Sumários

The Laplace equation (II): harmonic functions, regularity and local estimates.

27 Setembro 2019, 11:00 • Manuel Duque Pereira Monteiro Marques

Uniqueness for the Poisson equation, by an energy method. 

 Regularity and harmonicity of functions with the mean value property. 
 Local estimates for the gradient and Liouville theorem.

Convolution, test functions and mollifiers

26 Setembro 2019, 11:00 • Manuel Duque Pereira Monteiro Marques

A survey on properties of test functions and approximation of functions by convolution with mollifiers.

Comments on the exercises of the first series.

The Laplace equation (I)

24 Setembro 2019, 10:00 • Manuel Duque Pereira Monteiro Marques

Harmonic functions of one and two real variables; link with holomorphic functions.

Mean value formulas. Weak and strong maximum principles and some of their consequences: 
uniqueness, stability and comparison results for boundary value problems.

The Transport equation

20 Setembro 2019, 11:00 • Manuel Duque Pereira Monteiro Marques

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


The Transport equation. Explicit solutions for the homogeneous and nonhomogeneous cases. A weak formulation using test functions.

Preliminaries.

19 Setembro 2019, 11:00 • Manuel Duque Pereira Monteiro Marques

Preliminaries: Lebesgue spaces, convolution, divergence theorem.