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We consider the linear water-wave problem in a periodic channel $$\Pi ^h \subset {{\mathbb {R}}}^3$$ Π h ⊂ 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 reflexive Bergman spaces A(p)(Omega) on polygonal domains Omega of the complex plane. With some restrictions to the angles of the boundary of Omega, we show that the boundedness of the Toeplitz operator T-g : A(p)(Omega) -> A(p)(Omega) with a positive symbol g is equivalent to the boundedness of the Berezin transform of g or to g times the area measure being a Carleson measure. The result is also formulated for more general simply connected domains. The main technical tool is a new weighted Forelli-Rudin-type estimate.

We consider weighted Bergman spaces $$A_\mu ^1$$ A μ 1 on the unit disc as well as the corresponding spaces of entire functions, defined using non-atomic Borel measures with radial symmetry. By extending the techniques from the case of reflexive Bergman spaces, we characterize the solid core of $$A_\mu ^1$$ A μ 1 . Also, as a consequence of a characterization of solid $$A_\mu ^1$$ A μ 1 -spaces, we show that, in the case of entire functions, there indeed exist solid $$A_\mu ^1$$ A μ 1 -spaces. The second part of the article is restricted to the case of the unit disc and it contains a characterization of the solid hull of $$A_\mu ^1$$ A μ 1 , when $$\mu$$ μ equals the weighted Lebesgue measure with the weight v. The results are based on the duality relation of the weighted $$A^1$$ A 1 - and $$H^\infty$$ H ∞ -spaces, the validity of which requires the assumption that $$-\log v$$ - log v belongs to the class $$\mathcal {W}_0$$ W 0 , studied in a number of publications; moreover, v has to satisfy the condition (b), introduced by the authors. The exponentially decreasing weight $$v(z) = \exp ( -1 /(1-|z|)$$ v ( z ) = exp ( - 1 / ( 1 - | z | ) provides an example satisfying both assumptions.

We construct "almost periodic'' unbounded domains, where a large class of elliptic spectral problems have essential spectra possessing peculiar structure: they consist of monotone, non-negative sequences of isolated points and thus have infinitely many gaps.

[EN] We consider the Cauchy problem for a general class of parabolic partial differential equations in the Euclidean space R-N. We show that given a weighted L-p-space L-w(p)(R-N) with 1 0 such that, if the initial data f belongs to the closed linear span of e(n) with n >= n(m), then the decay rate of the solution of the problem is at least t(-m) for large times t. The result generalizes the recent study of the authors concerning the classical linear heat equation. We present variants of the result having different methods of proofs and also consider finite polynomial decay rates instead of unlimited m., The research of Bonet was partially supported by the projects MTM2016-76647P and GV Prometeo/2017/102 (Spain). The research of Taskinen was partially supported by a research grant from the Faculty of Science of the University of Helsinki.

[EN] There are recent results concerning the boundedness and also unboundedness of Bergman projections on weighted spaces of the unit disc in special cases of rapidly decreasing weights, i.e. "large" Bergman spaces. The aim of our paper is to show that the cases of boundedness are largely exceptional: in general the Bergman projections are unbounded. In addition we give a new, more functional analytic proof for the known central boundedness case which also enables us to transfer our results to harmonic Bergman spaces., The research of Bonet was partially supported by the project MCIN PID2020-119457GB-I00/AEI/10.13039/501100011033. The research of Taskinen was partially supported by the Vaisala Foundation of the Finnish Academy of Sciences and Letters.

[EN] We determine the solid hull for 2 < p < infinity and the solid core for 1 < p < 2 of weighted Bergman spaces A(mu)(p) , 1 < p < infinity, of analytic functions on the disk and on the whole complex plane, for a very general class of nonatomic positive bounded Borel measures mu . New examples are presented. Moreover, we show that the space A(mu)(p) , 1 < p < infinity, is solid if and only if the monomials are an unconditional basis of this space., Bonet's work was partially supported by the MTM2016-76647-P and GV Prometeo/2017/102 projects (Spain). Taskinen's work was partially supported by a research grant from the Faculty of Science of the University of Helsinki.

