Sumários

Exercise class 4

15 Março 2018, 11:00 • Hugo Tavares

Laplace and Poisson's equation.

Solution of exercises 1b)c), 2, 3, 4, 5, 6 of Exercise Sheet 3.

Lesson 7: The Laplace equation, part VI. The Heat Equation, part I.

13 Março 2018, 14:00 • Hugo Tavares

The Dirichlet Principle for the Poisson Problem. The drawbacks of working in the space of C^2 functions.The heat/diffusion equation: initial and boundary conditions. The parabolic cylinder, and the space of C^{2,1} functions. The weak maximum principle and consequences: uniqueness, stability and comparison.

Lesson 6: The Laplace Equation, part V

9 Março 2018, 11:00 • Hugo Tavares

Green's function and Poisson's formula for the ball.

Consequences: Harnack's inequality.

Exercise class 3

8 Março 2018, 11:00 • Hugo Tavares

The transport equation: solution of exercises 2a)b)c).
Laplace and Poisson's equation: solution of exercises 3 and 4 of Exercise Sheet 2.

Lesson 5: The Laplace equation, part IV.

6 Março 2018, 14:00 • Hugo Tavares

Comments about the uniqueness of solution to the Poisson equation in the whole space.

Green's function and representation formula. Example: the case of the ball in dimension 3 or higher.