FRIDAY, September 1, 2000

CNA/MATH COLLOQUIUM: 4:30 P.M., WeH 7500, Augusto Visintin, Universita' degli Studi di Trento, Dipartimento diMatematica, Trento, Italy.

TITLE: "Quasilinear Equations with Hysteresis"

CNA SEMINAR: 1:30 P.M., PPB 300, Paolo Podio-Guidugli, Univ. di Roma.

TITLE: Strain and Superconductivity

ABSTRACT: TBA


TUESDAY, October 3, 2000

CNA SEMINAR: 1:30 P.M., PPB 300, Gianpietro DelPiero, Department of Engineering, University of Ferrara.

TITLE: ``On the role of interface energies in the description of material behavior"

ABSTRACT: Research progresses in the subject of material behavior have been very rapid. A common aspect in this development is a tendency to by-passing some of the basic assumptions of classical continuum mechanics. Based on the concept of interface energy, a possibility of a unified view in a broader context, but still at the macroscopic level and within the limits of the continuum scheme, has recently emerged. Indeed, starting from a purely elastic scheme and allowing for the creation of discontinuities at the price of an interface energy, it is possible to obtain a variety of responses, depending on the shape chosen for the function describing the dependence of the interface energy on the amplitude of the discontinuities. This approach has the advantage of using the direct methods of calculus of variations to determine the stable or metastable equilibrium configurations, leaving the way open to traditional techniques for numerical solution. Moreover, it is conceivable to relate the shape chosen for the interface energy to the fine properties of the structure of matter.

CNA SEMINAR: 1:30 P.M., PPB 300, Arie Leizarowitz, Technion, Israel.

TITLE: On second order two-dimensional variational problems on unbounded domains.

CNA SEMINAR: 1:30 P.M., PPB 300, A. Pisztora, Department of Mathematical Sciences, Carnegie Mellon University.

TITLE: ``On the equlibrium theory of phase coexistence: a microscopic point of view"

ABSTRACT: The phenomenological theory of coexisting phases assumes the existence of a quantity, called the ``surface energy'', associated with the phase boundaries. This quantity can be computed as a surface integral of a (direction dependent) scalar quantity (called the surface tension) over the phase boundaries. The theory asserts that in a system in equilibrium the shapes of the phases are determined so as to minimize the total surface energy. (In dimensions more than two the arising variational problems are very hard and remain partly unresolved.)

In this talk I will focus on an other mathematical challange associated with this physical phenomenon: namely, how can we justify the phenomenological picture based on statistical mechanics/probability theory. More precisely I will consider an archetypal microscopic model of a two-component material exhibiting phase coexistence and explain results which imply the aforementioned phenomenological assumptions and answer questions which can't even be addressed remaining on the macroscopic level (such as what the phases are exactly).

I will try to provide all the background necessary to understand the results and will not go into probabilistic details. The talk should be understandable for graduate students.

CNA SEMINAR: 1:30 P.M., PPB 300, A. Pisztora, Department of Mathematical Sciences, Carnegie Mellon University.

TITLE: ``On the equlibrium theory of phase coexistence: a microscopic point of view, part 2"

ABSTRACT: The phenomenological theory of coexisting phases assumes the existence of a quantity, called the ``surface energy'', associated with the phase boundaries. This quantity can be computed as a surface integral of a (direction dependent) scalar quantity (called the surface tension) over the phase boundaries. The theory asserts that in a system in equilibrium the shapes of the phases are determined so as to minimize the total surface energy. (In dimensions more than two the arising variational problems are very hard and remain partly unresolved.)

In this talk I will focus on an other mathematical challange associated with this physical phenomenon: namely, how can we justify the phenomenological picture based on statistical mechanics/probability theory. More precisely I will consider an archetypal microscopic model of a two-component material exhibiting phase coexistence and explain results which imply the aforementioned phenomenological assumptions and answer questions which can't even be addressed remaining on the macroscopic level (such as what the phases are exactly).