A first order differential equation with a periodic operator coefficient acting in a pair of Hilbert spaces is considered. This setting models both elliptic equations with periodic coefficients in a cylinder and parabolic equations with time periodic coefficients. Our main results are a construction of a pointwise projector and a spectral splitting of the system into a finite-dimensional system of ordinary differential equations with constant coefficients and an infinite-dimensional part whose solutions have better properties in a certain sense. This complements the well-known asymptotic results for periodic hypoelliptic problems in cylinders and for elliptic problems in quasicylinders obtained by P. Kuchment and S. A. Nazarov, respectively. As an application, a center manifold reduction is presented for a class of nonlinear ordinary differential equations in Hilbert spaces with periodic coefficients. This result generalizes the known case with constant coefficients explored by A. Mielke.

We consider Toepliz operators with integrable symbols acting on Bergman spaces A(p), 1

We derive a one-dimensional model for an elastic shuttle, that is, a thin rod with rounded ends and small fixed terminals, by means of an asymptotic procedure of dimension reduction. In the model, deformation of the shuttle is described by a system of ordinary differential equations with variable degenerating coefficients, and the number of the required boundary conditions at the end points of the one-dimensional image of the rod depends on the roundness exponent m is an element of(0,1). Error estimates are obtained in the case m is an element of(0,1/4) by using an anisotropic weighted Korn inequality, which was derived in an earlier paper by the authors. We also briefly discuss boundary layer effects, which can be neglected in the case m is an element of(0,1/4) but play a crucial role in the formulation of the limit problem for m >= 1/4.

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.

[EN] Given a continuous, radial, rapidly decreasing weight v on the complex plane, we study the solid hull of its associated weighted space H¿v(C) of all the entire functions f such that v|f| is bounded. The solid hull is found for a large class of weights satisfying the condition (B) of Lusky. Precise formulations are obtained for weights of the form v(r)= exp(¿arp),a>0,p>0. Applications to spaces of multipliers are included., The research of Bonet was partially supported by MTM2013-43540-P, GVA Prometeo II/2013/013 and GVA ACOMP/2015/186. The research of Taskinen was partially supported by the Magnus Ehrnrooth and the Vaisala Foundations.

[EN] We determine the solid hull and solid core of weighted Banach spaces H-upsilon(infinity) of analytic functions functions f such that upsilon vertical bar f vertical bar is bounded, both in the case of the holomorphic functions on the disc and on the whole complex plane, for a very general class of radial weights upsilon. Precise results are presented for concrete weights on the disc that could not be treated before. It is also shown that if H-upsilon(infinity) is solid, then the monomials are an (unconditional) basis of the closure of the polynomials in H-upsilon(infinity). As a consequence H-upsilon(infinity) does not coincide with its solid hull and core in the case of the disc. An example shows that this does not hold for weighted spaces of entire functions., The research of Bonet was partially supported by the project MTM2016-76647-P. The research of Taskinen was partially supported by the Vaisala Foundation of the Finnish Academy of Sciences and Letters.

We construct asymptotic expansions as epsilon -> +0 for an eigenvalue embedded into the continuous spectrum of water-wave problem in a cylindrical three-dimensional channel with a thin screen of thickness O(epsilon). The screen may be either submerged or surface-piercing, and its wetted part has a sharp edge. The channel and the screen are mirror symmetric so that imposing the Dirichlet condition in the middle plane creates an artificial positive cut-off-value Lambda(dagger) of the modified spectrum. Depending on a certain integral characteristic I of the screen profiles, we find two types of asymptotics of eigenvalues, lambda(epsilon) = Lambda(dagger) - O(epsilon(2)) and lambda(epsilon) = Lambda(dagger) - O(epsilon(4)) in the cases I > 0 and I = 0, respectively. We prove that in the case I = 0. For the justification of these result, the main tools are a reduction to an abstract spectral equation and the use of the max-min principle.

