Course Description - Bo Li
Weak Convergence Methods for Variational Problems
Modeling Crystalline Solids
Math 285J: Seminar in Applied Mathematics, Spring, 1998, UCLA
MW 4:00 pm -- 5:15 pm, MS 5128
Instructor: Bo Li (Office: MS 7370,
In this course we shall first
review the basics of the calculus of multiple
integrals and the theory of compensated compactness.
We shall then focus on a recently developed mathematical
theory of crystalline microstructure based on the
principle of energy minimization.
The following is a list of tentative topics to be covered in this course.
The theory of quasi-convexity: sequential weak lower
semi-continuity of multiple integrals; existence theorems in the calculus
of variations; partial regularity of minimizers; quasi-convexification.
Various concepts of convexity and their relations, characterization of
The theory of compensated compactness: the Div-Curl lemma and its
generalizations; necessary conditions; sufficient conditions;
Young measures and gradient Young measures. Some applications in
nonlinear partial differential equations.
Some background of the mechanics of crystalline solids: Bravais
lattices; martensitic phase transformation;
the Cauchy-Born rule; twinning; Hadamard compatibility condition;
classification of interfaces.
The Ball-James theory of microstructure: the two-well problem;
the reduction from a multi-well problem to a two-well one;
simple laminates for a six-well problem.
Kinematics of various microstructures, restrictions on microstructure.
Most of our references are recently published research articles. But
the following books contain some subjects listed above.
B. Dacorogna, Direct methods in the calculus of variations,
L. C. Evans, Weak convergence methods for nonlinear partial
differential equations, CBMS Regional conference series in mathematics
74, AMS, 1990.
P. G. Ciarlet, Mathematical elasticity, Vol. 1: three-dimensional
elasticity, North-Holland, 1988.
This course is designed for graduate students who are interested
in one of the areas of partial differential equations, the calculus of
variations, applied analysis, and mathematical aspects of materials science.
A graduate real analysis course will be a prerequisite. But no
knowledge on crystalline solids is assumed.
The material presented in this course will be essentially self-closed.