CNA SEMINARS

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"
ABSTRACT: Let ${\cal F}$ be a hysteresis operator, and A a second order elliptic operator. The following equation arisesin elasto-plasticity, ferromagnetism, ferroelectricity:

\begin{displaymath}{\partial^2\over \partial t^2} \big[u +{\cal F}(u)\big] +Au =f. \eqno(1) \end{displaymath}

For ${\cal F}$ equal to a (possibly discontinuous) scalar Preisach operator, existence of a solution is proved for an associated initial- and boundary-value problem. Processes in ferromagnetic metals and in ferromagnetic insulators can be represented by the following vector equations, respectively,

\begin{displaymath}{\partial\over\partial t}\big[\vec H +\vec{\cal F}(\vec H)\big] +\nabla \times\nabla \times\vec H = \vec f, \eqno(2) \end{displaymath}


\begin{displaymath}{\partial^2\over \partial t^2}\big[\vec H +\vec{\cal F}(\vec H)\big] +\nabla\times\nabla \times\vec H = \vec f \eqno(3) \end{displaymath}

(here written with normalized coefficients). Existence of a solution is proved for an initial- and boundary-value problem associated to (2), for $\vec{\cal F}$ equal to a (possibly discontinuous) vector estension of the Preisach operator. A similar result is proved for (3), for a smaller class of (discontinuous) hysteresis operators.

TUESDAY, September 5, 2000

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.

TUESDAY, October 10, 2000

CNA SEMINAR: 1:30 P.M., PPB 300, Arie Leizarowitz, Technion, Israel.
TITLE: On second order two-dimensional variational problems on unbounded domains.
ABSTRACT: We consider the energy functional

\begin{displaymath}J_{\Omega }=\int _{\Omega }[(\Delta u)^2-b\vert\nabla u\vert^2+\psi (u)]dxdy\end{displaymath}

where $\Omega $ is a bounded domain in R2 and $\psi $ is a C2(R2) super quadratic potential. Associated with this functional is the notion of mean energy, which is meaningful for a class of configurations u on the whole plane, and we study the corresponding minimization problem. Minimizers of this problem are called equilibrium configurations. We establish apriori bounds for this unbounded-domain problem, and use it to establish existence of equilibrium configurations. We also discuss the existence of radially symmetric equilibrium configurations. Moreover, we remark on the connection between radially symmetric configurations and 1-dimensional configurations.

TUESDAY, October 17, 2000


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.

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: TBA

TUESDAY, November 14, 2000

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

TUESDAY, December 5, 2000

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