#### Objective

The objective of the course is to give students a solid bases in mathematical analysis techniques used in the study of linear and nonlinear partial differential equations

#### Content

Part I (upgrading, 20 h): This part is an upgrade of the students. The first chapter concerns a few reminders of functional analysis, including the weak and the weak* topology, the complex interpolation theory, Sobolev spaces $ W ^ {m, p} (Ω), $ injections and compactness theorems. The second chapter is devoted to study of linear elliptic problems: existence of weak solutions and regularity. In the last chapter we introduce the real Sobolev spaces $ H ^ s (R ^ N) $ and study their important properties.
Part II (14 h): In this section, we introduce the direct method of nonlinear functional minimization via the Euler-Lagrange equation and the problem of existence of minimizing sequences, the equation of Schrödinger will be taken for exemple. The next topic is the question of regularity of solutions of the equation of Euler - Lagrange. The Nash - Degiorgi theorem will be discussed very briefly without proof. Stability of critical points and their relationship with the spectral theory will also be briefly discussed. Examples of the nonlinear Schrödinger equation Ginzburg and Landau will be discussed. We also consider the minimization problem under different types of stresses (full, timely and differential). Finally we give the Mountain Pass theorem and it will be applied in the case of a semi-linear elliptic equation
Part III (14 h): It is divided into three chapters. The first one concern the integration theory in a Banach space: Bochner integral, L ^ p (0, T; X) spaces and vector distributions.
The second deals with unbounded operators: m-dissipative operators, extrapolations, self-adjoint and symmetric operators. After we introduce introducing the theory of semi-groups, and we proove the Hille-Yosida and Lumer-Phillips theorems.
The third chapter concerns the applications of the semi-group method to the study of some classical linear PDE.