Downloads For MATH516-101
Download 1: Course Outline
Download 2: First Homework (Due Date: September 27th, 2016)
Download 3: Second Homework (Due Date: October 14th, 2016, by 5:30pm)
Download 4: Solutions to First Homework
Download 5: Third Homework (Due Date: October 27th, 2016, by 5:30pm)
Download 6: Solutions to Second Homework
Download 7: Solutions to Third Homework
Download 8: Fourth Homework (Due Date: November 10, 2016, by 5:30pm)
Download 9: Fifth Homework (Due Date: November 22, 2016, by 5:30pm)
Download 10: Moser iterations
Download 11: Solutions to Fourth Homework
Download 12: Modica Estimates
Download 13: Sixth Homework (Due Date: December 9, 2016, by 5:30pm)
Download 14: Solutions to Fifth Homework
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.
October 18: Sobolev inequalities.
October 20: Morrey's estimate. Compact Embeddings.
Oct. 25: Poincare's inequalities. $H^{-1} $ space. Lax-Milgram. Weak solutions.
Oct. 27: Existence of weak solution. Lax-Milgram Theorem.
Nov. 1: Regularity of Weak Solutions. H^2 estimates.
Nov. 4: H^2 estimates. L^p-theory. Moser's iteration.
Nov. 8: Moser's iterations.
Nov. 10: Moser's iterations. maximum principle.
Nov. 15: Maximum Principle and Applications.
Nov. 17: Applications of Maximum Principle. BernsteinEstimates. Modica's Estimates
Nov. 22: Method of Moving Planes.
Nov. 24: Method of sub-super solutions. Uniqueness.
Nov. 29: Uniquenss. Direct Minimization. Nehari's Manifold Method. Mountain-Pass Lemma.
Dec. 1: Nehari's Manifold Method. Mountain-Pass Lemma.
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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October 13: Properties of Sobolev spaces: approximations, extensions, trace theorems.
October 18: Sobolev inequalities.
October 20: Morrey's estimate. Compact Embeddings.
Oct. 25: Poincare's inequalities. $H^{-1} $ space. Lax-Milgram. Weak solutions.
Oct. 27: Existence of weak solution. Lax-Milgram Theorem.
Nov. 1: Regularity of Weak Solutions. H^2 estimates.
Nov. 4: H^2 estimates. L^p-theory. Moser's iteration.
Nov. 8: Moser's iterations.
Nov. 10: Moser's iterations. maximum principle.
Nov. 15: Maximum Principle and Applications.
Nov. 17: Applications of Maximum Principle. BernsteinEstimates. Modica's Estimates
Nov. 22: Method of Moving Planes.
Nov. 24: Method of sub-super solutions. Uniqueness.
Nov. 29: Uniquenss. Direct Minimization. Nehari's Manifold Method. Mountain-Pass Lemma.
Dec. 1: Nehari's Manifold Method. Mountain-Pass Lemma.
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).
Back to My Home Page
October 18: Sobolev inequalities.
October 20: Morrey's estimate. Compact Embeddings.
Oct. 25: Poincare's inequalities. $H^{-1} $ space. Lax-Milgram. Weak solutions.
Oct. 27: Existence of weak solution. Lax-Milgram Theorem.
Nov. 1: Regularity of Weak Solutions. H^2 estimates.
Nov. 4: H^2 estimates. L^p-theory. Moser's iteration.
Nov. 8: Moser's iterations.
Nov. 10: Moser's iterations. maximum principle.
Nov. 15: Maximum Principle and Applications.
Nov. 17: Applications of Maximum Principle. BernsteinEstimates. Modica's Estimates
Nov. 22: Method of Moving Planes.
Nov. 24: Method of sub-super solutions. Uniqueness.
Nov. 29: Uniquenss. Direct Minimization. Nehari's Manifold Method. Mountain-Pass Lemma.
Dec. 1: Nehari's Manifold Method. Mountain-Pass Lemma.
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).
Back to My Home Page
October 20: Morrey's estimate. Compact Embeddings.
Oct. 25: Poincare's inequalities. $H^{-1} $ space. Lax-Milgram. Weak solutions.
Oct. 27: Existence of weak solution. Lax-Milgram Theorem.
Nov. 1: Regularity of Weak Solutions. H^2 estimates.
Nov. 4: H^2 estimates. L^p-theory. Moser's iteration.
Nov. 8: Moser's iterations.
Nov. 10: Moser's iterations. maximum principle.
Nov. 15: Maximum Principle and Applications.
Nov. 17: Applications of Maximum Principle. BernsteinEstimates. Modica's Estimates
Nov. 22: Method of Moving Planes.
Nov. 24: Method of sub-super solutions. Uniqueness.
Nov. 29: Uniquenss. Direct Minimization. Nehari's Manifold Method. Mountain-Pass Lemma.
Dec. 1: Nehari's Manifold Method. Mountain-Pass Lemma.
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).
Back to My Home Page
Oct. 25: Poincare's inequalities. $H^{-1} $ space. Lax-Milgram. Weak solutions.
Oct. 27: Existence of weak solution. Lax-Milgram Theorem.
Nov. 1: Regularity of Weak Solutions. H^2 estimates.
Nov. 4: H^2 estimates. L^p-theory. Moser's iteration.
Nov. 8: Moser's iterations.
Nov. 10: Moser's iterations. maximum principle.
Nov. 15: Maximum Principle and Applications.
Nov. 17: Applications of Maximum Principle. BernsteinEstimates. Modica's Estimates
Nov. 22: Method of Moving Planes.
