MATH-305 / 5 crédits
Enseignant: Nobile Fabio
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;
Measure and Integration; Functional Analysis I
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
- Use a work methodology appropriate to the task.
- Demonstrate a capacity for creativity.
- Demonstrate the capacity for critical thinking
Ex cathedra lectures, exercises in classroom
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
- Partial Differential Equations / Fritz John
- Functional Analysis, Sobolev Spaces and Partial Differential Equation / Haïm Brézis
- Partial Differential Equations / Lawrence C. Evans
- Elliptic Partial Differential Equations of Second Order / David Gilbarg & Niel S. Trudinger
Lecture notes available on the webpage
- 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