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Topics in advanced analysis I

Course Type: 
PhD Course
Academic Year: 
October - January
48 h
  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

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