793 research outputs found

    A model for the quasi-static growth of brittle fractures based on local minimization

    Full text link
    We study a variant of the variational model for the quasi-static growth of brittle fractures proposed by Francfort and Marigo. The main feature of our model is that, in the discrete-time formulation, in each step we do not consider absolute minimizers of the energy, but, in a sense, we look for local minimizers which are sufficiently close to the approximate solution obtained in the previous step. This is done by introducing in the variational problem an additional term which penalizes the L2L^2-distance between the approximate solutions at two consecutive times. We study the continuous-time version of this model, obtained by passing to the limit as the time step tends to zero, and show that it satisfies (for almost every time) some minimality conditions which are slightly different from those considered in Francfort and Marigo and in our previous paper, but are still enough to prove (under suitable regularity assumptions on the crack path) that the classical Griffith's criterion holds at the crack tips. We prove also that, if no initial crack is present and if the data of the problem are sufficiently smooth, no crack will develop in this model, provided the penalization term is large enough.Comment: 20 page

    Thin waveguides with Robin boundary conditions

    Full text link
    We consider the Laplace operator in a thin three dimensional tube with a Robin type condition on its boundary and study, asymptotically, the spectrum of such operator as the diameter of the tube's cross section becomes infinitesimal. In contrast with the Dirichlet condition case, we evidence different behaviors depending on a symmetry criterium for the fundamental mode in the cross section. If that symmetry condition fails, then we prove the localization of lower energy levels in the vicinity of the minimum point of a suitable function on the tube's axis depending on the curvature and the rotation angle. In the symmetric case, the behavior of lower energy modes is shown to be ruled by a one dimensional Sturm-Liouville problem involving an effective potential given in explicit form

    Gamma-convergence of quadratic functionals perturbed by bounded linear functionals

    Get PDF
    Given a bounded open set Ω⊂Rn, we study sequences of quadratic functionals on the Sobolev space H01(Ω), perturbed by sequences of bounded linear functionals. We prove that their Γ-limits, in the weak topology of H01(Ω), can always be written as the sum of a quadratic functional, a linear functional, and a non-positive constant. The classical theory of G- and H-convergence completely characterises the quadratic and linear parts of the Γ-limit and shows that their coefficients do not depend on Ω. The constant, which instead depends on Ω and will be denoted by −Μ(Ω), plays an important role in the study of the limit behaviour of the energies of the solutions. The main result of this paper is that, passing to a subsequence, we can prove that Îœ coincides with a non-negative Radon measure on a sufficiently large collection of bounded open sets Ω. Moreover, we exhibit an example that shows that the previous result cannot be obtained for every bounded open set. The specific form of this example shows that the compactness theorem for the localisation method in Γ-convergence cannot be easily improved

    On weak convergence of locally periodic functions

    Full text link
    We prove a generalization of the fact that periodic functions converge weakly to the mean value as the oscillation increases. Some convergence questions connected to locally periodic nonlinear boundary value problems are also considered.Comment: arxiv version is already officia

    Effective macroscopic dynamics of stochastic partial differential equations in perforated domains

    Full text link
    An effective macroscopic model for a stochastic microscopic system is derived. The original microscopic system is modeled by a stochastic partial differential equation defined on a domain perforated with small holes or heterogeneities. The homogenized effective model is still a stochastic partial differential equation but defined on a unified domain without holes. The solutions of the microscopic model is shown to converge to those of the effective macroscopic model in probability distribution, as the size of holes diminishes to zero. Moreover, the long time effectivity of the macroscopic system in the sense of \emph{convergence in probability distribution}, and the effectivity of the macroscopic system in the sense of \emph{convergence in energy} are also proved

    Power calculation for gravitational radiation: oversimplification and the importance of time scale

    Full text link
    A simplified formula for gravitational-radiation power is examined. It is shown to give completely erroneous answers in three situations, making it useless even for rough estimates. It is emphasized that short timescales, as well as fast speeds, make classical approximations to relativistic calculations untenable.Comment: Three pages, no figures, accepted for publication in Astronomische Nachrichte

