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 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 |
|
| 7 |
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 |
|
| 8 |
Oct 26 (Tue) |
- Gagliardo-Nirenberg-Sobolev 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) : 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) |
||