Young-Heon Kim’s  Teaching              Home           Research

(Term 1, 2010/2011: September, 2010 -- December, 2010)

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

Course syllabus


HW 1 (basic PDE's), HW 2 (Sobolev Spaces), HW 3 (Elliptic and Parabolic PDE).


Nov 30, 2010. Teaching evaluation is NOW available online in the UBC system: Pleaes take it seriously and please complete the evaluations before the deadline: 11PM on December 5th, 2010. (Survey results will not be released to instructors until course grades have been submitted.)

Schedule / Plan / Progress / Summary (Subject to change)

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 p187--p192]

2 Sept 14 (Tue)
Duhamel's principle for heat  [Evans, p49-50] 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, p67-68]. 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: Mean-value 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: mean-value 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

Oct 19 (Tue)
HW 1 is  due
 - Extension.
Oct 21 (Thur)
 -Traces (Evans 5.5) - Charcterization of trace-zero functions: W^{1,p}_0 space.
- Sobolev imbeddings: Gagliardo-Nirenberg-Sobolev inequality

Oct 26 (Tue)
 - Gagliardo-Nirenberg-Sobolev inequality.  - Poincare inequality
- Morrey inequality
Oct 28 (Thur)
No class

Nov 2 (Tue)
No class
Nov 4 (Thur)
 - Morrey inequality
- Compact Sobolev imbedding (Evans 5.7) : Rellich-Kondrachov compactness.

10 Nov 9 (Tue)
 - Proof of Rellich-Kondrachov compactness. Poincare inequality (Evans 5.8.1)
- Elliptic equations (Evans Ch 6.) Weak solutions. Lax-Milgram 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)