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
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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