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In this paper, the asymptotic behavior of the solutions of a monotone problem posed in a locally periodic oscillating domain is studied. Nonlinear monotone boundary conditions are imposed on the oscillating part of the boundary whereas the Dirichlet condition is considered on the smooth separate part. Using the unfolding method, under natural hypothesis on the regularity of the domain, we prove the weak $L^2$-convergence of the zero-extended solutions of the nonlinear problem and their flows to the solutions of a limit distributional problem., 16 pages, 2 figures

We consider a general second order elliptic operator in a planar waveguide perforated by small holes distributed along a curve and subject to classical boundary conditions on the holes. Under weak assumptions on the perforation, we describe all possible homogenized problems.

We consider a boundary value problem for a homogeneous elliptic equation with an inhomogeneous Steklov boundary condition. The problem involves a singular perturbation, which is the Dirichlet condition imposed on a small piece of the boundary. We rewrite such problem to a resolvent equation for a self-adjoint operator in a fractional Sobolev space on the boundary of the domain. We prove the norm convergence of this operator to a limiting one associated with an unperturbed problem involving no Dirichlet condition. We also establish an order sharp estimate for the convergence rate. The established convergence implies the convergence of the spectra and spectral projectors. In the second part of the work we study perturbed eigenvalues converging to limiting simple discrete ones. We construct two-terms asymptotic expansions for such eigenvalues and for the associated eigenfunctions.

This work deals with the homogenization of two dimensions&rsquo, tidal equations. We study the asymptotic behavior of the sequence of the solutions using the sigma-convergence method. We establish the convergence of the sequence of solutions towards the solution of an equivalent problem of the same type.

In the current work, we are performing the asymptotic analysis, beyond the periodic setting, of the Cahn-Hilliard-Navier-Stokes system. Under the general deterministic distribution assumption on the microstructures in the domain, we find the limit model equivalent to the heterogeneous one. To this end, we use the sigma-convergence concept which is suitable for the passage to the limit., 28 pages

We study the spectrum of a quadratic operator pencil with a small P & xdcab;& x1d4af;& xdcaf;-symmetric periodic potential and a fixed localized potential. We show that the continuous spectrum has a band structure with bands on the imaginary axis separated by usual gaps, while on the real axis, there are no gaps but at certain points, the bands bifurcate into small parabolas in the complex plane. We study the isolated eigenvalues converging to the continuous spectrum. We show that they can emerge only in the aforementioned gaps or in the vicinities of the small parabolas, at most two isolated eigenvalues in each case. We establish sufficient conditions for the existence and absence of such eigenvalues. In the case of the existence, we prove that these eigenvalues depend analytically on a small parameter and we find the leading terms of their Taylor expansions. It is shown that the mechanism of the eigenvalue emergence is different from that for small localized perturbations studied in many previous works.

We carry out the deterministic homogenization of nonlinear elliptic operators beyond periodicity. To proceed with, we prove the existence of nonlinear correctors in the usual distributional sense. This lays the foundation for the study of regularity results in the nonlinear deterministic homogenization theory beyond the periodic setting.

We consider a waveguide-like domain consisting of two thin straight tubular domains connected through a tiny window. The perpendicular size of this waveguide is of order e. Under the assumption that the window is appropriately scaled we prove that the Neumann Laplacian on this domain converges in (a kind of) norm resolvent sense as e → 0 to a one-dimensional Schrodinger operator corresponding to a δ ′ -interaction of a non-negative strength. We estimate the rate of this convergence, also we prove the convergence of spectra.

