MATH-305 / 5 credits

Teacher: De Nitti Nicola

Language: 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;

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

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

## Notes/Handbook

Lecture notes available on the webpage

## 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;

## In the programs

• Semester: Fall
• Exam form: Oral (winter session)
• Subject examined: Introduction to partial differential equations
• Lecture: 2 Hour(s) per week x 14 weeks
• Exercises: 2 Hour(s) per week x 14 weeks
• Type: optional

## Reference week

Tuesday, 8h - 10h: Lecture MAB111

Tuesday, 10h - 12h: Exercise, TP MAB111

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