PDE


Introduction to Partial Differential Equations (MATH 557)

Spring 2024

Syllabus



Class time: TuTh: 10:05AM - 11:20AM

Office hours: TuTh: 01:30PM - 02:30PM

Office: 106 Physics building

Location: Gross Hall 318



Exam: There will be a midterm exam and a final exam. The midterm is on February 29 (leap day). 

Final is on April 23, 10:05 am to 12:15 pm, Gross Hall 318.

Grading:   

• Weekly homework (1/3). 

• Midterm (1/3) 

• Final exam (1/3)   

Attendance is required. 



Topics Covered: Introduction to PDE, ODE: uniqueness, ODE: existence of solutions, Linear PDE: Laplace equation,  Poisson equation, Heat equation, Wave equation, Non-linear 1st order PDE: envelopes, Non-linear 1st order PDE: method of characteristics, Holder and Sobolev spaces, Sobolev inequalities and compactness theorems, Curve-Shortening flow, 2nd order elliptic equations: existence of weak solutions, 2nd order elliptic equations: regularity, Fourier transform, Laplace transform, Linear evolution equation I: parabolic, Linear evolution equation II: hyperbolic, Geometric Applications: Ricci flow on surfaces, Navier-Stokes equations.


Prerequisites: Real analysis, multivariable calculus, and ordinary differential equations.


The main references:

1) L. C. Evans, Partial Differential Equations

2) Class lecture notes, here (updated on April 19)

References (recommended): 

3) D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, (2nd ed)

4) J. Kazdan, Applications of Partial Differential Equations to Problems in Geometry, Lecture Notes

5) S. Donaldson, Geometric Analysis, Lecture Notes

6) T. Walpuski, Introduction to Partial Differential Equations, Lecture Notes.


Homework: There will be weekly homework sets on Gradescope, due on Friday 11:59PM and to be submitted via Gradescope. Late homework will not be accepted.

Homework 1 (due date Feb 09). Questions, Solutions.

Homework 2 (due date Feb 16). Questions, Solutions.

Homework 3 (due date Feb 23). Questions, Solution can be found in Evans, pages 70 to 73.

Homework 4 (due date Mar 01). Questions, Solutions. 

Homework 5 (due date Mar 08). Questions. Solved in the class.

Break.

Homework 6 (due date April 01). Questions. Solutions.

Homework 7 (due date April 12). Questions.

Homework 8 (due date April 26). Questions.


(Possible) topics of discussion:


Midterm



Final