We consider an incompressible Bingham flow in a thin domain with rough boundary, under the action of given external forces and with no-slip boundary condition on the whole boundary of the domain. In mathematical terms, this problem is described by non linear variational inequalities over domains where a small parameter $\epsilon$ denotes the thickness of the domain and the roughness periodicity of the boundary. By using an adapted linear unfolding operator we perform a detailed analysis of the asymptotic behavior of the Bingham flow when $\epsilon$ tends to zero. We obtain the homogenized limit problem for the velocity and the pressure, which preserves the nonlinear character of the flow, and study the effects of the microstructure in the corresponding effective equations. Finally, we give the interpretation of the limit problem in terms of a non linear Darcy law., Comment: 23 pages, 2 figures

The macroscopic behavior of the solution of a coupled system of partial differential equations arising in the modeling of reaction–diffusion processes in periodic porous media is analyzed. Our mathematical model can be used for studying several metabolic processes taking place in living cells, in which biochemical species can diffuse in the cytosol and react both in the cytosol and also on the organellar membranes. The coupling of the concentrations of the biochemical species is realized via various properly scaled nonlinear reaction terms. These nonlinearities, which model, at the microscopic scale, various volume or surface reaction processes, give rise in the macroscopic model to different effects, such as the appearance of additional source or sink terms or of a non-standard diffusion matrix.

We consider the spectral Neumann problem for the Laplace operator in an acoustic waveguide $\Pi_{l}^{\varepsilon}$ obtained from a straight unit strip by a low box-shaped perturbation of size $2l\times\varepsilon,$ where $\varepsilon>0$ is a small parameter. We prove the existence of the length parameter $l_{k}^{\varepsilon}=\pi k+O\left( \varepsilon\right) $ with any $k=1,2,3,...$ such that the waveguide $\Pi_{l_{k}^{\varepsilon}}^{\varepsilon }$ supports a trapped mode with an eigenvalue $\lambda_{k}^{\varepsilon}% =\pi^{2}-4\pi^{4}l^{2}\varepsilon^{2}+O\left( \varepsilon^{3}\right) $ embedded into the continuous spectrum. This eigenvalue is unique in the segment $\left[ 0,\pi^{2}\right] $ and is absent in the case $l\neq l_{k}^{\varepsilon}.$ The detection of this embedded eigenvalue is based on a criterion for trapped modes involving an artificial object, the augmented scattering matrix. The main technical difficulty is caused by corner points of the perturbed wall $\partial\Pi_{l}^{\varepsilon}$ and we discuss available generalizations for other piecewise smooth boundaries., Comment: 36 pages, 6 figures

In recent years, the widespread availability of smartphones provided with sensors has enabled the possibility of harvesting large quantities of data in urban areas exploiting user devices, thus enabling so-called mobile crowd sensing (MCS). While many efforts have been made to improve specific techniques for MCS - spanning from signal processing to the assignment of data collection campaigns to users, and to the entire data processing spectrum - to the best of our knowledge, thus far there have been no active experiments of MCS that involve all these techniques in a large-scale real-world experiment. Based on these considerations, we started the ParticipAct Living Lab testbed, an ongoing experiment at the University of Bologna involving 300 students for one year in crowd sensing campaigns that can passively access smartphone sensors and also require active user collaboration. In this article we describe the guidelines behind the design of ParticipAct, as well as its features and architecture. Moreover, we report some of the seminal results gathered during the first three months of its deployment, including accuracy of the classifier provided by ParticipAct client and user inclination to successfully complete tasks depending on the level of active collaboration required to executing them.

We consider an infinite strip perforated along a curve by small holes. In this perforated domain, we consider a scalar second-order elliptic differential operator subject to classical boundary conditions on the holes. Assuming that the perforation is non-periodic, we describe possible homogenized problems and prove the norm-resolvent convergence of the perturbed operator to a homogenized one. We also provide estimates for the rate of the convergence.

We consider the spectral Dirichlet problem for the Laplace operator in the plane $\Omega^{\circ}$ with double-periodic perforation but also in the domain $\Omega^{\bullet}$ with a semi-infinite foreign inclusion so that the Floquet-Bloch technique and the Gelfand transform do not apply directly. We describe waves which are localized near the inclusion and propagate along it. We give a formulation of the problem with radiation conditions that provides a Fredholm operator of index zero. The main conclusion concerns the spectra $\sigma^{\circ}$ and $\sigma^{\bullet}$ of the problems in $\Omega^{\circ}$ and $\Omega^{\bullet},$ namely we present a concrete geometry which supports the relation $\sigma^{\circ}\varsubsetneqq\sigma^{\bullet}$ due to a new non-empty spectral band caused by the semi-infinite inclusion called an open waveguide in the double-periodic medium., Comment: 25 pages, 5 figures