I will try to provide all the background necessary to understand the results and will not go into probabilistic details. The talk should be understandable for graduate students.

October 31, 2000

CNA SEMINAR: 1:30 P.M., PPB 300, Stuart Hastings, Dept. of Mathematics, University of Pittsburgh.

TITLE: ``Layers, chaos, and global attractors for some one-dimensional reaction diffusion problems"

Abstract

TUESDAY, November 7, 2000

CNA SEMINAR: 1:30 P.M., PPB 300, J. Necas, Charles University at Prague, Czech Republic.

TITLE: ``Incompressible fluids in 3d with wiscosity depending on pressure"

ABSTRACT: The incompressible fluid models where the viscosity can depend both on the second invariant of the symmetric velocity gradient and on the pressure were documentated to be important in particular at high pressure. The aim of the lecture is to show that they are physically reasonable viscosity forms, for which the global in time existence of weak solution can be proved. Three-dimensional flows are analyzed.

CNA SEMINAR: 1:30 P.M., PPB 300, Giovanni P. Galdi, Department of Mechanical Engineering, University of Pittsburgh

TITLE: TBA

ABSTRACT: TBA

CNA SEMINAR: 1:30 P.M., PPB 300, Michal Kowalczyk, Department of Mathematical Sciences, Carnegie Mellon University.

TITLE: Finite dimensional nonlinear problems and localized solutions to elliptic equations and systems

ABSTRACT: Solving a wide range of semilinear elliptic problems can be reduced to solving a system of nonlinear equations in n real or complex variables (this is sometimes called finite dimensional reduction). Some known examples are elliptic problems involving critical Sobolev exponent, Ginzburg-Landau equation, stationary reaction-diffusion systems and others. Finite dimensional reduction for these equations not only yields the existence of solutions but also allows to describe their important properties such as location of their singularities, vortices or local maxima. Unfortunately solving the finite dimensional problem corresponding to a PDE might be a challenge in itself-for instance showing that the vortices in Ginzburg-Landau equation concentrate along Abrikosov lattice seems to be still an open problem. In my talk I will discuss a reaction-diffusion system in which location of local maxima of one of the components can be obtained by solving a system of nonlinear equations in n complex variables. Exploiting the invariance of the finite dimensional problem with respect to isometries of and using the Principle of Symmetric Criticality we will be able to establish the existence of interesting patterns such as honeycomb or a snowflake.

TITLE: "Micromagnetics : Equilibria and Relaxation

CNA SEMINAR: 1:30 P.M., PPB 300, Morton Gurtin, Carnegie Mellon University Department of Mathematical Sciences.

TITLE: ``A gradient theory of single-crystal plasticity that accounts for geometrically necessary dislocations"

ABSTRACT: This talk discusses the kinematics of geometrically necessary dislocations (GNDs) and develops a concomitant gradient theory that accounts for GNDs within a thermomechanical framework. The theory is based on classical macroforces; microforces for each slip system consistent with a microforce balance; a mechanical version of the second law that includes, via the microforces, work performed during slip; a rate-dependent constitutive theory that includes dependences on a tensorial measure of geometrically necessary dislocations. The microforce balances are equivalent to nonlocal (pde) yield conditions for the individual slip systems. To make contact with classical dislocation theory, the microstresses are shown to represent counterparts of the Peach-Koehler force on a single dislocation.

CNA SEMINAR: 1:30 P.M., PPB 300, Leonid Berlyand, , Department of Mathematics and Materials Research Institute, Penn State, University Park, PA.

