Exams:
There will have only one take-home project.

REFERENCES:
There are some books on this topic which may serve well as a references for the course:

• M. Sihavy, The Mechanics and Thermodynamics of Continuous Media, Springer, 1997.
• I. Ekeland and R. Temam, Convex analysis and Variational Problems, North-Holland, 1976
• B. Dacorogna, Direct Methods in the Calculus of Variations, Springer-Verlag, Berlin, 1989.
• J. T. Oden, Qualitative Methods in Nonlinear Mechanics.
• J.T. Oden and J.N. Reddy, Variational methods in theoretical mechanics

Prerequisites and Corequisites:
Some experience with advanced calculus, linear algebra and partial differential equations. Some previous experience with functional analysis and continuum mechanics would be helpful but is not essential.

Objectives:
The fields of nonconvex analysis and optimization have experieced significant development during the recent decads. Many nonlinear problems arising in applied mathematics, physics, economics and engineering sciences require the consideration of nonconvexity and nondifferentiablity. Since the 1980's, the theories of nonconvex analysis and variational mathods have become the important mathematical tools for nonlinear analysis and mechanics. The objective of this second part of the two-semester course offering is to provide a clear and unified presentation of the modern mathematical skills and qualitative variational methods needed for applied mathematicians and engineers in solving nonlinear problems in mathematical physics, optimizations, continuum mechanics, and economics, etc. The emphasis is on understanding and applying not proving.

Course Content:
The course will discuss the following themes (these can be adjusted to suit the interests of the students):

Chapter 1. Introduction to Nonconvex Analysis

• 1.0 Review: Dynamical Systems and Convex Analysis
• 1.1 Symmetrical Hamiltonian Systems
• 1.2 Clarke Duality
• 1.3 Minimax Theory in Nonconvex Variational Problems
• 1.4 Classification of the critical points
• 1.5 Complementary Formulations in Dynamics
• 1.6 Eigenvalue problems
• 1.7 Applications in linear elastic dynamics.

Chapter 2. Nonconvex Analysis in Fully Nonlinear Systems

• 2.1 Framework and virtual work principles
• 2.2 Potential extremum principles
• 2.3 Dual extremum principles
• 2.4 Minimax Theory and Triality extremum principle
• 2.5 Lagrangian and complementary energy principles
• 2.6 Complementary principles
• 2.7 Relaxation methods

Chapter 3. Applications in Finite Deformation Theory

• 3.1 Finite deforamtion theory and super potentials
• 3.2 Constitive inequalities and conjugate tensor measures
• 3.3 Quasiconvexity, Polyconvexity, and Rank-1 convexity.
• 3.4 Null Lagrangians and Weak Sequential Continuity.
• 3.5 Complementary-dual variational principles
• 3.6 Bifurcation theory, degree theory, and nonlinear eigenvalue problems.
• 3.7 Applications in Phase transitions, Mathematical theory of hysteresis.

Chapter 4. Numerical Methods and Algorithms

• 4.1 Duality theory in mathematical programming
• 4.2 Mixed finite element methods
• 4.3 Hybrid finite element methods
• 4.4 Complementary finite element methods
• 4.5 Nonconvex programming
• 4.6 Other problems of nonlinear programming.

General:
I reserve the option of modifying these policies where appropriate as the course develops.

Virginia Tech Honor System Information

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http://www.math.vt.edu/people/gao/class_home/5495_7679.html

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Mathematics Department
Virginia Polytechnic Institute & State University
Last updated on September 10, 1996.