Sumários

Lecture 8: The Heat Equation, part II

19 Março 2019, 10:30 • Hugo Tavares

Proof of the uniqueness result for the initial/boundary value problem using energy methods.

The fundamental solution of the heat equation. Solution to the initial value problem in the whole space.

Lesson no. 7: The Laplace equation, part VI. The Heat equation, part I

15 Março 2019, 12:00 • Hugo Tavares

Poisson equation: The Dirichlet principle. Counterexample by Weierstrass on the existence of a minimizer.

The heat/diffusion equation: initial and boundary conditions. The parabolic cylinder and the space of C^{2,1}-functions. The weak maximum principle and its consequences: uniqueness, stability and comparison principle.

Solution to exercise 3 of Exercise Sheet 2.

Lesson no. 6: The Laplace Equation, part V

12 Março 2019, 14:00 • Hugo Tavares

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

Consequences: Harnack's inequality.

Lesson no. 5: The Laplace equation, part IV

8 Março 2019, 12:00 • Hugo Tavares

Comments about the uniqueness of solution to the Poisson equation in the whole space: asymptotic behavior at infinity.
Green's function and representation formula. Example: the case of the ball in dimension 3 or higher.

Lesson no. 4: The Laplace equation, part III

1 Março 2019, 12:00 • Hugo Tavares

Local estimates for the gradient, Liouville Theorem.

The fundamental solution of the Laplace Equation. Solution of the Poisson equation in the whole space.