602 research outputs found
A Short Presentation of Emmanuele’s Work
No full textAs the title suggests, this is a short presentation of DiBenedetto’s mathematical work. In the first part Ugo Gianazza gives a general overview, without entering too much into details of specific papers; in his contribution Daniele Andreucci focuses on DiBenedetto’s accomplishments in BioMathematics
Homogenization and concentration of capacity in the rod outer segment with incisures
No full textWe present a quantitative model of the spatio-temporal dynamics of second messengers mediating phototransduction in retinal rods. The spatial domain (the rod outer segment) has a quite complex geometry, involving different “thin” domains, whose thickness is three orders of magnitude smaller than the other dimensions. The model relies on a “pointwise” application of first principles leading to a system of evolution equations set in such a structured geometry. Then, exploiting an idea first presented in [Andreucci, D., Bisegna, P. and Dibenedetto, E., 2002, homogenization and concentrated capacity in reticular almost disconnected structures. Comptes rendus mathematique. Academie des sciences. Paris, séries I, 335, 329–332], the diffusion problem is reduced to one with a simpler geometry, still preserving the essential features of the original one. This is achieved by an homogenization and concentration limit. However, here we take into account for the first time the presence of “incisures”, which are important for phototransduction, and introduce new mathematical features mainly in the concentration limit
Homogenization of a parabolic problem with alternating boundary conditions.
No full textWe consider the homogenization of a parabolic problem in a perforated domain with Robin–Neumann boundary conditions oscillating in time. Boundary conditions alternating in time appear in biological applications, for example in the modeling of ion channels, see [1]. Our approach relies upon a generalization of the unfolding technique, see e.g., [2], to the time-periodic case. To this end we show how the method of periodic unfolding can be applied to classical homogenization problem for a parabolic equation with diffusion and capacity-like coefficients in the diffusion equation oscillating both in space and time, with general independent scales.
From an analytical point of view, in the present case such oscillations must compensate the blow up of the boundary measure of the holes.We obtain a macroscopic parabolic problem containing an extra linear term due to the absorption determined by the Robin condition; this term keeps memory of the underlying temporal and spatial microstructures.
[1] D. Andreucci, D. Bellaveglia. Permeability of interfaces with alternating pores in
parabolic problems. Asymptotic Analysis. 79 (2012), 189–227.
[2] D. Cioranescu, A. Damlamian and G. Griso. The periodic unfolding method in
homogenization. SIAM Journal on Mathematical Analysis. 40(4) (2008), 1585–1620
Existence, uniqueness, and error estimates for a model of polymer crystallization
No full textWe perform a mathematical analysis of a model for the crystallization
of polymers. Essentially, the model is a system of both second order
and first order evolutionary partial differential equations. The main
novelty here is the fact that non-Lipschitz continuous functions of the
unknown variables may appear in the constitutive equations. This lack of
smoothness must be allowed in order for the model to account for the
isokinetic assumption. Using monotonicty and techniques we prove
existence and continuous dependence on the data of the weak solution to
the mathematical problem. Moreover we introduce a time-discrete approximation
to the problem, and we prove the convergence of the semidiscrete solutions
to the continuous one, also giving optimal a priori error estimates
Asymptotic Estimates for the p-Laplacian on Infinite Graphs with Decaying Initial Data
No full textWe consider the Cauchy problem for the evolutive discrete p-Laplacian in infinite graphs, with initial data decaying at infinity. We prove optimal sup and gradient bounds for nonnegative solutions, when the initial data has finite mass, and also sharp evaluation for the confinement of mass, i.e., the effective speed of propagation. We provide estimates for some moments of the solution, defined using the distance from a given vertex. Our technique relies on suitable inequalities of Faber-Krahn type, and looks at the local theory of continuous nonlinear partial differential equations. As it is known, however, not all of this approach can have a direct counterpart in graphs. A basic tool here is a result connecting the supremum of the solution at a given positive time with the measure of its level sets at previous times. We also consider the case of slowly decaying initial data, where the total mass is infinite
Asymptotic properties of solutions to the Cauchy problem for degenerate parabolic equations with inhomogeneous density on manifolds
No full textWe consider the Cauchy problem for doubly nonlinear degenerate parabolic equations with inhomogeneous density on noncompact Riemannian manifolds. We give a qualitative classification of the behavior of the solutions of the problem depending on the behavior of the density function at infinity and the geometry of the manifold, which is described in terms of its isoperimetric function. We establish for the solutions properties as: stabilization of the solution to zero for large times, finite speed of propagation, universal bounds of the solution, blow up of the interface. Each one of these behaviors of course takes place in a suitable range of parameters, whose definition involves a universal geometrical characteristic function, depending both on the geometry of the manifold and on the asymptotics of the density at infinity
Pro iOS Geo: building apps with location based services
No full textDeepen your app development skills with Pro iOS Geo. This book shows you how to use geolocation-based tools to enhance the iOS apps you develop. Author Giacomo Andreucci describes different ways to integrate geo services, depending on the kind of app you're looking to develop: a web app, a hybrid app, or a native app. You'll discover how to use the Google Maps API features to integrate powerful geo capabilities in your apps with a little effort. You'll learn how to: Design geographic features for your apps while respecting usability criteria Design touristic geo apps Use HTML5 and the Google
Homogenization and concentrated capacity in reticular almost disconnected structures
No full textWe compute the homogenized-concentrated limit for a pair of non-linearly coupled diffusion equations in a perforated cylindric domain with coaxial cylindric holes periodically distributed along its axis. This problem arises from visual transduction. 2002 Academie des sciences/editions scientifiques et medicales Elsevier SAS
Extinction in a Finite Time for Parabolic Equations of Fast Diffusion Type on Manifolds
No full textWe prove extinction in a finite time for a singular parabolic equation on a Riemannian manifold, under suitable assumptions on the Riemannian metric and on the inhomogeneous coefficient appearing in the equation. The result relies on a suitable embedding theorem, of which we present a new proof
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