We find a concrete integral formula for the class of generalized Toeplitz operators $$T_a$$ in Bergman spaces $$A^p$$ , $$1

International audience; 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 consider a class of Toeplitz operators with special radial symbols on weighted holomorphic Orlicz space. For an operator from such class we prove necessary and sufficient conditions of boundedness on holomorphic Orlicz space in terms of behaviour of averages of the radial symbol of the operator. In the case when either symbol or its certain average is nonnegative, we obtain characterization for boundedness of the corresponding Toeplitz operator.

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 ansatze. 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.

[EN] Let v be a radial weight function on the unit disc or on the complex plane. It is shown that for each analytic function f0 in the Banach space Hv all analytic functions f such that v|f| is bounded, the distance of f0 to the subspace Hv0 of Hv of all the functions g such that v|g| vanishes at infinity is attained at a function g0Hv0. Moreover a simple, direct proof of the formula of the distance of f to Hv0 due to Perfekt is presented. As a consequence the corresponding results for weighted Bloch spaces are obtained., The authors are very thankful to the referees for their careful reading of the manuscript and their suggestions. The research of Bonet was partially supported by the Projects MTM2016-76647-P and GV Prometeo 2017/102. The research of Taskinen was partially supported by the research grant from the Faculty of Science of the University of Helsinki.

[EN] We prove sufficient conditions for the boundedness and compactness of Toeplitz operators T-a in weighted sup-normed Banach spaces H-v(infinity) of holomorphic functions defined on the open unit disc D of the complex plane; both the weights v and symbols a are assumed to be radial functions on D. In an earlier work by the authors was shown that there exists a bounded, harmonic (thus non-radial) symbol a such that T-a is not bounded in any space H-v(infinity) with an admissible weight v. Here, we show that a mild additional assumption on the logarithmic decay rate of a radial symbol a at the boundary of D guarantees the boundedness of T-a. The sufficient conditions for the boundedness and compactness of T-a, in a number of variations, are derived from the general, abstract necessary and sufficient condition recently found by the authors. The results apply for a large class of weights satisfying the so called condition (B), which includes in addition to standard weight classes also many rapidly decreasing weights. (c) 2020 Elsevier Inc. All rights reserved., The research of Bonet was partially supported by the projects MTM2016-76647-P and GV Prometeo/2017/102

We characterize the boundedness and compactness of Toeplitz operators T a with radial symbols a in weighted H ∞ -spaces H v ∞ on the open unit disc of the complex plane. The weights v are also assumed radial and to satisfy the condition (B) introduced by the second named author. The main technique uses Taylor coefficient multipliers, and the results are first proved for them. We formulate a related sufficient condition for the boundedness and compactness of Toeplitz operators in reflexive weighted Bergman spaces on the disc. We also construct a bounded harmonic symbol f such that T f is not bounded in H v ∞ for any v satisfying mild assumptions. As a corollary, the Bergman projection is never bounded with respect to the corresponding weighted sup-norms. However, we also show that, for normal weights v, all Toeplitz operators with a trigonometric polynomial as the symbol are bounded on H v ∞ .

In this paper we consider Toepliz operators with (locally) integrable symbols acting on Bergman spaces $A^p$ ($1

We consider the Robin Laplacian in the domains Ω and Ωε, ε > 0, with sharp and blunted cusps, respectively. Assuming that the Robin coefficient a is large enough, the spectrum of the problem in Ω is known to be residual and to cover the whole complex plane, but on the contrary, the spectrum in the Lipschitz domain Ωε is discrete. However, our results reveal the strange behaviour of the discrete spectrum as the blunting parameter ε 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 neighbourhood of a point for any small ε > 0 while the second ones move at a high rate O(| ln ε|) downwards along the real axis ℝ to −∞. At the same time, any point λ ∈ ℝ is a ‘blinking eigenvalue’, i.e., it belongs to the spectrum of the problem in Ωε almost periodically in the | ln ε|-scale. Besides standard spectral theory, we use the techniques of dimension reduction and self-adjoint extensions to obtain these results.

