Introduction to Free Boundary Problems for Parabolic Partial Differential Equations
Hours: from 10:30 to 12:10 on every Friday
From October 2, 2009 to January 22, 2010
Classroom: RM 801 of Godoto Building
Partial differential equations for functions of the time variable and spatial variables
are considered under the initial condition and the boundary conditions. Usually the spatial
variables are in a fixed domain. However, when we consider the mixture of ice and water,
the domain occupied by the ice changes with time since the ice melt into water and we cannot
specify the boundary of the two domains beforehand. In this case the boundary between ice
and water is an unknown. Such a problem is called a free boundary problem.
In these series of lectures, we learn about the existence, uniqueness and asymptotic behavior
of solutions of the free boundary problem for the equation of the heat conduction, in
particular, the Stefan problem. We learn also about the phase-field model which is an
approximation method suitable for numerical simulations.
* To learn basic methods to construct a classical solution of the Stefan problem.
* To understand the relationship between free boundary problems and phase-field models.
1. Classical solutions of the Stefan problem
1.1. Fundamental properties of solutions of parabolic partial differential equations of second order (one lecture)
1.2. Construction of approximate solutions of the one-phase Stefan problem (two lectures)
1.3. Existence of a classical solution of the one-phase Stefan problem (two lectures)
1.4. Asymptotic behavior (one lecture)
2. Variational Inequalities (four lectures)
2.1. Variational approach to the one-phase Stefan problem
2.2. Some properties of the free boundary of the Stefan problem
3. Phase field models: transition layer to sharp interface (three lectures)
4. Numerical solutions (one lecture)
 Avener Friedman, "Variational Principles and Free-Boundary Problems", John Wiley and Sons, New York, 1982.
 $B2O86ED=(IW(B, $B!V<+M36-3&LdBj!W(B, $BEl5~Bg3X=PHG2q!$(B1989.
 David Kinderlehrer and Guido Stampacchia, "An Introduction to Variational Inequalities and Their Applications", Academic Press, New York, 1980.
 Anvarbek M. Meirmanov, "The Stefan Problem", Walter de Gruyter, Berlin/New York, 1992.
Office hours: after the lecture