MENU

You are here

Topics in advanced analysis I

Lecturer: 
Course Type: 
PhD Course
Academic Year: 
2020-2021
Period: 
October - January
Duration: 
48 h
Description: 
  1. First order equations:
    • the method of characteristics
    • the Hamilton-Jacobi equation:
      • non existence of smooth solutions
      • Hopf-Lax formula and comments about the correct notion of solution
    • Conservation laws
      • Non-existence of smooth solutions and non-uniqueness of distributional ones
      • Lax-Oleinik formula and comments comments about the correct notion of solution
  2. Elliptic equations:
    1. Laplace equation and harmonic functions:
      • Properties of harmonic functions: mean value property, regularity maximum principle, Harnack inequality
      • Existence: Perron’s method and the variational approach
    2. General elliptic equations:
      • Sobolev spaces
      • Weak formulation and uniqueness
      • Existence: Lax-Milgram theorem and Fredholm alternative
      • Regularity estimates in the interior and at the boundary
      • Maximum principle and Harnack inequality
  3. Parabolic equations:
    1. Weak formulation and uniqueness
    2. Existence: the Galerkin method
    3. Maximum principle and Harnack inequality
    4. Bits of semigroup theory and the theorem of Hille-Yosida
Location: 
A-005
Next Lectures: 
Thursday, October 29, 2020 - 16:00 to 18:00
Friday, October 30, 2020 - 14:00 to 16:00
Thursday, November 5, 2020 - 16:00 to 18:00
Friday, November 6, 2020 - 14:00 to 16:00
Thursday, November 12, 2020 - 16:00 to 18:00
Friday, November 13, 2020 - 14:00 to 16:00
Thursday, November 19, 2020 - 16:00 to 18:00
Friday, November 20, 2020 - 14:00 to 16:00
Thursday, November 26, 2020 - 16:00 to 18:00
Friday, November 27, 2020 - 14:00 to 16:00
Thursday, December 3, 2020 - 16:00 to 18:00
Friday, December 4, 2020 - 14:00 to 16:00
Thursday, December 10, 2020 - 16:00 to 18:00
Friday, December 11, 2020 - 14:00 to 16:00
Thursday, December 17, 2020 - 16:00 to 18:00
Friday, December 18, 2020 - 14:00 to 16:00
Thursday, January 7, 2021 - 16:00 to 18:00
Friday, January 8, 2021 - 14:00 to 18:00
Thursday, January 14, 2021 - 16:00 to 18:00
Friday, January 15, 2021 - 14:00 to 18:00

Sign in