Math 820 Partial Differential Equations (Spring 2010)

Lectures

Recommended Textbooks

1. Elliptic Partial Differential Equations of Second Order by Gilbarg-Trudinger.

Mailing List

Homework

1. G-T Chapter 2: 2-9, 14, 15, 18.
2. Prove that P_k = H_k \oplus |x|^2 P_{k-2}.
3. G-T Chapter 4: 9 (a) and (b).

Lectures

• Lecture 1: Tuesday, January 19
  
  1. Introduction and outline.
• Lecture 2: Thursday, January 21
  
  1. Mean value theorem for harmonic functions.
  2. Maximum principle and uniqueness.
  3. Harnack inequality.
  4. Higher derivative estimates.
  5. Green's representation formula.
• Lecture 3: Tuesday, January 26
  
  1. Green's function of a domain.
  2. Explicit Green's function for a ball.
  3. Poisson integral formula.
  4. Continuity up to the boundary.
  5. Harmonic iff continuous and spherical MVP in balls.
  6. Limit of uniformly convergent sequence of harmonic functions is harmonic.
• Lecture 4: Thursday, January 28
  
  1. Weak solutions.
  2. Harmonic iff measurable and solid MVP in balls.
  3. Weyl's lemma: weakly harmonic implies harmonic.
• Lecture 5: Tuesday, February 2
  
  1. Harmonic polynomials.
  2. Laplacian in spherical coordinates.
  3. Separation of variables.
  4. Eigenfunctions on S^n.
  5. L^2(S^n) spanned by eigenfunctions.
• Lecture 6: Thursday, February 4
  
  1. Removable singularity theorem (if blows-up less than fundamental solution).
  2. Laplacian in inverted coordinates.
  3. Direct proof of Kelvin transform.
  4. Proof of Kelvin using conformal geometry.
• Lecture 7: Tuesday, February 9
  
  1. Maximum principle for general elliptic operators.
  2. Hopf boundary point lemma.
  3. Strong maximum principle.
• Lecture 8: Thursday, February 11
  
  1. Applications of maximum principle to nonlinear equations.
• Lecture 9: Tuesday, February 16
  
  1. C^{1,\alpha} estimates for Newtonian Potential for f in L^p, p = n / (1 - \alpha).
• Lecture 10: Thursday, February 18
  
  1. C^2 estimates for Newtonian Potential for f Dini continuous.
• Lecture 11: Tuesday, February 23
  
  1. C^{2,\alpha} estimates for Newtonian Potential, f \in C^{\alpha}.
• Lecture 12: Thursday, February 25
  
  1. Boundary C^{2,\alpha} estimates in a ball for \Delta u = f in C^{\alpha} globally.
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