MATH-305 / 5 crédits

Enseignant: Nobile Fabio

Langue: Anglais


This is an introductory course on Elliptic Partial Differential Equations. The course will cover the theory of both classical and generalized (weak) solutions of elliptic PDEs.


  • Laplace equation; mean value property; maximum principle; fundamental solution; Dirichlet problem; Poisson integral and Newtonian potential; regularity theory in Holder spaces;
  • General second order linear elliptic equations; maximum principle; a priori bounds;
  • Sobolev spaces; weak derivatives and their properties; density results; extension results; traces; imbedding theorems; Poincaré inequalities;
  • Weak solutions of general elliptic equations;  Lax Milgram theorem; existence and uniqueness results; regularity theory in Sobolev spaces; compactness results and non coercive problems;

Learning Prerequisites

Required courses

Analysis I-IV

Recommended courses

Measure and Integration; Functional Analysis I

Learning Outcomes

By the end of the course, the student must be able to:

  • Classify different types of PDEs
  • Define different notions of solutions
  • Analyze the properties of solutions of PDEs
  • Prove existence and regularity results of solutions of elliptic PDEs

Transversal skills

  • Use a work methodology appropriate to the task.
  • Demonstrate a capacity for creativity.
  • Demonstrate the capacity for critical thinking

Teaching methods

Ex cathedra lectures, exercises in classroom

Assessment methods

Oral exam


Dans le cas de l'art. 3 al. 5 du Règlement de section, l'enseignant décide de la forme de l'examen qu'il communique aux étudiants concernés


Virtual desktop infrastructure (VDI)



  • David Gilbarg, Niel S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, 2nd edition, 2001.
  • Lawrence C. Evans. Partial Differential Equations, AMS-Graduate Studies in Mathematics, 2nd edition, 2010.
  • Haïm Brézis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, 2011
  • Fritz John, Partial Differential Equations, Springer-Verlag, 4th edition, 1982

Ressources en bibliothèque


Lecture notes available on the webpage

Moodle Link

Prerequisite for

  • Master courses on theory of PDEs: Equations aux dérivées partielles d'évolution, Calculus of variations, Optimal Transport, Dispersive PDEs, Theory of stochastic calculus, Nonlinear Schrödinger equations, Distributions and interpolation spaces; Introduction to general relativity; Introduction to stochastic PDEs
  • Bachelor / Master courses on numerical approximation of PDEs: Numerical Approximation of PDEs; Numerical methods for conservation laws; Computational finance; Numerical integration of stochastic differential equations; Numerics for fluids, structures & electromagnetics;

Dans les plans d'études

  • Semestre: Automne
  • Forme de l'examen: Oral (session d'hiver)
  • Matière examinée: Introduction to partial differential equations
  • Cours: 2 Heure(s) hebdo x 14 semaines
  • Exercices: 2 Heure(s) hebdo x 14 semaines
  • Type: optionnel

Semaine de référence

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