University of Utah - Department of Mathematics
MATH 5750 & MATH 6880 -
Topics in Applied Mathematics:
Homogenization and Optimal Design
Instructor: Marian BOCEA
Office: JWB 203
Office Hours: By appointment. Room: JWB 203
Class meets: MWF 10:45AM-11:35AM. Room: ST (William Stewart Building ) 215
Text: Doina CIORANESCU and Patrizia DONATO, An
Introduction to Homogenization, Oxford University Press (1999)
Course Web Page: http://www.math.utah.edu/~bocea/fall2005.html
Prerequisites: MATH 5440 Introduction
to PDEs, MATH 6210 Real Analysis, MATH 6420 Partial Differential Equations,
or consent of instructor.
Course Audience: Graduate students and advanced undergraduate
students interested in Applied Mathematics.
The course provides an introduction to the modern
theory of Homogenization and its applications to the Optimal Design
of composite materials in the Conductivity and Linear Elasticity settings.
The topics covered include: H-convergence, effective properties, bounds
on effective coefficients, the G-closure problem, compensated compactness
and correctors, multiple scale convergence.
In addition, and if time permits, the study of the asymptotic
behavior of certain classes of nonlinear problems arising in the analysis
of thin films of active materials and fluid-fluid phase transitions is undertaken
by means of De Giorgi's Gamma-convergence.
The basic facts about weak and weak* convergence
in Banach spaces, Sobolev spaces, and the variational formulation of
elliptic problems will be recalled without proofs, as needed throughout
the semester. A good review of these topics may be found in the first
four chapters of the textbook.
In addition to the textbook, we will use the following monographs:
Gregoire ALLAIRE, Shape Optimization
by the Homogenization Method, Springer (2002)
Methods for Structural Optimization,
Andrej CHERKAEV and
Robert KOHN (Editors),
Topics in the Mathematical Modelling
of Composite Materials,
Gianni DAL MASO, An Introduction to Gamma-convergence,
Theory of Composites,
University Press (2002)