Differential Analysis : 18.156 (Spring 2004)

Course Description: dvi , ps or pdf

Lectures

Textbooks

Examinations and Homework

• HW #1, due Thursday, February 26: Jost Chapter 1: 1, 4, 5, 7, 8, 9.
• HW #2, due Thursday, March 18: Gilbarg-Trudinger Chapter 4: 4.5, 4.9.

Brief Lecture notes

• Lecture 1: February 3
  
  1. Course overview.
  2. Examples of harmonic functions.
  3. Fundamental solutions for Laplacian and heat operator.
• Lecture 2: February 5
  
  1. Harmonic functions and mean value theorem.
  2. Maximum principle and uniqueness.
  3. Harnack inequality.
  4. Derivative estimates for harmonic functions.
  5. Green's representation formula.
• Lecture 3: February 10
  
  1. Definition of Green's function for general domains.
  2. Green's function for a ball.
  3. The Poisson kernel and Poisson integral.
  4. Solution of Dirichlet problem in balls for continuous boundary data.
  5. Continuous + Mean value property <-> harmonic.
• Lecture 4: February 12
  
  1. Weak solutions
  2. Further properties of Green's functions.
  3. Weyl's lemma: regularity of weakly harmonic functions.
• Lecture 5: February 19
  
  1. A removable singularity theorem.
  2. Laplacian in general coordinate systems.
  3. Asymptotic expansions.
• Lecture 6: February 24
  
  1. Kelvin transform I : direct computation.
  2. Harmonicity at infinity, and decay rates of harmonic functions.
  3. Kelvin II: Poission integral formula proof.
  4. Kelvin III: Conformal geometry proof.
• Lecture 7: February 26
  
  1. Weak maximum princple for linear elliptic operators.
  2. Uniqueness of solutions to Dirichlet problem.
  3. A priori C^0 estimates for solutions to Lu = f, c \leq 0.
  4. Strong maximum principle.
• Lecture 8: March 2
  
  1. Quasilinear equations (minimal surface equation).
  2. Fully nonlinear equations (Monge-Ampere equation).
  3. Comparison principle for nonlinear equations.
• Lecture 9: March 4
  
  1. If \Delta u \in L^{\infty}, then u \in C^{1,\alpha}, any 0 < \alpha < 1.
  2. If \Delta u \in L^{p}, p > n, then u \in C^{1,\alpha}, p = n/(1 - \alpha).
• Lecture 10: March 9
  
  1. If \Delta u \in C^{\alpha}, \alpha > 0, then u \in C^{2}.
  2. Moreover, if \alpha < 1, then u \in C^{2, \alpha} (proof to be completed next lecture).
• Lecture 11: March 16
  
  1. Interior C^{2, \alpha} estimate for Newtonian potential.
  2. Interior C^{2, \alpha} estimates for Poisson's equation.
  3. Boundary estimate on Newtonian potential: C^{2,\alpha} estimate up to the boundary for domain with flat boundary portion.
• Lecture 12: March 18
  
  1. Schwartz reflection reviewed.
  2. Green's function for upper half space reviewed.
  3. C^{2,\alpha} boundary estimate for Poisson's equation for flat boundary portion.
  4. Global C^{2, \alpha} estimate for Poisson's equation in a ball for zero boundary data.
  5. C^{2,\alpha} regularity of Dirichlet problem in a ball for C^{2, \alpha} boundary data.
• Lecture 13: March 30
  
  1. Global C^{2,\alpha} solution of Poisson's equation \Delta u = f \in C^{\alpha}, for C^{2,\alpha} boundary values in balls.
  2. Constant coefficient operators.
  3. Interpolation between Holder norms.
• Lecture 14: April 1
  
  1. Interior Schauder Estimate.
• Lecture 15: April 6
  
  1. Global Schauder estimate.
  2. Banach Spaces and Contraction Mapping Principle
• Lecture 16: April 8
  
  1. Continuity method.
  2. Can solve Dirichlet problem for general L provided can solve for Laplacian.
  3. Corollary: Solution of C^{2\alpha} Dirichlet problem in balls for general L.
  4. Solution of Dirichlet problem in C^{2\alpha} for continuous boundary values, in balls.
• Lecture 17: April 13
  
  1. Elliptic regularity: if f and coefficients of L \in C^{k,\alpha}, Lu = f, then u \in C^{k+2, \alpha}.
  2. C^{2,\alpha} regularity up to the boundary.
• Lecture 18: April 15
  
  1. C^{k,\alpha} regularity up to the boundary.
  2. Hilbert Spaces and Riesz Representation Theorem.
  3. Weak solution of Dirichlet problem for Laplacian in W^{1,2}_0
  4. Weak derivatives.
  5. Sobolev spaces.
• Lecture 19: April 22
  
  1. Sobolev imbedding theorem p < n.
  2. Morrey's inequality.
• Lecture 20: April 27
  
  1. Sobolov imbedding for p > n, Holder continuity.
  2. Kondrachov compactness theorem.
  3. Characterization of W^{1,p} in terms of difference quotients.
• Lecture 21: April 29
  
  1. Characterization of W^{1,p} in terms of difference quotients.
  2. Interior W^{2,2} estimates for W^{1,2}_0 solutions of Lu =f \in L^2.
• Lecture 22: May 4
  
  1. Interior W^{k+2,2} estimates for solutions of Lu =f \in W^{k,2}.
  2. Global (up to the boundary) W^{k+2,2} estimates for solutions of Lu =f \in W^{k,2}
• Lecture 23: May 6
  
  1. Weak L^2 maximum principle.
  2. Global a priori W^{k+2,2} estimate for L u = f, f \in W^{k,2}, c(x) \leq 0
• Lecture 24: May 11
  
  1. Cube decomposition.
  2. Marcinkiewicz interpolation theorem.
  3. L^p estimate for the Newtonian potential.
  4. W^{1,p} estimate for N.P.
  5. W^{2,2} estimate for N.P.
• Lecture 25: May 13
  
  1. W^{2,p} estimate for N.P., 1 < p < \infty.
  2. W^{2,p} estimate for operators L with continuous leading order coefficients.