TITLE: "Network Approximation for Effective Properties of a High Contrast Random Composites"

ABSTRACT: We present a new approach for calculation of effective properties of high contrast random composites (random particles in a matrix or host when properties of the particles and the hosting medium are very different) and provide its rigorous mathematical justification. The main idea of this approach is the reduction of the original continuum problem, which is described by PDE with rough coefficients, to a discrete random network. We introduce the the interparticle distance parameter for nonperiodic (irregular) distributions of particles and show asymptotic equivalence of the effective (homogenized) coefficients for the discrete and continuum models in the limit when the relative interparticle distance goes to zero. Our method is based on variational techniques and it provides an exact error estimate in which all constant are explicitly defined. Such results are rare in homogenization: most of the error estimates provide the order of magnitude only and involve some unknown constants. Our study was motivated by a problem of optimizing effective properties of polymer/ceramic composites. We use the discrete network to compute the effective coefficients numerically and draw physical predictions about polydispersed composites (particles of different sizes) in the percolation regime. The work was done jointly with A. Kolpakov [1]. 1. L. Berlyand, A. Kolpakov. To appear in ``Archive of Rational Mecahnics and Analysis''.

CNA SEMINAR: 1:30 P.M., PPB 300, Michel Jabbour, Department of Mathematical Sciences, Carnegie Mellon University.

TITLE: "On the equations of anisotropic curvature flow in three-dimensions with curvature-dependent energy"

ABSTRACT: In the context of crystal growth or grain boundary motion, the evolution of a phase interface or grain boundary is sometimes kinetically driven, i.e., it is essentially independent of the behavior of the bulk phases. Moreover, for realistic interfacial energies, the anisotropic motion-by-curvature equation exhibits backward-parabolic behavior over intervals of their domain, thus inducing the formation of facets and wrinkles. In this talk, a thermodynamically consistent regularized model is derived, which can be used to analyze such interfacial phenomena. The main ingredient is a constitutive dependence of the interfacial energy density on the curvature tensor which, in our framework, requires the introduction of interfacial configurational moments and yields a generalized Eshelby relation for the surface configurational stress tensor.

CNA SEMINAR: 1:30 P.M., PPB 300, Luc Tartar, Carnegie Mellon University, Department of Mathematical Sciences.

TITLE: ``On traces of W^2,1"

ABSTRACT: The traces on a hyperplane x_N = 0 of functions in W^1,1(R^N) is exactly L¹(R^N-1) (Emilio GAGLIARDO, 1957), but the space of traces of functions in W^2,1(R^N) is strictly smaller than W^1,1(R^N-1) (Francoise DEMENGEL, 1984). I will describe what I know on the question of characterizing the space of traces.

[Joint work with Francoise DEMENGEL, Departement de Mathematiques, Universite de Cergy-Pontoise, France]

TUESDAY, February 27, 2001

CNA SEMINAR: 1:30 P.M., PPB 300, Alfred Carasso, Mathematical and Computational Sciences Division, National Institute of Standards and Technology, Gaithersburg, MD

TITLE: ``Direct Blind Deconvolution of Images and Levy Stable Laws"

ABSTRACT: Blind deconvolution seeks to deblur an image without knowing the cause of the blur. Iterative methods are commonly applied to that problem, but the iterative process is slow, uncertain, and often ill-behaved. This talk considers a significant but limited class of blurs that can be expressed as convolutions of 2-D radially symmetric Levy `stable' probability density functions. This class includes Gaussian and Lorentzian (Cauchy) distributions. For such blurs, methods are developed that can detect the point spread function from 1-D Fourier analysis of the blurred image. A separate image deblurring technique uses this detected point spread function to deblur the image. Each of these two steps uses direct non-iterative methods, and requires interactive adjustment of parameters. As a result, blind deblurring of 512X512 images can be accomplished in minutes of CPU time on current desktop workstations. Numerous blind experiments on synthetic data show that for a given blurred image, several distinct point spread functions may be detected that lead to useful, yet visually distinct reconstructions. Application to real blurred images will also be demonstrated.

CNA SEMINAR: 1:30 P.M., PPB 300, M. Foss, Carnegie Mellon University, Department of Mathematical Sciences.

