MATH516-101 :       Partial Differential Equations   (First term 2018/2019)


Lecture I: Monday, 1:00--2:00 pm, MATH-202

Lecture II: Wednesday, 1:00--2:00 pm, MATH-202

Lecture III: Friday, 1:00--2:00 pm, MATH-202

Office Hours, Every MWF: 4:30-5:30pm, or by appointment


Downloads For MATH516-101


Download 1: Course Outline

Download 2: Perron's Method

Download 3: Assignment One (due: Sept. 21)

Download 4: Assignment Two (due: October 10)

Download 5: Assignment Three (due: October 26)

Download 6: Assignment Four (due: November 9)

Download 7: Note on Moser's Lemma

Download 8: Assignment Five(due: November 30th)

Download 9: Note on Maximum Principle


Updates For MATH 516-101


First class; Sept. 5, 2018

Sept 5: 2.2.1 of L. Evans.

Sept 7: Green's representation formula for bounded domain. 2.2.4 of L. Evans.

Sept 10: Poisson's formula in a ball.

Sept. 12: Properties of harmonic functions. Towards Perron's method for Dirichlet BVP.

Sept. 14: Properties of subharmonic functions.

Sept. 17: Proof of Perron's method.

Sept 19:Local barrier, regular points. C^2, $ exterior cone condition.

Sept 21: Energy method. Hadamard's example. Heat equation formula using Fourier transform.

Sept. 24: Initial value Problem for heat equation: Existence.

Sept. 26: Maximum Principle, Uniqueness of heat equation.

Sept. 28: Uniqueness for heat equation. Tikohnov's example.

Oct. 1: inhomogeneous heat equation. regularity of heat equation.

Oct. 3: interior regularity of solutions for heat equation. D'Almbert's formula for wave.

Oct. 5: n=1, d'Alembert's formula, n=3, Kirchhoff formula.

Oct. 10: n=2, Poisson's formula for wave equation. Inhomogeneous wave, Duhamel's formula.

Oct. 12: definition of weak derivatives, 4 examples.

Oct. 15: Another characterization of weak derivatives. Computation of weak derivatives of composition functions. Computation of $ D u^{+}, D u^{-1}$.

Oct. 17: Sobolev spaces. Density Theorems.

Oct. 19: Extension theorems.

Oct. 22: Trace theorem.

Oct. 24: Gagliaro-Nirenberg inequality and Sobolev inequality.

Oct. 26: Morrey's estimates.

Oct. 29: Loss of compactness in the embedding. Compactness of Sobolev embedding.

Oct. 29: Non-compactness note.

Oct. 31: Compactness Proof. Poincare inequalities.

Nov. 2: Characterization of $H^{-1} (\Omega)$. General Elliptic Dirichlet Problem. Bilinear forms.

Nov. 5: Lax-Milgram Theorem. Existence of weak solution.

Nov. 7: Fredholm Alternatives I, II.

Nov. 9: Fredholm Alternative III. H^2-estimate.

Nov. 14: Global $H^2$-estimates. $W^{2,p}$-estimates.

Nov. 16: Local $H^2$ estimates. $W^{2,p}$ and Schauder estimates.

Nov. 19: Moser's iteration lemma. Applications.

Nov. 21: Moser's iteration Lemma.

Nov. 23: Maximum Principle. Weak and strong.

Nov. 26: Hopf boundary lemma. Maximum Principle with sign conditions.

Nov. 28:Applications of Maximum Principle. Bernstein Methods.

Nov. 30: Bernstein Method on gradient estimates (local and global). Distributions of final papers.


Announcements For MATH 516-101


The 4:30-5:30pm office hour on Oct.1 is cancelled.

Final report due: Dec. 19.


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