Nov. 24: Method of sub-super solutions. Uniqueness.
Nov. 29: Uniquenss. Direct Minimization. Nehari's Manifold Method. Mountain-Pass Lemma.
Dec. 1: Nehari's Manifold Method. Mountain-Pass Lemma.
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).
Back to My Home Page
Oct. 27: Existence of weak solution. Lax-Milgram Theorem.
Nov. 1: Regularity of Weak Solutions. H^2 estimates.
Nov. 4: H^2 estimates. L^p-theory. Moser's iteration.
Nov. 8: Moser's iterations.
Nov. 10: Moser's iterations. maximum principle.
Nov. 15: Maximum Principle and Applications.
Nov. 17: Applications of Maximum Principle. BernsteinEstimates. Modica's Estimates
Nov. 22: Method of Moving Planes.
Nov. 24: Method of sub-super solutions. Uniqueness.
Nov. 29: Uniquenss. Direct Minimization. Nehari's Manifold Method. Mountain-Pass Lemma.
Dec. 1: Nehari's Manifold Method. Mountain-Pass Lemma.
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).
Back to My Home Page
Nov. 1: Regularity of Weak Solutions. H^2 estimates.
Nov. 4: H^2 estimates. L^p-theory. Moser's iteration.
Nov. 8: Moser's iterations.
Nov. 10: Moser's iterations. maximum principle.
Nov. 15: Maximum Principle and Applications.
Nov. 17: Applications of Maximum Principle. BernsteinEstimates. Modica's Estimates
Nov. 22: Method of Moving Planes.
Nov. 24: Method of sub-super solutions. Uniqueness.
Nov. 29: Uniquenss. Direct Minimization. Nehari's Manifold Method. Mountain-Pass Lemma.
Dec. 1: Nehari's Manifold Method. Mountain-Pass Lemma.
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).
Back to My Home Page
Nov. 4: H^2 estimates. L^p-theory. Moser's iteration.
Nov. 8: Moser's iterations.
Nov. 10: Moser's iterations. maximum principle.
Nov. 15: Maximum Principle and Applications.
Nov. 17: Applications of Maximum Principle. BernsteinEstimates. Modica's Estimates
Nov. 22: Method of Moving Planes.
Nov. 24: Method of sub-super solutions. Uniqueness.
Nov. 29: Uniquenss. Direct Minimization. Nehari's Manifold Method. Mountain-Pass Lemma.
Dec. 1: Nehari's Manifold Method. Mountain-Pass Lemma.
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).
Back to My Home Page
Nov. 8: Moser's iterations.
Nov. 10: Moser's iterations. maximum principle.
Nov. 15: Maximum Principle and Applications.
Nov. 17: Applications of Maximum Principle. BernsteinEstimates. Modica's Estimates
Nov. 22: Method of Moving Planes.
Nov. 24: Method of sub-super solutions. Uniqueness.
Nov. 29: Uniquenss. Direct Minimization. Nehari's Manifold Method. Mountain-Pass Lemma.
Dec. 1: Nehari's Manifold Method. Mountain-Pass Lemma.
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).
Back to My Home Page
Nov. 10: Moser's iterations. maximum principle.
Nov. 15: Maximum Principle and Applications.
Nov. 17: Applications of Maximum Principle. BernsteinEstimates. Modica's Estimates
Nov. 22: Method of Moving Planes.
Nov. 24: Method of sub-super solutions. Uniqueness.
Nov. 29: Uniquenss. Direct Minimization. Nehari's Manifold Method. Mountain-Pass Lemma.
Dec. 1: Nehari's Manifold Method. Mountain-Pass Lemma.
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).
Back to My Home Page
Nov. 15: Maximum Principle and Applications.
Nov. 17: Applications of Maximum Principle. BernsteinEstimates. Modica's Estimates
Nov. 22: Method of Moving Planes.
Nov. 24: Method of sub-super solutions. Uniqueness.
Nov. 29: Uniquenss. Direct Minimization. Nehari's Manifold Method. Mountain-Pass Lemma.
Dec. 1: Nehari's Manifold Method. Mountain-Pass Lemma.
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).
Back to My Home Page
Nov. 17: Applications of Maximum Principle. BernsteinEstimates. Modica's Estimates
Nov. 22: Method of Moving Planes.
Nov. 24: Method of sub-super solutions. Uniqueness.
Nov. 29: Uniquenss. Direct Minimization. Nehari's Manifold Method. Mountain-Pass Lemma.
Dec. 1: Nehari's Manifold Method. Mountain-Pass Lemma.
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).
Back to My Home Page
Nov. 22: Method of Moving Planes.
Nov. 24: Method of sub-super solutions. Uniqueness.
Nov. 29: Uniquenss. Direct Minimization. Nehari's Manifold Method. Mountain-Pass Lemma.
Dec. 1: Nehari's Manifold Method. Mountain-Pass Lemma.
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).
Back to My Home Page
Nov. 24: Method of sub-super solutions. Uniqueness.
Nov. 29: Uniquenss. Direct Minimization. Nehari's Manifold Method. Mountain-Pass Lemma.
Dec. 1: Nehari's Manifold Method. Mountain-Pass Lemma.
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).
Back to My Home Page
Nov. 29: Uniquenss. Direct Minimization. Nehari's Manifold Method. Mountain-Pass Lemma.
Dec. 1: Nehari's Manifold Method. Mountain-Pass Lemma.
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).
Back to My Home Page
Dec. 1: Nehari's Manifold Method. Mountain-Pass Lemma.
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).
Back to My Home Page
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).
Back to My Home Page