Homogenization for strongly anisotropic nonlinear elliptic equations

In this paper we present homogenization results for elliptic degenerate differential equations describing strongly anisotropic media. More precisely, we study the limit as e? 0 \epsilon \to 0 of the following Dirichlet problems with rapidly oscillating periodic coefficients:¶¶ . \cases {{ -div(\alpha(\frac{x}{\epsilon}}, \nabla u) A(\frac{x}{\epsilon}) \nabla u) = f(x) \in L^{\infty}(\Omega) \atop u = 0 su \eth\Omega\ } ¶¶where, p > 1,     a: \Bbb Rⁿ ×\Bbb Rⁿ ? \Bbb R,     a(y,x) ? áA(y)x,x?^p/2-1, A ? M^{n ×n}(\Bbb R) p>1, \quad \alpha : \Bbb R^n \times \Bbb R^n \to \Bbb R, \quad \alpha(y,\xi) \approx \langle A(y)\xi,\xi \rangle ^{p/2-1}, A \in M^{n \times n}(\Bbb R) , A being a measurable periodic matrix such that A^t(x) = A(x) 3 0A^t(x) = A(x) \ge 0 almost everywhere.¶¶The anisotropy of the medium is described by the following structure hypothesis on the matrix A:¶¶l^2/p(x) |x|² ￡ áA(x)x,x? ￡ L ^2/p(x) |x|², \lambda^{2/p}(x) |\xi|^2 \leq \langle A(x)\xi,\xi \rangle \leq \Lambda ^{2/p}(x) |\xi|^2, ¶¶where the weight functions l \lambda and L \Lambda (satisfying suitable summability assumptions) can vanish or blow up, and can also be "moderately" different. The convergence to the homogenized problem is obtained by a classical compensated compactness argument, that had to be extended to two-weight Sobolev spaces.     

Three-dimensional strain field measurement on cylindrical vessels by computer analysis of laser interferograms

The problem of obtaining strain values by numerical analysis of interferograms is considered. A research activity in laser holographic interferometry, started at the Joint Research Center of Ispra some years ago as part of a program for the development of diagnostic methods for structural reliability assessment, has led to the implementation and validation of the numerical code ELISA (experimental laser interferometry strain analysis). The code uses three (properly digitized) interferograms and obtains the maps of the absolute strains of the inspected zone. A graphic post-processor allows the representation of hoop, axial and shear strains.     

Convergence of a class of degenerate Ginzburg-Landau functionals and regularity for a subelliptic harmonic map equation

In this paper, we consider a class of Ginzburg-Landau functionalsE _ε associated with a couple of non-commuting vector fields which yield a “degenerate” energy. We study the asymptotic behavior of the minimizers, showing that it does not depend on the topological degree of the boundary datum; and we prove uniqueness and regularity of the minimizer of the limit problem, in spite of the lack of lifting theorems in the natural function spaces for the limit functional.

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Résumé  Dans cet article, nous considérons une classe de fonctionnellesE _ε du type Ginzburg-Landau associée a un couple de champs de vecteurs définissant une énergie dégénérée. Nous étudions le comportement asymptotique des minimiseurs. Nous démontrons que ce comportement ne dépend pas du degré topologique de la donnée a la frontiere et nous prouvons l’unicité et la régularité du minimiseur du probléme limite, malgré l’absence d’un théorème de lifting dans les espaces de Sobolev naturels pour la même fonctionnelle.

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The authors were supported by University of Bologna, funds for selected research topics, and by GNAMPA of the INDAM, Italy, project “Analysis in metric spaces and subelliptic equations.”     

An initial-boundary value problem with mixed lateral conditions for heat equation

Sunto Viene studiato un problema con condizioni laterali miste per l'equazione del calore, nel caso in cui la ? superficie di separazione ? abbia un punto caratteristico. Viene dato un teorema di esistenza e di unicità in spazi con peso a regolarità variabile.

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Entrata in Redazione l'8 marzo 1978.

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Partially supported by Istituto Nazionale di Fisica Nucleare, Sezione di Bologna, Bologna, Italy.     

Sharp Geometric Poincare Inequalities for Vector Fields and Non-Doubling Measures

We derive Sobolev–Poincaré inequalities that estimatethe L^q(d µ) norm of a function on a metric ball when µis an arbitrary Borel measure. The estimate is in terms of theL¹(d ) norm on the ball of a vector field gradient of the function,where d dx is a power of a fractional maximal function of µ.We show that the estimates are sharp in several senses, andwe derive isoperimetric inequalities as corollaries. 1991 MathematicsSubject Classification: 46E35, 42B25.     

Agent-based interpretations of classic network models

The paper is aimed at developing agent-based variants of traditional network models that make full use of concurrency. First, we review some classic models of the static structure of complex networks with the objective of developing agent-based models suited for simulating a large-scale, technology-enabled social network. Secondly, we outline the basic properties that characterize such networks. Then, we briefly discuss some classic network models and the properties of the networks they generate. Finally, we discuss how such models can be converted into agent-based models (i) to be executed more easily in heavily concurrent environments and (ii) to serve as basic blocks for more complex agent-based models. We evidence that many implicit assumptions made by traditional models regarding their execution environment are too expensive or outright impossible to maintain in concurrent environments. Consequently, we present the concurrency issues resulting from the violation of such assumptions. Then, we experimentally show that, under reasonable hypothesis, the agent-based variants maintain the main features of the classic models, notwithstanding the change of environment. Eventually, we present a meta-model that we singled out from the individual classic models and that we used to simplify the agent-oriented conversion of the traditional models. Finally, we discuss the software tools that we built to run the agent-based simulations.     

Destination of titanium particles detached from titanium plasma sprayed implants

Small titanium particles may detach from titanium plasma sprayed (TPS) implants during implant insertion, when no preliminary tapping is used, probably for the frictional force between titanium coating and host bone. Aim of this study was to investigate the destination of these titanium particles observed in the peri-implant environment. Twenty-four TPS screws were implanted in tibiae of two sheep. Fourteen and 90 days after implantation the implants with the surrounding bone were removed and processed to be analyzed by light microscope and scanning electron microscope (secondary electron and back-scattered electron probes). Small titanium particles detached from the unloaded TPS implants were observed both in the newly-formed bone matrix and in marrow tissue. Histomorphometric analysis showed that both at 14 and 90 days after implantation the titanium particles appeared more concentrated in marrow tissue than in calcified bone matrix, decreasing by 66.4% over time. In particular, smaller particles (<250 microm(2)) decreased by 81.5%, whereas the larger ones (250-2000 microm(2)) did not show any significant variations over time, suggesting that most of the smaller particles may undergo to ionic dissolution, probably migrating into the peri-implant marrow lacunae. A slight migration of titanium particles from the implant surface towards the more distant peri-implant tissues was also demonstrated over time.     

Nitroxide biradicals as thread units in paramagnetic cucurbituril-based rotaxanes

The first example of paramagnetic rotaxane containing cucurbit[6]urils has been reported and characterized both by ESR and NMR spectroscopy.     

Curvature Diffusions in General Relativity

We define and study on Lorentz manifolds a family of covariant diffusions in which the quadratic variation is locally determined by the curvature. This allows the interpretation of the diffusion effect on a particle by its interaction with the ambient space-time. We will focus on the case of warped products, especially Robertson-Walker manifolds, and analyse their asymptotic behaviour in the case of Einstein-de Sitter-like manifolds.