Fiche de cours

Introduction to partial differential equations

MATH-305

Fiche de cours

Enseignant(s) :

Nobile Fabio

Langue:

English

Summary

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.

Content

• 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

Resources

Virtual desktop infrastructure (VDI)

No

Bibliography

• 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

Notes/Handbook

Lecture notes available on the webpage

Moodle Link

• https://go.epfl.ch/MATH-305

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;