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.

International audience; We consider scattering of surface waves modelled by the linear water wave equation in an unbounded two-dimensional domain of finite depth, at a given frequency and a given incidence. Using asymptotic analysis for small perturbations of the bottom shape, we build a fixed-point equation whose unique solution is a shape which cannot be detected by a distant observer. The method works at any incidence except π/4.

We construct a family of periodic piezoelectric waveguides Πɛ, 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 Πɛ grows unboundedly.

[EN] We consider the classical Cauchy problem for the linear heat equation and integrable initial data in the Euclidean space R-N. We show that given a weighted L-p-space L-w(p)(R-N) with 1 0 such that, if the initial data f belong to the closed linear space of e(n) with n >= n(m), then the decay rate of the solution of the heat equation is at least t(-m). Such a basis can be constructed as a perturbation of any given Schauder basis. The proof is based on a construction of a basis of L-w(p)(R-N), which annihilates an infinite sequence of bounded functionals., Open access funding provided by University of Helsinki including Helsinki University Central Hospital. The authors would like to thank Thierry Gallay (Grenoble) for discussions which helped in the final formulation of our results. The research of Bonet was partially supported by the projects MTM2016-76647-P and GV Prometeo 2017/102. The research of Taskinen was partially supported by the research grant from the Faculty of Science of the University of Helsinki.

[EN] It is shown that the monomials A = (z(n))(n=0)(infinity) are a Schauder basis of the Frechet spaces A(+)(-gamma), gamma >= 0, that consists of all the analytic functions f on the unit disc such that (1 - vertical bar z vertical bar)(mu)vertical bar f(z)vertical bar is bounded for all mu > gamma. Lusky proved that A is not a Schauder basis for the closure of the polynomials in weighted Banach spaces of analytic functions of type H-infinity. A sequence space representation of the Frechet space A(+)(-gamma) is presented. The case of (LB)-spaces A(-)(-gamma), gamma > 0, that are defined as unions of weighted Banach spaces is also studied., Research of the third author was partially supported by the Vaisala Foundation of the Finnish Academy of Sciences and Letters.

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 study the spectral linear elasticity problem in an unbounded periodic waveguide, which consists of a sequence of identical bounded cells connected by thin ligaments of diameter of order $ h >0$. The essential spectrum of the problem is known to have band-gap structure. We derive asymptotic formulas for the position of the spectral bands and gaps, as $h \to 0$., 28 pages, 5 figures

We show that the linear water wave problem in a bounded liquid domain may have continuous spectrum, if the interface of a two-layer liquid touches the basin walls at zero angle. The reason for this phenomenon is the appearance of cuspidal geometries of the liquid phases. We calculate the exact position of the continuous spectrum. We also discuss the physical background of wave propagation processes, which are enabled by the continuous spectrum. Our approach and methods include constructions of a parametrix for the problem operator and singular Weyl sequences.

We consider an elastic rod with rounded ends and diameter proportional to a small parameter h > 0. The roundness of the ends is described by an exponent m ∈ (0,1). We derive for the rod an asymptotically sharp Korn inequality with a special weighted anisotropic norm. Weight factors with m-dependent powers of h appear both in the anisotropic norm and the Korn inequality itself, and we discover three critical values m = 1 ∕ 4, m = 1 ∕ 2 and m = 3 ∕ 4 at which these exponents are crucially changed. Copyright © 2014 John Wiley & Sons, Ltd.

We consider a problem on the interaction of surface waves with a freely floating submerged body, which combines a spectral Steklov problem with a system of algebraic equations. We reduce this spectral problem to a quadratic pencil and then to the standard spectral equation for a self-adjoint operator in a certain Hilbert space. In addition to general properties of the spectrum, we investigate the asymptotics of eigenvalues and eigenvectors with respect to an intrinsic small parameter.

