MATH516-101 :       Partial Differential Equations   (First term 2019/2020)


Lecture I: Monday, 1:00--2:00 pm, MATHAnnex-1102

Lecture II: Wednesday, 1:00--2:00 pm, MATHAnnex-1102

Lecture III: Friday, 1:00--2:00 pm, MATHAnnex-1102

Office Hours, Every MW: 3:30-5:30pm, TU-Thur: 1-2:00pm or by appointment


Downloads For MATH516-101


Download 1: Course Outline

Download 2: HW1 (due: by 6pm of Sept. 20, 2019)

Download 3: Perron's Method

Download 4: HW2 (due: by 6pm of Oct. 7, 2019)

Download 5: HW2, Q8 solution

Download 6: HW3 (due: by 6pm of Oct. 23, 2019)

Download 7: HW4 (due: by 6pm of Nov. 8, 2019)

Download 8: HW5 (due: by 6pm of Dec. 3, 2019)

Download 9: Lecture Notes on Moser Iterations

Download 10: Lecture Notes on Maximum Principle


Updates For MATH 516-101


First class; Sept. 4, 2019

Sept 4: 1st order linear PDEs (transport equation); Poisson's Formula (stated).

Sept 6: Representation formula for Laplace equation. Mean-Value-Properties for harmonic function. (Evans' book, 2.2.1, 2.2.2.)

Sept 9: Properties of harmonic functions: smoothness, gradient estimates, analyticity. (Evans 2.2.3)

Sept. 11: Analyticity, maximum principle, Harnack inequality.

Sept. 13: Green's function, Green's representation formula.

Sept. 16: Green's function. Poisson's formula in a ball.

Sept 18: Poisson formula in a ball. Perron's method.

Sept. 20: Perron's method. Complete. Barrier construction.

Sept. 23: Barrier construction for exterior cone condition. Dirichlet Principle. Completion of Laplace equation.

Sept 25: Derivation of formula for heat equation. Discussion the bound for the initial conditions.

Sept. 27: Rigorous verification of heat equation formula. Parabolic maximum principle.

Sept. 30: Uniqueness of solutions to heat equation under growth condition. Tychohov's example. Smoothness of heat equation.

Oct. 2: d'Alembert's formula in dimension n=1; Kitchhoff's formula for dimension n=3.

Oct. 4: n=2, Poisson's formula. Duhammel's principle.

Oct. 7: Duhammel's principle. wave with source in dimension 1 and 3. Energy method.

Oct. 9: Definition of weak derivatives. Examples of weak derivatives.

Oct. 11:Definition of Sobolev spaces. Density theorems for weak derivatives.

Oct. 16: Computation of D f(u). Density theorems for Sobolev spaces. Extension Theorem.

Oct. 18: Extension and trace operator.

Oct. 21: Characterization of $W^{1,p}_0$. Trace Operator. Gagliaro-Nirenberg-Sobolev inequality.

Oct. 23: Gagliaro-Nirenberg-Sobolev inequalities. Sobolev inequalities in bounded domain. Poincare inequality.

Oct. 25: Morrey estimates. Examples of Loss of compactness.

Oct. 28: Compactness of Embeddings.

Oct. 30: Poincare inequalities. Differentiabilities of $W^{1,p}, p>n$ functions. Dual space of $H_0^1$.

Nov. 1: Characterization of $H^{-1}$. existence of weak solutions by Riesz representation theorem. definition of weak solutions for general elliptic opertaor.

Nov. 4: Lax-Milgram Theorem. Existence of Weak Solution I.

Nov. 6: Existence of Weak Solutions II. Fredholm Alternatives.

Nov. 8: Existence Theorem III. Eigenvalues. H^2 estimates.

Nov. 13: Local $H^2$ estimates.

Nov. 15: Global $H^2$-estimates. Regularity Theorems.

Nov. 18: Moser iterations and Applications. Part I.

Nov. 20: Moser iterations. De Giorgi's proof.

Nov. 22: Hopf Boundary Lemma.

Nov. 25: Strong Maximum Principle. The case of $c(x)\geq 0$.

Nov. 27: A priori estimates. Bernstein estimates.

Nov. 29: Harnack inequalities. Modica's estimates.


Announcements For MATH 516-101


New office Hours on September 27: 3-6pm

Office Hours in the week of September 30-Oct. 4: Oct. 2, 2-3pm, Oct. 3, 10-4pm, Oct. 4 2-3pm.

Office Hours in the last week: Tuesday, Thursday, all day; Wed, 3-5:30pm

Office Hours in the exam period: Dec. 2-20, except, Dec. 4,5,6.


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