TITLE: ``Lavrentiev's Phenomenon in Nonlinear Elasticity"

ABSTRACT: Lavrentiev's phenomenon is associated with the sensitivity of a variational problem's infimum to the regularity required of the competing mappings. We present one of a class of physically natural stored energy densities where the energy functional exhibits the Lavrentiev phenomenon.

CNA SEMINAR: 1:30 P.M., PPB 300, Martin Feinberg, Department of Chemical Engineering, Ohio State University.

TITLE: "Lecture 1. Chemical Engineering for Mathematicians"

ABSTRACT: This first lecture is intended to provide background for the second. Although a great deal of hard-core chemical engineering is mathematically complex, the questions themselves are fairly easy to understand. I'll try to describe, in mathematical terms, how chemical engineers think about systems of chemical reactions and how those reactions enter into processes designed to enhance the production of good things and suppress the production of bad things. At the very least, it should become clear how the occurrence of chemical reactions gives rise to systems of nonlinear equations. With luck, it will also become clear why - in light of the great freedom one has in designing chemical processes - questions of process design are mathematically rich.

CNA SEMINAR: 1:30 P.M., PPB 300, Martin Feinberg, Department of Chemical Engineering, Ohio State University.

TITLE: "Lecture 2. Toward a Theory of Process Design"

ABSTRACT: Given a network of chemical reactions and given a set of reactant streams, one wants to know how best to utilize those streams in order to enhance the production rate of a desired product while suppressing the production rates of unwanted by-products. The set of qualitative design strategies is so vast that it becomes difficult to know not only which design outcome is optimal but also which outcomes are even feasible - that is, which outcomes are attainable by some design and which are not. There is beginning to emerge a theory for assessing outer limits on what can be attained from all possible designs, even those that remain unimagined. The theory brings together parts of chemical engineering and parts of mathematics (e.g, convexity theory, dynamical systems, differential geometry) in surprising ways. I'll try to survey a little of what's known and some of what's not.

CNA SEMINAR: 1:30 P.M., PPB 300, Gilles Clermont, University of Pittsburgh.

TITLE: "Organ system cooperation and disease"

ABTRACT: In health, the organ systems of the body interact with one another and respond in a coordinated way to environmental perturbations or changes in activity level. This coordinated and inter-related activity of organ systems leads to variability in ``output" (e.g., fluctuations in heart rate or the rate of secretion of bile). Under pathophysiological conditions, the normal variability in the measure ``output" from organ systems can be diminished or lost entirely. This loss of variability may be a sign of "decomplexification", a concept by which disease results in loss of appropriate organ-to-organ communication and cooperation. Preliminary data suggest that decomplexification is present in critically ill patients and portends a poorer outcome. Accordingly, being able to detect and quantitate loss of variability could be valuable for prognostication and identifying opportunities for early intervention. This seminar will review physiologic bases of this hypothesis and suggest data mining tools that might be useful in exploring the presence of decomplexification.

CNA Seminar: 1:30 P.M., PPB 300, Antonio Gaudiello.

TITLE:Coupled and Uncoupled Limits for a N-Dimensional Multidomain Neumann Problem.
A. Gaudiello, B. Gustafsson, C. Lefter and J. Mossino.

ABSTRACT: We consider a quasilinear problem with exponent $p\in (1,+\infty)$, in a multidomain of $R^N$, $N\geq 2$, consisting of two vertical cylinders, one placed upon the other: the first one with given height and small cross section, the other one with small height and given cross section.

Assuming that the volumes of the two cylinders tend to zero with same rate, we prove that the limit problem is well posed in the union of the limit domains, with respective dimension $1$ and $N-1$. Moreover, the limit problem is coupled if $p>N-1$ and uncoupled if $1

THURSDAY, April 19, 2001

CNA SEMINAR: 1:30 P.M., PPB 300, Giovanni Leoni,

TITLE: ``Interfacial energies for incoherent inclusions"

ABSTRACT: We study a variational problem describing an incoherent interface between a rigid inclusion and a linearly elastic matrix. The elastic material is allowed to slip relative to the inclusion along the interface, and the resulting mismatch is penalized by an interfacial energy term, which depends on the surface gradient of the relative displacement.