    Random parking, Euclidean functionals, and rubber elasticity

    Get PDF
    We study subadditive functions of the random parking model previously analyzed by the second author. In particular, we consider local functions SS of subsets of Rd\mathbb{R}^d and of point sets that are (almost) subadditive in their first variable. Denoting by Ο\xi the random parking measure in Rd\mathbb{R}^d, and by ΟR\xi^R the random parking measure in the cube QR=(−R,R)dQ_R=(-R,R)^d, we show, under some natural assumptions on SS, that there exists a constant Sˉ∈R\bar{S}\in \mathbb{R} such that % lim⁥R→+∞S(QR,Ο)∣QR∣ = lim⁥R→+∞S(QR,ΟR)∣QR∣ = Sˉ \lim_{R\to +\infty} \frac{S(Q_R,\xi)}{|Q_R|}\,=\,\lim_{R\to +\infty}\frac{S(Q_R,\xi^R)}{|Q_R|}\,=\,\bar{S} % almost surely. If ζ↊S(QR,ζ)\zeta \mapsto S(Q_R,\zeta) is the counting measure of ζ\zeta in QRQ_R, then we retrieve the result by the second author on the existence of the jamming limit. The present work generalizes this result to a wide class of (almost) subadditive functions. In particular, classical Euclidean optimization problems as well as the discrete model for rubber previously studied by Alicandro, Cicalese, and the first author enter this class of functions. In the case of rubber elasticity, this yields an approximation result for the continuous energy density associated with the discrete model at the thermodynamic limit, as well as a generalization to stochastic networks generated on bounded sets.Comment: 28 page

    Coupling techniques for nonlinear hyperbolic equations. III. The well-balanced approximation of thick interfaces

    Full text link
    We continue our analysis of the coupling between nonlinear hyperbolic problems across possibly resonant interfaces. In the first two parts of this series, we introduced a new framework for coupling problems which is based on the so-called thin interface model and uses an augmented formulation and an additional unknown for the interface location; this framework has the advantage of avoiding any explicit modeling of the interface structure. In the present paper, we pursue our investigation of the augmented formulation and we introduce a new coupling framework which is now based on the so-called thick interface model. For scalar nonlinear hyperbolic equations in one space variable, we observe that the Cauchy problem is well-posed. Then, our main achievement in the present paper is the design of a new well-balanced finite volume scheme which is adapted to the thick interface model, together with a proof of its convergence toward the unique entropy solution (for a broad class of nonlinear hyperbolic equations). Due to the presence of a possibly resonant interface, the standard technique based on a total variation estimate does not apply, and DiPerna's uniqueness theorem must be used. Following a method proposed by Coquel and LeFloch, our proof relies on discrete entropy inequalities for the coupling problem and an estimate of the discrete entropy dissipation in the proposed scheme.Comment: 21 page

    A second order minimality condition for the Mumford-Shah functional

    Full text link
    A new necessary minimality condition for the Mumford-Shah functional is derived by means of second order variations. It is expressed in terms of a sign condition for a nonlocal quadratic form on H01(Γ)H^1_0(\Gamma), Γ\Gamma being a submanifold of the regular part of the discontinuity set of the critical point. Two equivalent formulations are provided: one in terms of the first eigenvalue of a suitable compact operator, the other involving a sort of nonlocal capacity of Γ\Gamma. A sufficient condition for minimality is also deduced. Finally, an explicit example is discussed, where a complete characterization of the domains where the second variation is nonnegative can be given.Comment: 30 page

    Hepatitis C virus and non-Hodgkin's lymphomas: Meta-analysis of epidemiology data and therapy options.

    Get PDF
    Hepatitis C virus (HCV) is a global health problem affecting a large fraction of the world\u2019s population: This virus is able to determine both hepatic and extrahepatic diseases. Mixed cryoglobulinemia, a B-cell \u201cbenign\u201d lymphoproliferative disorders, represents the most closely related as well as the most investigated HCV-related extrahepatic disorder. Since this virus is able to determine extrahepatic [non-Hodgkin\u2019s lymphoma (NHL)] as well as hepatic malignancies (hepatocellular carcinoma), HCV has been included among human cancer viruses. The most common histological types of HCV-associated NHL are the marginal zone, the lymphoplasmacytic and diffuse large cell lymphomas. The role of the HCV in the pathogenesis of the B-cell lymphoproliferative disorders is confirmed also by the responsiveness of the NHL to antiviral therapy. The purpose of this review is to provide an overview of the recent literature and a meta analysis of the epidemiology data, to explain the role of HCV in the development of NHL\u2019s lymphoma. Furthermore, the possibility to treat these HCV-related NHL with the antiviral therapy or with other therapeutic options, like chemotherapy, is also discussed
    • 

    corecore