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
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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