Miguel Rodrigues
The first part of the course is devoted to Sobolev spaces as covered by
- chapters 8 and 9 of Haïm Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations;
- chapter 5 of Craig Evans, Partial Differential Equations.
The 5 minutes Lebesgue videoclip (unfortunately in French language) entitled Comment mesurer la taille d'une fonction ? also provides some insights on the latter. The definitive statements about Sobolev embeddings may be found in the research paper by Haïm Brezis and Petru Mironescu, Where Sobolev interacts with Gagliardo-Nirenberg, J. Funct. Anal., Vol. 277 (2019), no. 8, p.2839-2864.
The second part of the course is focused on (mostly linear) elliptic equations as covered by
- chapter 6 of Craig Evans, Partial Differential Equations;
- David Gilbarg and Neil Trudinger, Elliptic Partial Differential Equations of Second Order;
- Nicolai Krylov, Lectures on elliptic and parabolic equations in Hölder spaces and
Lectures on elliptic and parabolic equations in Sobolev spaces.
Prerequisites
Prerequisites include measure theory and integration, differential calculus and ordinary differential equations, functional analysis and elementary ditribution theory. A good way to get an overall grasp on those is to study
- appendices of Craig Evans, Partial Differential Equations.
- chapter 10 of Thomas Alazard and Claude Zuily, Tools and problems in partial differential equations.
To cover the material in details one may combine the following books
- Sylvie Benzoni-Gavage, Calcul différentiel et équations différentielles (in French);
- first twelve chapters of Walter Rudin, Real and Complex Analysis;
- first five chapters of Haïm Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations;
- Claude Zuily, Distributions et équations aux dérivées partielles (in French).
Tests
The course includes two tests: one homework (CC1), one final in-class evaluation (CC2).
The final course grade is then obtained through max((CC1+CC2)/2,CC2).