Test 1 
Wed, Oct 9,
in class 
Final Exam 
Takehome 
Date 
Material
Covered 
Suggested HW problems 
Assignments  Remarks 
Aug 21  Introduction.  
Aug 23  Review of multivariable calculus, chapter 1, appendix C.  HW1  
Aug 26  More review. Intro to PDEs. Basic definitions. General form of a PDE. Linear PDEs. 
If possible, read pages 20 to 22 of the textbook prior to Wed class.  
Aug 28  Existence theorem for Poisson's equaion.  
Aug 30  Existence theorem for Poisson's equaion.  
Sept 2  Mean value formulas and maximum principle.  Problems 2, 3, 4, 5, 6, 10 and 11
of section 2.5. Prove theorems 14 on p. 37, and 15 on p. 41. 
Problems 2, 3, 4, and 6 of section
2.5. Due: Sept 11. 

Sept 4  Mollifiers,
regularization. Regularity of harmonic functions. 
Prove theorem 7 on p. 29 and
theorem 10 on p. 31. Due: Sept 18th. 

Sept 6  Green's function for the Laplacian, analyticity of harmonic functions, Harnack inequality.  
Sept 9  Fundamental solution for the heat equation. Initial value problem.  
Sept 11  Student presentation: distributions.  Do these
problems. Due: Sept 18th. 

Sept 13  Wave equation in 1d; D'Alembert's formula.  
Sept 16  Wave equation in even and odd dimensions.  Problems 13, 14, 15, 18, 19, 21,
24 of section 2.5. Do also these
problems. Due Sept 25th. 

Sept 18  Finite propagation speed for the wave equation; uniqueness. Review of measure theory.  
Sept 20  Review of functional analysis.  Extra
credit. Due: Sept 30th. 

Sept 23  Weak derivatives.  
Sept 25.  Sobolev spaces.  Problems 1, 3, 4, 9, 10, 11, 14,
15 of chapter 5. Also these
problems. Due: Oct 4th. Extra credit: 2, 5, 6 of chapter 5. Due: Oct 4th. 

Sept 27  Sobolev embedding.  
Sept 30  Sobolev embedding.  
Oct 2  Sobolev
embedding and other properties of Sobolev functions. 

Oct 4  Sobolev
embedding and other properties of Sobolev functions. 

Oct 7  Extension and difference quotients for Sobolev functions.  
Oct 9  Test.  
Oct 11  Fall Break.  
Oct 14  Test
corrections Classification of equations in elliptic, parabolic and hyperbolic. 

Oct 16 
Definition
and generalities about weak solutions of second order
linear elliptic equations. 

Oct 18 
Existence
of weak solutions for second order linear elliptic
equations. 

Oct 21 
Fredholm
alternative. Statement and applications. 
Problems 1, 2, and 3 of
section 6.6. Due: Oct 28th. 

Oct 23 
Fredholm
alternative. 

Oct 25 
Fredholm
alternative. 

Oct 28 
Elliptic
regularity. 

Nov 1  Elliptic
regularity. Weak maximum principle. 
HW. Due Nov 11. 

Nov 6  Boundary maximum principle  
Nov 8 
Strong
maximum principle 

Nov 11 
Negative
norm Sobolev spaces 
Notes. 

Nov 13 
Applications
of negative norm Sobolev spaces: necessary and sufficient
condition for existence of linear PDEs. 

Nov 15 
Seminar 
The PDE
seminar (4:10pm, SC 1307) will replace the class. 

Dec 2nd 
Local
solvability. Egorov's counterexample. 
Notes. 

Dec 4 
Discussion
of the takehome final exam. 