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Seminar Abstracts

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 arises in elasto-plasticity, ferromagnetism, ferroelectricity:

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

\begin{displaymath}\frac{\partial^2}{ \partial t^2}[\vec{H} +\vec{\cal F}(\vec{H})] +\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.