APMAE4990 - Section 002
Calculus of Variations & Applications
Spring 2022
Spring 2022
Time: MW 1:10-2:25
Place: 327 Mudd Building
Instructor: Dr. James Scott (he/him)
Affiliation: Applied Physics and Applied Mathematics, SEAS
Email: See "Home" on this website
Office Hours: T 2pm-3pm, W 11am-12pm, or by appointment
Office Location: 287D Engineering Terrace (accessible by going through the APAM offices)
Prerequisites: Multivariable Calculus (APMA E2001 or equivalent) is required. Introduction to Real Variables / Real Analysis (MATH UN2000 or equivalent) is highly recommended but not required. PDEs (APMA E4200 or equivalent) is also helpful but not required.
Course Content: A modern introduction to the Calculus of Variations, with both theory and applications. Topics included are existence of solutions, variational formulations, relaxation, and Gamma-convergence. Settings will range from one-dimensional convex problems to multi-dimensional nonconvex problems. Applications will include minimal surfaces, the isoperimetric inequality, solid mechanics and elasticity, composite materials, vibrations and wave propagation.
Textbooks: No book is required for the course, but the course content will follow the presentation from the following books:
Introduction to the Calculus of Variations. Bernard Dacorogna. 2008 Imperial College Press.
Variational Methods with Applications in Science and Engineering. Kevin W. Cassell. 2013 Cambridge University Press, ISBN: 9781107022584
Direct Methods in the Calculus of Variations. Bernard Dacorogna. 2008 Springer-Verlag, 2nd edition.
An Introduction to the Calculus of Variations. Charles Fox. Dover Publications, ISBN: 9780486654997
Gamma-Convergence for Beginners. Andrea Braides. 2002 Oxford University Press. Available electronically.
Grades: Homework (40%), Exam 1 (20%), Exam 2 (20%), Final (20%).
Homework will be assigned every two weeks and submitted via Gradescope (on Canvas) by 11:59 pm on the due date. The lowest homework score will be dropped. If you have questions about the homework problems, please come to office hours or email me. You are welcome to work together on homework. However, each student must turn in their own assignment.
There will be two non-cumulative take-home exams, and one non-cumulative take-home final. The online link for each exam will be available only for the duration of the exam. You will have 48 hours for the two exams, and one week for the final. The exams will be submitted via Gradescope.
Uploading Files to Gradescope: See this PDF and this video.
Additional Resources
Paul's Online Math Notes for Multivariable Calculus
Paul's Online Math Notes for Differential Equations
Mathematical Foundations of Elasticity Theory - Lecture notes by John Ball
Lecture Notes
Lectures 1 and 2 - Introduction
Lecture 3 - Classical Methods
Lecture 4 and 5 - Preliminaries, Direct Methods Preliminaries
Lecture 6 and 7 - Sobolev Spaces, Direct Methods
Lecture 8 - Direct Methods, Euler-Lagrange Equations, Lavrentiev Phenomenon
Lecture 9 - Relaxation Theory in the Scalar Case, Isoperimetric Inequality
Lecture 10 - Isoperimetric Inequality, The Case of Vector-Valued Functions
Lecture 11 - An example: Hyperelastic Materials
Lecture 12 and Lecture 13 - Relaxation and Quasiconvexity
Lecture 14 and Lecture 15 - Hyperelastic Materials II
Lecture 16 and Lecture 17 - Introduction to Gamma Convergence
Lecture 18 - Gamma Convergence of Integrals
Lecture 19 - Gamma Convergence - Phase Transition (A better SNL sketch)
Lecture 20 - Gamma Convergence - Mumford-Shah Functionals
Lecture 21 - Image Restoration
Lecture 22 - Gradient Descent, Image Segmentation
Lecture 23 and Lecture 24 - Constrained Optimization - Introduction
Lecture 25 - Optimal Control
Lectures 26 and 27 - Optimization and Deep Learning
Lecture 28 - Other Topics