Hysteresis and Phase Transitions

1. Front Matter

   Pages i-x

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2. Introduction

   • Martin Brokate, Jürgen Sprekels

   Pages 1-9

3. Some Mathematical Tools

   • Martin Brokate, Jürgen Sprekels

   Pages 10-21

4. Hysteresis Operators

   • Martin Brokate, Jürgen Sprekels

   Pages 22-121

5. Hysteresis and Differential Equations

   • Martin Brokate, Jürgen Sprekels

   Pages 122-149

6. Phase Transitions and Hysteresis

   • Martin Brokate, Jürgen Sprekels

   Pages 150-174

7. Hysteresis Effects in Shape Memory Alloys

   • Martin Brokate, Jürgen Sprekels

   Pages 175-217

8. Phase Field Models with Non-Conserving Kinetics

   • Martin Brokate, Jürgen Sprekels

   Pages 218-270

9. Phase Field Models With Conserved Order Parameters

   • Martin Brokate, Jürgen Sprekels

   Pages 271-303

10. Phase Transitions in Eutectoid Carbon Steels

    • Martin Brokate, Jürgen Sprekels

    Pages 304-331

11. Back Matter

    Pages 332-358

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Hysteresis is an exciting and mathematically challenging phenomenon that oc­ curs in rather different situations: jt, can be a byproduct offundamental physical mechanisms (such as phase transitions) or the consequence of a degradation or imperfection (like the play in a mechanical system), or it is built deliberately into a system in order to monitor its behaviour, as in the case of the heat control via thermostats. The delicate interplay between memory effects and the occurrence of hys­ teresis loops has the effect that hysteresis is a genuinely nonlinear phenomenon which is usually non-smooth and thus not easy to treat mathematically. Hence it was only in the early seventies that the group of Russian scientists around M. A. Krasnoselskii initiated a systematic mathematical investigation of the phenomenon of hysteresis which culminated in the fundamental monograph Krasnoselskii-Pokrovskii (1983). In the meantime, many mathematicians have contributed to the mathematical theory, and the important monographs of 1. Mayergoyz (1991) and A. Visintin (1994a) have appeared. We came into contact with the notion of hysteresis around the year 1980.

• Approximation
• Convexity
• Phase Transition
• Phase Transitions
• differential equation
• function space
• linear optimization
• wave equation

• Mathematisches Seminar, Christian-Albrechts-Universität zu Kiel, Kiel, Germany

  Martin Brokate

• Weierstrass Institute for Applied Analysis and Stochastics, Berlin, Germany

  Jürgen Sprekels

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