We consider the linear water-wave problem in a bounded water-basin with a shallow beach. In spite of the boundedness of the domain, the spectrum of the problem may have a continuous component, if the beach of the basin has a cuspidal form. Following the approach and methods of an earlier paper by S.A.Nazarov and the second named author, we improve the results of the citation by determining the spectrum in an open borderline case under weaker geometric assumptions.

A waveguide in which the essential spectrum of the Laplace–Dirichlet problem consists of a countable set of points in the real positive semiaxis is constructed. The waveguide is built from a family of identical cells, which are connected by apertures in their common walls, while the sizes of the apertures decrease with distance from the “central” cell. It is shown that the lowest point of the essential spectrum is the limit of an infinite sequence of eigenvalues in the discrete spectrum. A hypothesis on the structure of the discrete spectrum inside gaps is stated, and some unsolved problems are mentioned. Bibliography: 8 titles.

[EN] Let v(r) = exp(¿a/(1 ¿ r) b ) with a > 0 and 0 < b ¿ 2 be an exponential weight on the unit disc. We study the solid hull of its associated weighted Banach space H¿ v (D) of all the analytic functions f on the unit disc such that v|f| is bounded., The research of Bonet was partially supported by the projects MTM2013-43540-P and MTM2016-76647-P. This paper was completed during the Bonet's stay at the Katholische Universitat Eichstatt-Ingolstadt (Germany). The support of the Alexander von Humboldt Foundation is greatly appreciated. The research of Taskinen was partially supported by the Vaisala Foundation of the Finnish Academy of Sciences and Letters.

We investigate the trapping of oblique water-waves by horizontal cylinders in a two-layer liquid. Two cases depending on a small parameter are studied: first, the liquid densities are close to each other, and second, the density of the upper layer is small in comparison with that of the lower layer. We give sufficient conditions for wave trapping for appropriate limit problems. The main results involve asymptotic formulas for the surface and interfacial trapped mode frequencies. In addition, we prove comparison principles and sufficient conditions for the existence of trapped modes in various situations.

Let X be a complex manifold, let A be a topological discrete valuation ring, and write for the sheaf of functions on X with values in A. We prove Cartan theorems A and B for coherent -modules, when X is a Stein manifold and A satisfies some requirements like being a nuclear direct limit of Banach algebras. The result is motivated by questions in the work of the second author with Kashiwara in the proof of the codimension-three conjecture for holonomic microdifferential systems.

The two-dimensional problem concerning oblique incident waves and a freely floating cylinder is reduced to the study of the spectrum of a suitable self-adjoint operator in a Hilbert space. Using tools from spectral measure theory, we estimates the difference between the eigenfrequencies of the original problem and those of a problem for an inert body unaffected by the buoyancy forces. We obtain a localization of the eigenfrequencies of the freely floating body and, in addition, derive a sufficient condition for the existence of a point spectrum in the corresponding boundary value problem. Bibliography: 24 titles.

Here, ∂ ν is the outward normal derivative. Problem (2) is obviously related to acoustic wave propagation in polluted media. Additionally, the separation of vari ables in the elasticity system in an isotropic space with cylindrical holes [1] or in the Steklov problem for sur face waves propagating over a liquid layer with vertical cylindrical columns [2] (Fig. 1b) also leads to a Neu mann spectral problem. The boundary value problem (2) is associated [3, Section 10.1] with a positive self adjoint operator in L2(ΠR) with the domain H(ΠR) (Lebesgue and Sobo ΠR 2 \ R α1 1 2 – α2 1 2 – , ⎝ ⎠ ⎛ ⎞

We show that the spectrum of the Dirichlet and Neumann problems for the Laplace operator in the plane perforated by a double–periodic family of circular holes contains gaps (even any a priori given number of gaps) of certain radii of holes. The result is obtained by asymptotic analysis of the cell spectral problem, interpreted as a problem in a domain with thin bridges. Some open questions are stated.