# MATH516-101 : Partial Differential Equations (First term 2016/2017)

### Lecture I: Tuesday 9:30--11:00 am, MATH-Annex 1102

### Lecture II: Thursday 9:30--11:00 am, MATH-Annex 1102

### Office Hours, Every MWF: 4:30-5:20pm, or by appointment

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# Downloads For MATH516-101

# Updates For MATH 516-101

Coverage of September 8th: Chapter 2.2 of Evans, Chapter 1 of Han-Lin, Chapter 2 of Gilbarg-Trudinger.

September 13rd: Chapter 2.2 of Evans, Chapter 1 of Han-Lin. Applications of MVP (MP, uniqueness, derivative estimates, Liouville, Harnack)

September 18th: MVP. Green's representation.

September 20th: Green's representation formula. Chapter 2.2 of Evans.

September 22nd: Perron's Method. Chapter 4.4 of F. John's book.

September 27th: Dirichlet energy method. Solution to heat equation. Section 2.2.5 and 2.3.1 of Evans.

September 29th: Uniqueness of solutions to heat equation. Tychonov's example. inhomogeneous equation.

October 4: Wave equation. 1D d'Alembert's formula. 3D Kirchnoff's formula. 2D Poisson's formula.

October 6: Weak derivatives. Definitions and examples. Approximations by smooth functions. Chapter 5 of Evans.

October 11: Definition of Sobolev spaces. Smooth approximation of Sobolev spaces. Chapter 5 of Evans. ###

### October 13: Properties of Sobolev spaces: approximations, extensions, trace theorems.

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### October 18: Sobolev inequalities.

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### October 20: Morrey's estimate. Compact Embeddings.

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### Oct. 25: Poincare's inequalities. $H^{-1} $ space. Lax-Milgram. Weak solutions.

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### Oct. 27: Existence of weak solution. Lax-Milgram Theorem.

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### Nov. 1: Regularity of Weak Solutions. H^2 estimates.

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### Nov. 4: H^2 estimates. L^p-theory. Moser's iteration.

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### Nov. 8: Moser's iterations.

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### Nov. 10: Moser's iterations. maximum principle.

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### Nov. 15: Maximum Principle and Applications.

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### Nov. 17: Applications of Maximum Principle. BernsteinEstimates. Modica's Estimates

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### Nov. 22: Method of Moving Planes.

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### Nov. 24: Method of sub-super solutions. Uniqueness.

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### Nov. 29: Uniquenss. Direct Minimization. Nehari's Manifold Method. Mountain-Pass Lemma.

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### Dec. 1: Nehari's Manifold Method. Mountain-Pass Lemma.

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# Announcements For MATH 516-101

The class on September 6th (Tuesday) is cancelled due to conflicts with the qualification exam and Imagine Day. The first class will be on Sept. 8th (Thursday).

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