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

Assignments

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

Announcements

Nov 30, 2010. Teaching evaluation is NOW available online in the UBC system: https://eval.olt.ubc.ca/science. 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 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)