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\noindent \underline{\bf FRIDAY, September 1, 2000}

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

\noindent {\bf TITLE}: "Quasilinear Equations with Hysteresis"

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

$$ \frac{\partial^2}{\partial t^2} [u +{\cal F}(u)] +Au =f. \eqno(1) $$

\noindent 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.

\noindent Processes in ferromagnetic metals and in ferromagnetic insulators
 can be represented by the following vector equations,
respectively, 

$$ \frac{\partial}{\partial t}[\vec{H} +\vec{\cal F}(\vec{H}) +\nabla \times
 \nabla \times \vec{H} = \vec{f}, \eqno(2) $$

$$ \frac{\partial^2}{ \partial t^2}[\vec{H} +\vec{\cal F}(\vec{H})] +\nabla \times \nabla \times \vec{H} = \vec{f} \eqno(3) $$

\noindent (here written with normalized coefficients).

\noindent 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.

\noindent A similar result is proved for (3), 
for a smaller class of (discontinuous) hysteresis operators.

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