Course web page: Available on CMU Canvas

Lectures: MWF 12:00 - 12:50pm online via Zoom --- linked in Canvas

Professor: Robert Pego      Email: rpego AT cmu.edu

• Office hours: via Zoom (initially M 8:30-9:30pm, TTh 5-6pm), or just arrange another time - after lunch is often ok.

Potentially useful reference books: (some available via SpringerLink)

• An Introduction to Partial Differential Equations by M. Renardy and R.C. Rogers (Texts in Applied Mathetmatics 13), Springer 1993.
• Elliptic Partial Differential Equations of Second Order by D. Gilbarg and N.S. Trudinger, Springer, 2nd ed. 1983.
• Partial Differential Equations in Action by Sandro Salsa, Springer, 3rd ed. 2016.
• Weak convergence methods for nonlinear partial differential equations by L. C. Evans (CBMS regional conf. ser. 74) Amer. Math. Soc. 1990.

Course Outline:

This semester's topics deal primarily with modern uses of function spaces to study solutions of PDE. A preliminary list is:

1. Elliptic PDE:
   1. Existence of weak solutions: Lax-Milgram, Fredholm Alternative.
   2. Regularity of weak solutions: interior regularity, regularity up to the boundary.
   3. Eigenvalues and eigenfunctions.
2. Parabolic PDE: Sobolev spaces involving time. Existence of weak solutions (Galerkin method). The Navier-Stokes equations.
3. Viscosity solutions of Hamilton-Jacobi equations. Application to control theory and dynamic programming.
4. Selected topics (may vary or differ)
   1. Semigroups and operator-theoretic methods
   2. Direct method of the calculus of variations
   3. Div-curl lemma, Young measures
   4. Concentration compactness
   5. Homogenization - basic examples

Prerequisites:   Essential: Measure theory, elements of Sobolev spaces, distribution theory. Very helpful: functional analysis, basic complex variables, Fourier transform

Learning objectives:   Students will become familiar with several important modern techniques to analyze partial differential equations as indicated above. As well as the elements of regularity theory for weak solutions (dating from the 1950s) they will gain expertise in working with viscosity solutions of Hamilton-Jacobi equations (from the 1980s) and recent strong solutions of Navier-Stokes equations (from the 2000s). Also they will directly experience the research literature through a project report and presentation.

Grading: Based on approximately 6 homework sets and a project report (at least 6 pages) with oral presentation (20 minutes)

Homework will be posted on the course Canvas page, and solutions should be uploaded into Gradescope.

Project and presentation You will be asked to do a project on a topic that goes beyond the material discussed in class. Most of the projects are based on reading articles in the research literature and/or chapters in research monographs. At the end of the semester everyone will give a presentation. Details on the project as well as a list of suggested topics and articles will be provided in class.

Academic integrity requires that your tests and homework solutions are your independent work and not copied from other sources. On homework you are encouraged, however, to discuss with others and consult other resources to improve your understanding.

Health and wellness: Graduate study is time-consuming and can be stressful. Make your health a priority, be smart about time management, seek social support, and do ask for help when needed.