \rightarrow 0$: the endpoints are determined within corrections of order h3/2${h}^{3/2}$. The width of the first spectral band is shown to be O(h)$O({h})$. [ABSTRACT FROM AUTHOR]

We present a 1‐D model of a junction of five thin elastic rods forming the shape of a bit brace (hand drill), or, a crankshaft. The distinguishing feature of this junction is the existence of the so‐called movable elements, which are rods and knots requiring modifications of the classical asymptotic ansätze. These consist of constant longitudinal displacements on the edges of the skeleton of the junction and affect the transmission conditions at its nodes. We provide asymptotic formulas for the displacements, stresses and elastic energy, as well as error estimates. An exact solution of the model is given for a particular loading. [ABSTRACT FROM AUTHOR]

We investigate the eigenvalue problem −div(σ∇u) = λu (P) in a 2D domain Ω divided into two regions Ω±. We are interested in situations where σ takes positive values on Ω+ and negative ones on Ω−. Such problems appear in time harmonic electromagnetics in the modeling of plasmonic technologies. In a recent work [L. Chesnel, X. Claeys and S.A. Nazarov, Asymp. Anal.88 (2014) 43–74], we highlighted an unusual instability phenomenon for the source term problem associated with (P): for certain configurations, when the interface between the subdomains Ω± 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 (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. [ABSTRACT FROM AUTHOR]

We study a 2D semi-linear equation in a domain with a small Dirichlet obstacle of size . Using the method of matched asymptotic expansions, we compute an asymptotic expansion of the solution as tends to zero. Its relevance is justified by proving a rigorous error estimate. Then we construct an approximate model, based on an equation set in the limit domain without the small obstacle, which provides a good approximation of the far field of the solution of the original problem. The interest of this approximate model lies in the fact that it leads to a variational formulation which is very simple to discretize. We present numerical experiments to illustrate the analysis. [ABSTRACT FROM AUTHOR]

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 −∞ and probes the resulting scattered field at −∞ and +∞. 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 ⋆ 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 +∞. In particular, if the perturbation is smooth and small (in amplitude and in width), k ⋆ 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 −∞ and +∞. We implement numerically the method to provide examples of such undetectable defects. [ABSTRACT FROM AUTHOR]

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. .. [ABSTRACT FROM AUTHOR]