Joint work with Paolo Cermelli

CNA SEMINAR: 1:30 P.M., PPB 300, Olga Ladyzhenskaya, Steklov Instiute of Mathematics, St. Petersburg, Russia.

TITLE: ``On multiplicators in nonhomogeneous spaces and their applications to some linear systems of hydrodynamical type"

ABSTRACT: (1) We describe sufficient conditions on a function $\tilde m: E^n\to C$ to be a multiplicator in the nonhomogenious Holder spaces. (2) We discuss some known and unknown results concerning multiplicators in nonhomogeneous Lebegue's space $L_q(E^m, L_r(\Omega))$, $\Omega \subset E^n$. (3) We give the application of the criterium from 1) to some linear problems of hydrodynamical type.

CNA SEMINAR: 1:30 P.M., PPB 300, Anna Vainchtein, University of Pittsburgh, Department of Mathematics.

TITLE: "Thermodynamics of martensitic phase transitions and hysteresis"

ABSTRACT: Materials undergoing stress-induced martensitic phase transformations(such as shape memory alloys) often exhibit hysteretic behavior when subjected to cyclic loading. The hysteresis consists of a rate-independent part, which persists at quasistatic loading, and rate-dependent hysteresis, whose size increases with applied strain rate. The hysteresis loops exhibit serrations accompanied by nucleation events and often by very irregular, ``jerky" motion of phase boundaries. To model hysteresis, we consider a (generally non-isothermal) one-dimensional dynamic model that incorporates a finite bar with a non-monotone temperature-dependent stress-strain law and nonzero latent heat. Inertia is taken into account and two dissipation mechanisms are considered: heat conduction and the internal viscous dissipation of kinetic origin, proportional to the strain rate. Time-dependent mechanical loading and ambient temperature are prescribed at the ends of the bar. In the first part of the talk, we will focus on the isothermal (purely mechanical) description. We will show that this model predicts rate-independent hysteresis which is caused primarily by metastability of equilibria and phase nucleation. The hysteresis loops persist even when the loading rate is very slow, and viscosity effects are minor. The hysteresis loops are serrated, and a stick-slip interface motion is observed. We will then consider placing the viscoelastic bar on an elastic foundation, to mimic interaction of phases in higher dimensions. This model results in tilted hysteresis loops with multiple serrations and reveals an interesting interplay between the foundation-favored step-by-step phase nucleation process and the inertia-favored interface slip and phase annihilation. In the second part, we will describe recent results for a more general, thermodynamic model. We will show numerical simulations that predict the rate-dependent portion of the hysteresis. When heat conductivity is large, or the applied loading is sufficiently slow, the results are similar to those for the isothermal case. At faster loading or smaller heat conductivity (or larger latent heat), the stick-slip interface motion is partially replaced by irregular fast-slow interface motion and relaxation oscillations in both released heat and end load, caused by the inability of the system to remove latent heat fast enough to relax to an equilibrium and the resulting competition between material instability and loading.

CNA SEMINAR: 1:30 P.M., PPB 300, Paolo Marcellini, University of Florence
TITLE: "Some sharp conditions for lower semicontinuity in $L^{1}$

CNA SEMINAR: 1:30 P.M., PPB 300, Nicola Fusco, University of Naples
TITLE: ``Functions of bounded variation and rearrangments"

CNA SEMINAR: 1:30 P.M., PPB 300, H. Christian Gromoll, University of California, San Diego.

TITLE: "Diffusion Approximation for a Processor Sharing Queue in Heavy Traffic"

ABSTRACT: We discuss a diffusion approximation result for the queue length process of a GI/GI/1 processor sharing queue in heavy traffic. The main approach is to study a certain measure valued process, which encodes information about the residual service times and from which standard measures of performance such as queue length, workload and sojourn time can be recovered.