MATH516-101 :       Partial Differential Equations   (First term 2021/2022)

Lecture I: Monday, 1:00--2:00 pm, SWNG-208

Lecture II: Wednesday, 1:00--2:00 pm, SWNG-208

Lecture III: Friday, 1:00--2:00 pm, SWNG-208

Office Hours, Monday,Friday: 2:30pm-3:30pm, 5-6pm; Wednesday: 2:30pm-3:30pm or by appointment

Downloads For MATH516-101

Download 1: Course Outline

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Download 4: Assignment One (deadline: Sept. 22)

Download 5: Lecture Note 3

Download 6: Lecture Note 4 (Perron's Method)

Download 7: Assignment Two (deadline: Oct. 6)

Download 8: Lecture Note 5

Download 9: Lecture Note 6

Download 10: Assignment Three (deadline: Oct. 20)

Download 11: Lecture Note 7

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Download 13: Assignment 4 (due Nov. 3)

Download 14: Lecture Note 9

Download 15: Assignment 5 (due Nov. 19)

Download 16: Lecture Note 10

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Download 18: Assignment 6 (due Dec. 8)

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Updates For MATH 516-101

First class; Sept. 8, 2021

Sept 8, 2021: Introduction to PDE, solutions to transport equation, solution to Poisson equation (stated). (Evans 2.1, 2.2)

Sept. 10: Solution to Poisson equation. Harmonic functions.

Sept. 13: Mean-Value-Property. Maximum Principle. Gradient Estimate.

Sept. 15: Applications of gradient estimates (analyticity). Harnack inequality. $C^\infty$ of harmonic functions.

Sept. 17: Green's function and Green's representation formula.

Sept. 20: Green's function for a ball, Poisson integral formula. Solvability of Poisson equation with $g=0$.

Sept. 22: Perron's Method I.

Sept. 24: Perron's Method II.

Sept. 27: Energy Method, Dirichlet Principle.

Oct. 4: Duhammel's Principle and inhomogeneous heat equation. Smoothness of heat equation.

Oct. 6: Wave equation for $ n=1$ and $ n=3$. Kirchhoff formula.

Oct. 8: Wave equation in dimension $n=2$. Inhomogeneous Wave (Duhammel's principle.) Uniqueness and finite speed of propogantion by energy method.

Oct. 13: smooth approximation of $L^p$ spaces. Weak derivatives: uniqueness and examples.

Oct. 15: Calculus on weak derivatives, Sobolev space $W^{k,p}$

Oct. 18: Density theorems. Extensions.

Oct. 20: Trace theorems and characterization of $W_0^{1,p}$.

Oct. 22: Sobolev inequalities.

Oct. 25: Morrey's estimates.

Oct. 27: Morrey's. Higher order Sobolev inequalities.

Oct: 29: Compactness. Poincare inquality

Nov. 1: Conjugate space of $H_0^1$. Weak solution.

Nov. 3:Lax-Milgram. First existence theorem.

Nov. 5: Second existence theorem.

Nov. 8: Third existence theorem.

Nov. 10, Nov. 12: Winter break

Nov. 15: $H^2$ theory (interior).

Nov. 17: $H^2$ theory (boundary). Local Boundedness.

Nov. 19: De Giorgi-Nash-Moser iteration (De Giorgi).

Nov. 22: Moser's iteration.

Nov. 24: Maximum Principle.

Nov. 26: Maximum Principle.

Nov. 29: Gradient estimates. Bernstein techniques.

Dec. 1: Harnack Inequality.

Dec. 3: Weak solutions of linear parabolic equations.

Dec. 6: Regularity of weak solutions of parabolic equations---the end.

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