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Networks & Heterogeneous Media

2011 , Volume 6 , Issue 1

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Spectral stiff problems in domains surrounded by thin stiff and heavy bands: Local effects for eigenfunctions
Delfina Gómez, Sergey A. Nazarov and Eugenia Pérez
2011, 6(1): 1-35 doi: 10.3934/nhm.2011.6.1 +[Abstract](90) +[PDF](776.9KB)
We consider the Neumann spectral problem for a second order differential operator, with piecewise constants coefficients, in a domain $\Omega_\varepsilon$ of $R^2$. Here $\Omega_\varepsilon$ is $\Omega \cup \omega_\varepsilon \cup \Gamma$, where $\Omega$ is a fixed bounded domain with boundary $\Gamma$, $\omega_\varepsilon$ is a curvilinear band of variable width $O(\varepsilon)$, and $\Gamma=\overline{\Omega}\cap \overline {\omega_\varepsilon}$. The density and stiffness constants are of order $O(\varepsilon^{-m-1})$ and $O(\varepsilon^{-1})$ respectively in this band, while they are of order $O(1)$ in $\Omega$; $m$ is a positive parameter and $\varepsilon \in (0,1)$, $\varepsilon\to 0$. Considering the range of the low, middle and high frequencies, we provide asymptotics for the eigenvalues and the corresponding eigenfunctions. For $m>2$, we highlight the middle frequencies for which the corresponding eigenfunctions may be localized asymptotically in small neighborhoods of certain points of the boundary.
A mathematical model for spaghetti cooking with free boundaries
Antonio Fasano, Mario Primicerio and Andrea Tesi
2011, 6(1): 37-60 doi: 10.3934/nhm.2011.6.37 +[Abstract](97) +[PDF](482.9KB)
We propose a mathematical model for the process of dry pasta cooking with specific reference to spaghetti. Pasta cooking is a two-stage process: water penetration followed by starch gelatinization. Differently from the approach adopted so far in the technical literature, our model includes free boundaries: the water penetration front and the gelatinization onset front representing a fast stage of the corresponding process. Behind the respective fronts water sorption and gelatinization proceed according to some kinetics. The outer boundary is also moving and unknown as a consequence of swelling. Existence and uniqueness are proved and numerical simulations are presented.
On the convergence rate in multiscale homogenization of fully nonlinear elliptic problems
Fabio Camilli and Claudio Marchi
2011, 6(1): 61-75 doi: 10.3934/nhm.2011.6.61 +[Abstract](103) +[PDF](429.6KB)
This paper concerns periodic multiscale homogenization for fully nonlinear equations of the form $u^\epsilon+H^\epsilon (x,\frac{x}{\epsilon},\ldots,\frac{x}{epsilon^k},Du^\epsilon,D^2u^\epsilon)=0$. The operators $H^\epsilon$ are a regular perturbations of some uniformly elliptic, convex operator $H$. As $\epsilon\to 0^+$, the solutions $u^\epsilon$ converge locally uniformly to the solution $u$ of a suitably defined effective problem. The purpose of this paper is to obtain an estimate of the corresponding rate of convergence. Finally, some examples are discussed.
Closed curves of prescribed curvature and a pinning effect
Matteo Novaga and Enrico Valdinoci
2011, 6(1): 77-88 doi: 10.3934/nhm.2011.6.77 +[Abstract](79) +[PDF](197.8KB)
We prove that for any $H: R^2 \to R$ which is $Z^2$-periodic, there exists $H_\varepsilon$, which is smooth, $\varepsilon$-close to $H$ in $L^1$, with $L^\infty$-norm controlled by the one of $H$, and with the same average of $H$, for which there exists a smooth closed curve $\gamma_\varepsilon$ whose curvature is $H_\varepsilon$. A pinning phenomenon for curvature driven flow with a periodic forcing term then follows. Namely, curves in fine periodic media may be moved only by small amounts, of the order of the period.
Non-standard dynamics of elastic composites
Maksym Berezhnyi and Evgen Khruslov
2011, 6(1): 89-109 doi: 10.3934/nhm.2011.6.89 +[Abstract](78) +[PDF](518.3KB)
An elastic medium with a large number of small axially symmetric solid particles is considered. It is assumed that the particles are identically oriented and under the influence of elastic medium they move translationally or rotate around symmetry axis but the direction of their symmetry axes does not change. The asymptotic behavior of small oscillations of the system is studied, when the diameters of particles and distances between the nearest particles are decreased. The equations, describing the homogenized model of the system, are derived. It is shown that the homogenized equations correspond to a non-standard dynamics of elastic medium. Namely, the homogenized stress tensor linearly depends not only on the strain tensor but also on the rotation tensor.
Homogenization of convection-diffusion equation in infinite cylinder
Iryna Pankratova and Andrey Piatnitski
2011, 6(1): 111-126 doi: 10.3934/nhm.2011.6.111 +[Abstract](95) +[PDF](362.9KB)
The paper deals with a periodic homogenization problem for a non-stationary convection-diffusion equation stated in a thin infinite cylindrical domain with homogeneous Neumann boundary condition on the lateral boundary. It is shown that homogenization result holds in moving coordinates, and that the solution admits an asymptotic expansion which consists of the interior expansion being regular in time, and an initial layer.
On the location of the 1-particle branch of the spectrum of the disordered stochastic Ising model
Michele Gianfelice and Marco Isopi
2011, 6(1): 127-144 doi: 10.3934/nhm.2011.6.127 +[Abstract](75) +[PDF](425.3KB)
We analyse the lower non trivial part of the spectrum of the generator of the Glauber dynamics for a $d$-dimensional nearest neighbour Ising model with a bounded random potential. We prove conjecture 1 in [1]: for sufficiently large values of the temperature, the first band of the spectrum of the generator of the process coincides with a closed non random segment of the real line.
A rate-independent model for permanent inelastic effects in shape memory materials
Michela Eleuteri, Luca Lussardi and Ulisse Stefanelli
2011, 6(1): 145-165 doi: 10.3934/nhm.2011.6.145 +[Abstract](117) +[PDF](488.0KB)
This paper addresses a three-dimensional model for isothermal stress-induced transformation in shape memory polycrystalline materials in presence of permanent inelastic effects. The basic features of the model are recalled and the constitutive and the three-dimensional quasi-static evolution problem are proved to be well-posed. Finally, we discuss the convergence of the model to reduced/former ones by means of a rigorous $\Gamma$-convergence analysis.

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