MATH
516:101
Partial
Differential
Equations
I:
Introduc
tion to Elliptic and Parabolic PDE (UBC
course page is here
)
Class:
Tu 11:20  12:30, Thur 11:20  12:30, 13:00 14:00. All
classes at MATX 1102.
Office
hours (subject to change): Tu 2:30  3:20, Thur 2:00  3:00.
Or by
appointment: please email at yhkim "at"
math "dot" ubc 'dot' ca
Assignments
Week  Date  Contents 
1  Sept 7 (Tue) 
Basic PDE's (Laplace/heat/wave equations) and their physical motivations. Fundamental solution for Laplace equation. Solution to Poisson equation: convolution with the fundamental solution. Dirac delta function. [Evans] p20 25. 
Sept 9 (Thur) 
finished the proof of [Evans, p23 Theorem 1]. Initial Value
Problem for heat equation. L^p spaces. Fourier transform and its
properties w.r.t. L^2 product, convolution, derivatives. Heat
kernel. Smoothing of heat equation. Continuity of solution to heat
equation at t=0. [Evanst, 2.3.1, 4.3.1 p187p192] 

2  Sept 14 (Tue) 
Duhamel's principle for heat [Evans, p4950] and wave
equations [Evans, p80. 2.4.2]. Solution to Initial Value Problem for
wave equations by Fourier transform [Evans, p194]. 1 Dimensional case
(d'Alembert's formula) [Evans, p6768]. Finite propagation speed of
solutions to wave equaiton. 
Sept 16 (Thur) 
Wave equations: computation of the fundamental solution:
Fourier transform method. Finite propagation speed. Domain of
dependence. Sharp Huygen's principle in odd dimensions. n=3:
Kirchhoff's formula. n= 2: Poisson's formula. Energy method for heat and wave equations. Uniqueness of solution to Initial/Boundary value problems. Finite propagation speed of wave. [Evans, 2.3.4, 2.4.3] 

3  Sept 21 (Tue) 
 Properties of Laplace's equation and Harmonic functions:
Meanvalue property, strong maximum principle, uniqueness of boundary
valude problems, Harnack inequality. [Evans, 2.2.2, 2.2.3.a. 2.2.3.f.] 
Sept 23 (Thur) 
 smoothness of harmonic functions [Evans, 2.2.3. b]  properties of heat equation: meanvalue property, parabolic maximum principle, . [Evans, 2.3.2, 2.3.3]  Sobolev spaces: weak derivatives, definition of Sobolev spaces, properties of weak derivatives [Evans, 5.2.1  5.2.3] 

Sept 24. Fri. 
HW #1 is assigned. 

4  Sept 28 (Tue) 
 Properties of weak derivates. Sobolev spaces are
Banach space. Approximation by smooth functions in R^n and in open
domains. 
Sept 30 (Thur) 
 Sobolev spaces are Banach space. Approximation by
smooth functions in R^n and in open domains. 

5  Oct 5 (Tue) 
 Approximation by funcitons smooth up to the boundary. 
Oct 7 (Thur) 
Approximation by funcitons smooth up to the boundary.

Extension. 

6  Oct 12 (Tue) 
No class 
Oct 14 (Thur) 
No class 

7 
Oct 19 (Tue) HW 1 is due 
 Extension. 
Oct 21 (Thur) 
Traces (Evans 5.5)  Charcterization of tracezero
functions: W^{1,p}_0 space.  Sobolev imbeddings: GagliardoNirenbergSobolev inequality 

8 
Oct 26 (Tue) 
 GagliardoNirenbergSobolev inequality. 
Poincare inequality  Morrey inequality 
Oct 28 (Thur) 
No class 

9 
Nov 2 (Tue) 
No class 
Nov 4 (Thur) 
 Morrey inequality  Compact Sobolev imbedding (Evans 5.7) : RellichKondrachov compactness. 

10  Nov 9 (Tue) 
 Proof of RellichKondrachov compactness. Poincare
inequality (Evans 5.8.1)  Elliptic equations (Evans Ch 6.) Weak solutions. LaxMilgram theorem. 
Nov 11 (Thur) 
No class (Remembrance Day). 

11  Nov 16 (Tue) HW 2 is due 
 Existence and uniqueness of weak solution in H^1_0: 
Nov 18 (Thur) 
 Fredholm alternatives  elliptic regularity (interior estimates) 

12  Nov 23 (Tue) 
 elliptic regularity (interior estimates) 
Nov 25 (Thur) 
 elliptic regularity (boundary estimates)  Maximum principles: Weak Maximum Principle 

13  Nov 30 (Tue) 
 Strong Maximum Principle: Hopf's lemma 
Dec 2 (Thur) 
 Parabolic PDE: Existence and uniqueness of weak
solution. Last class 

No final exam (HW 3 is Due Dec. 10) 