Variational methods for PDEs

Lecturer(s): Martine MARION
Course ⋅ 16 hStudy ⋅ 12 h


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)


Partial differential equations, weak solutions, linear and non linear problems, variational methods


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

Learning Outcomes

  • To learn the analysis tools at the basis of the study of PDEs To be able to apply them to actual problems


Final mark = 70% Knowledge + 30% Know-how Knowledge N1 = 100% final exam Know-how N2 = 100% continuous assessment