793 research outputs found
A model for the quasi-static growth of brittle fractures based on local minimization
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 -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
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
Given a bounded open set ΩâRn, we study sequences of quadratic functionals on the Sobolev space H01(Ω), perturbed by sequences of bounded linear functionals. We prove that their Î-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 Î-limit and shows that their coefficients do not depend on Ω. The constant, which instead depends on Ω and will be denoted by âÎœ(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 Î-convergence cannot be easily improved
On weak convergence of locally periodic functions
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
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
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
We study subadditive functions of the random parking model previously
analyzed by the second author. In particular, we consider local functions
of subsets of and of point sets that are (almost) subadditive in
their first variable. Denoting by the random parking measure in
, and by the random parking measure in the cube
, we show, under some natural assumptions on , that there
exists a constant such that % % almost surely. If is the counting measure of in , 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
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
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 , 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 . 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.
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
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