University of Utah - Department of Mathematics
MATH 5750 & MATH 6880
- Fall 2005
Topics in Applied Mathematics: Homogenization and Optimal Design


Instructo
r: Marian BOCEA 

Office: JWB 203

Phone
: (801)585-6967

E-mail
: bocea@math.utah.edu

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.

Course Description:
    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)

Andrej CHERKAEV, Variational Methods for Structural Optimization, Springer (2000)

Andrej CHERKAEV and Robert KOHN
(Editors), Topics in the Mathematical Modelling of Composite Materials, Birkhäuser (1997)

Gianni DAL MASO, An Introduction to Gamma-convergence, Birkhauser (1993).

Graeme W. MILTON,
The Theory of Composites, Cambridge University Press (2002)