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
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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.