We consider a family $\{\Omega^\varepsilon\}_{\varepsilon>0}$ of periodic domains in $\mathbb{R}^2$ with waveguide geometry and analyse spectral properties of the Neumann Laplacian $-\Delta_{\Omega^\varepsilon}$ on $\Omega^\varepsilon$. The waveguide $\Omega^\varepsilon$ is a union of a thin straight strip of the width $\varepsilon$ and a family of small protuberances with the so-called "room-and-passage" geometry. The protuberances are attached periodically, with a period $\varepsilon$, along the strip upper boundary. For $\varepsilon\to 0$ we prove a (kind of) resolvent convergence of $-\Delta_{\Omega^\varepsilon}$ to a certain ordinary differential operator. Also we demonstrate Hausdorff convergence of the spectrum. In particular, we conclude that if the sizes of "passages" are appropriately scaled the first spectral gap of $-\Delta_{\Omega^\varepsilon}$ is determined exclusively by geometric properties of the protuberances. The proofs are carried out using methods of homogenization theory., Comment: 24 Pages, 1 Figure

We study the homogenization of elliptic equations stated in L2-space with degenerate weight. Both coefficients of the differential operator and the weight are ε-periodic and highly oscillating as ε tends to zero. Under minimal hypotheses on the coefficients and the weight we prove estimates of order ε and ε2 for L2-norm of the difference between the exact solution and its appropriate approximations by L2-norm of the right-side function. The spectral method based on Bloch decomposition is used. In the case of nonunique solution provided that the weight is not regular we consider estimates for any of so-called variational solutions.

We consider a two-dimensional quantum waveguide composed of two semi-strips of width 1 and 1 - epsilon, where epsilon > 0 is a small real parameter, i.e. the waveguide is gently converging. The width of the junction zone for the semi-strips is 1 + O(root epsilon). We will present a sufficient condition for the existence of a weakly coupled bound state below pi(2), the lower bound of the continuous spectrum. This eigenvalue in the discrete spectrum is unique and its asymptotics is constructed and justified when epsilon -> 0(+). RI Cardone, Giuseppe/I-2998-2012 OI Cardone, Giuseppe/0000-0002-5050-8908

The existence of an eigenvalue embedded in the continuous spectrum is proved for the Neumann problem for Helmholtz's equation in a two-dimensional waveguide with two outlets to infinity which are half-strips of width 1 and 1 - epsilon, where epsilon > 0 is a small parameter. The width function of the part of the waveguide connecting these outlets is of order root epsilon; it is defined as a linear combination of three fairly arbitrary functions, whose coefficients are obtained from a certain nonlinear equation. The result is derived from an asymptotic analysis of an auxiliary object, the augmented scattering matrix.

The problem about a body in a three dimensional infinite channel is considered in the framework of the theory of linear water-waves. The body has a rough surface characterized by a small parameter epsilon > 0 while the distance of the body to the water surface is also of order epsilon. Under a certain symmetry assumption, the accumulation effect for trapped mode frequencies is established, namely it is proved that, for any given d > 0 and integer N > 0, there exists a number epsilon(d, N) > 0 such that the problem has at least N eigenvalues in the interval (0, d) of the continuous spectrum in the case epsilon is an element of (0, epsilon (d, N)). The corresponding eigenfunctions decay exponentially at infinity, have finite energy, and imply trapped modes. (C) 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

In a three-dimensional solid with arbitrary periodic Lipschitz perforation the Korn inequality is proved with a constant independent of the perforation size. The convergence rate of homogenization as a function of the Sobolev-Slobodetskii smoothness of data is also estimated. We improve foregoing results in elasticity dropping customary restrictions on the shape of the periodicity cell and superfluous smoothness and smallness assumptions on the external forces and traction.