The course is composed of fours chapters:

- Harmonic functions and Laplace equation;

- Second order Elliptic equations: existence, properties of weak solutions;

- Variational and nonvariational methods for elliptic PDE;

- Linear heat equations: existence of weak solutions and regularity.

More precisely, in chapter 1 we study the harmonic functions in order to obtain qualitative properties (as Maximum principle; regularity or uniqueness) of solutions of Laplace equation. These properties are the goals of chapter 2 considering a more general class of elliptic PDEs and the weak solutions of these equations and discuss properties (regularity, maximum principles and uniqueness). For the last chapters, we are going to give methods to solve elliptic and parabolic PDEs. In the case of elliptic PDE, for the variational method we study the minimization method (without or with constraints) and concerning the nonvariational method, we consider the method of sub- and supersolution and Fixed point methods. Finally, we study the Heat equation treating the questions of existence, energy estimates by discretization methods and we discuss the regularity of weak solutions.

- Teacher: Jacques Giacomoni
- Teacher: Guillaume Warnault