Avanced Partial Differential Equations

A.Y. 2021/2022
Overall hours
Learning objectives
Getting acquainted with some regularity results for partial differential equations.
Expected learning outcomes
Basics methods which are used in the study of partial differential equations in order to obtain a priori estimates and which are at the base of the regularity theory.
Course syllabus and organization

Single session

Lesson period
Second semester
Videos with lessons and exercises collections will be made available at the official web space of this course and they will cover the topic of each week. Live meetings during the classes hours may be scheduled by using Zoom.us. The details of these meetings will be available in the course web page of the Ariel platform, as well as the videos and every kind of material needed for the course.
The exams will be done by following the ways suggested on the web page of the university. The exams will have the same structure as the ones done in presence.
Course syllabus
1. PDE examples
2. The basic model: harmonic functions.
3. Linear equations: Harnack's inequality, maximum principles and Schuder's estimates.
4. Nonlinear variational problems: regularity and De Giorgi-Nash-Moser theory.
5. Some recent results and open problems.
Prerequisites for admission
Basic facts of real analysis.
Sobolev and Holder spaces (if needed, these topics will be briefly presented during the course).
Teaching methods
Traditional lessons.
Teaching Resources
1. Q. Han and F.H. Lin, Elliptic Partial Differential Equations, Courant Lecture Notes in Math., v.1, 1997.
2. L. Ambrosio, A. Carlotto and A. Massaccesi, Lectures on Elliptic Partial Differen- tial Equations, Appunti. Sc. Norm. Super. Pisa (N. S.) 18, Edizioni della Normale, Pisa, 2019.
Assessment methods and Criteria
The exam consists of an oral examination. During the exam the student will have to give a seminar or discuss some results from the course's program as well as to solve some exercise in order to evaluate the knowledge of the student as well as his ability to apply them.
MAT/05 - MATHEMATICAL ANALYSIS - University credits: 6
Lessons: 42 hours
Professor: Ciraolo Giulio