Downloads For MATH516-101
Download 1: Course Outline
Download 2: Lecture Note 1
Download 3: Lecture Note 2
Download 4: Lecture Note 3
Download 5: Homework One (due Sept 21)
Download 6: Lecture Note 4
Download 7: Homework Two (due Oct. 5)
Download 7: Talk in Columbia
Download 7: Talk in Brazil
Updates For MATH 516-101
First class; Sept. 7, 20222
Sept 7, 2022: Introduction to PDE, solutions to transport equation, solution to Poisson equation (started). (Evans 2.1, 2.2)
Sept 9, 2022: Solution to Poisson equation $ -\Delta u= f, f\in C^1_0$. Statements of harmonic functions and Mean-Value-Property.
Sept 12, 2022: Equivalence of harmonic functions and Mean-Value-Property. Applications of MVP: Maximum Principle and Gradient Estimates
Sept. 14, 2022: Harnack inequality. Green's function and Green's representation formula. Reduction to $f=0$.
Sept. 16, 2022. Poisson integral formula for a ball. Start Perron's method.
Sept. 21, 2022. Perron's method: sub-harmonic functions, harmonic lifting, Perron's method.
Sept. 23, 2022. Barrier and regular points. Complete proof of Perron's method. Start the Dirichlet energy method.
Sept. 26, 2022: Dirichlet energy method. Hadamard's example. Fundamental solution for heat equation by Fourier method.
Sept. 28, 2022: Poisson Formula for heat equation. Maximum Principle. Uniqueness of heat equation (under growth condition).
Oct. 3, 2022. Uniqueness and non-uniqueness of heat equation. Tihkonv's example. Duhhamel's Principle
Oct. 5, 2022. Inhomogeneous heat equation. Mean-Value-Property. Regularity. Energy decreasing. backward uniqueness.
Oct. 7, 2022. Backward uniqueness for heat equation. d'Alembert's formula. Kirchhoff formula in dimension 3.
Oct. 12, 2022. Poisson formula in dimension two. Duhammel's principle.
Oct. 14, 2022. Finite speed of propagation. Weak derivatives.
Oct. 17, 2022. Weak derivatives. Sobolev space. Density theorems.
Oct. 19, 2022. Extension theorems
Oct. 21, 2022. Trace Theorem.
Oct. 24, 2022. Gagliarno-Nirenberg inequality
Oct. 26, 2022. Morrey's estimates. Embedding theorems for $W^{k,p}$.
Oct. 28, 2022. Three cases of loss of compactness. Compactness of Sobolev embedding.
Oct. 31, 2022: Poincare-Wirtinger inequality. Conjugate space of $H_0^1 (\Omega)$. Conjugate space of $ H^1 ( (a,b))$.
Nov. 2, 2022: Weak Solutions. Lax-Milgram Theorem.
Nov. 4. First existence theorem. Fredholm Alternatives
Nov. 7. Fredholm Alternatives. Third existence theorem.
Nov. 14. Properties of finite difference of quotients. $H^2$-interior regularity.
Nov. 16: $H^2$-interior regularity. Global $H^2$-regularity.
Nov. 18: Global $H^2$-regularity. Discussion of W^{2,p} and Schauder theory. Statement of Local Boundedness Theorem and Applications.
Nov. 21. Nash-Moser iteration. Local Boundedness.
Nov. 23. Nash-Moser iteration II.
Nov. 25. De Giorgi-Nash-Moser technique. Summary of De Giorgi-Nash-Moser Results. Maximum Principle.
Nov. 28. Weak and Strong Maximum Principle. Hopf Boundary Lemma.
Nov. 30. Narrow domain principle. Bernstein Techniques.
Dec. 2. Bernstein estimates for interior gradient estimates. Harnack inequality (Li-Yau).
Dec. 5. Weak solutions of parabolic equations. Galerkin Method
Dec. 7. Existence and uniqueness of weak solutions by Galerkin method. Regularity of weak solutions of parabolic equations.
Announcements For MATH 516-101
Back to My Home Page
Sept. 28, 2022: Poisson Formula for heat equation. Maximum Principle. Uniqueness of heat equation (under growth condition).
Oct. 3, 2022. Uniqueness and non-uniqueness of heat equation. Tihkonv's example. Duhhamel's Principle
Oct. 5, 2022. Inhomogeneous heat equation. Mean-Value-Property. Regularity. Energy decreasing. backward uniqueness.
Oct. 7, 2022. Backward uniqueness for heat equation. d'Alembert's formula. Kirchhoff formula in dimension 3.
Oct. 12, 2022. Poisson formula in dimension two. Duhammel's principle.
Oct. 14, 2022. Finite speed of propagation. Weak derivatives.
Oct. 17, 2022. Weak derivatives. Sobolev space. Density theorems.
Oct. 19, 2022. Extension theorems
Oct. 21, 2022. Trace Theorem.
Oct. 24, 2022. Gagliarno-Nirenberg inequality
Oct. 26, 2022. Morrey's estimates. Embedding theorems for $W^{k,p}$.
Oct. 28, 2022. Three cases of loss of compactness. Compactness of Sobolev embedding.
Oct. 31, 2022: Poincare-Wirtinger inequality. Conjugate space of $H_0^1 (\Omega)$. Conjugate space of $ H^1 ( (a,b))$.
Nov. 2, 2022: Weak Solutions. Lax-Milgram Theorem.
Nov. 4. First existence theorem. Fredholm Alternatives
Nov. 7. Fredholm Alternatives. Third existence theorem.
Nov. 14. Properties of finite difference of quotients. $H^2$-interior regularity.
Nov. 16: $H^2$-interior regularity. Global $H^2$-regularity.
Nov. 18: Global $H^2$-regularity. Discussion of W^{2,p} and Schauder theory. Statement of Local Boundedness Theorem and Applications.
Nov. 21. Nash-Moser iteration. Local Boundedness.
Nov. 23. Nash-Moser iteration II.
Nov. 25. De Giorgi-Nash-Moser technique. Summary of De Giorgi-Nash-Moser Results. Maximum Principle.
Nov. 28. Weak and Strong Maximum Principle. Hopf Boundary Lemma.
Nov. 30. Narrow domain principle. Bernstein Techniques.
Dec. 2. Bernstein estimates for interior gradient estimates. Harnack inequality (Li-Yau).
Dec. 5. Weak solutions of parabolic equations. Galerkin Method
Dec. 7. Existence and uniqueness of weak solutions by Galerkin method. Regularity of weak solutions of parabolic equations.
Announcements For MATH 516-101
Back to My Home Page