Lecturer:

Course Type:

PhD Course

Academic Year:

2020-2021

Period:

October - January

Duration:

48 h

Description:

- 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

- Elliptic equations:
- 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

- 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

- Laplace equation and harmonic functions:
- Parabolic equations:
- Weak formulation and uniqueness
- Existence: the Galerkin method
- Maximum principle and Harnack inequality
- Bits of semigroup theory and the theorem of Hille-Yosida

Research Group:

Location:

A-134