Sumários

Lesson 4: The Laplace equation, part III.

2 Março 2018, 11:00 • Hugo Tavares

Regularity Theorem (2nd part).

Local estimates for the gradient, Liouville Theorem.
The fundamental solution of the Laplace Equation. Solution of the Poisson equation in the whole space.

Exercise class 2

1 Março 2018, 11:00 • Hugo Tavares

Review of know facts. discussion about exercises 1c), 2, 3, 4ii)iii)iv) of Exercise Sheet 1.
The transport equation: solution of exercises 1 of Exercise Sheet 2.

Lesson 3: The Laplace equation, part II.

27 Fevereiro 2018, 14:00 • Hugo Tavares

The weak and Strong maximum principles, and some of its consequences: uniqueness, stability and comparison results.

Energy methods: uniqueness of solution.
Regularity theorem for harmonic function.

Lesson 2. The transport Equation. The Laplace equation, part I.

23 Fevereiro 2018, 11:00 • Hugo Tavares

Equation for a general conservation law. Deduction of the Transport and Diffusion/Heat equations.

The transport equation in the homogeneous and nonhomogeneous case: explicit solutions. Definition of weak solution using the test functino space.
The Laplace equation. Mean Value Theorem

Exercise class 1

22 Fevereiro 2018, 11:00 • Hugo Tavares

Survey about known facts: functions with compact support, the space of test functions, convolution and mollifiers (Young's inequality, regularization, approximation), the divergence theorem and some of its consequences.

Discussion about exercises 1a)b) and 4(i)(ii) of Exercise Sheet 1.