EDP modelisation and mathematic analysis

Variational methods, also called energy methods, are a major tool in the study of partial differential equations (PDEs) for linear and nonlinear problems. They rely on estimates of the solutions in well chosen functional spaces and the use of powerful methods borrowed from the theory of functional analysis. The aim of this course is twofold :

• to study the tools in analysis underlying these methods
• to apply them to the study of stationary PDEs (elliptic problems) as well as unsteady problems (parabolic problems). Various problems borrowed from mathematical physics will be investigated. Perquisites : it is recommended to have some background on Functional Analysis (at the level of the S8 course on this topic)

Chapter 1 : Sobolev spaces

• Introduction to the theory of distributions
• Density and trace theorems Chapter 2 : Linear elliptic problems
• Variational methods
• Eigenvalue problems Chapter 3 : Nonlinear elliptic problems
• Weak topology
• Galerkin method Chapter 4 : Parabolic problems
• Vector valued functions
• Variational formulation of some model problem