Advanced Partial Differential Equations
A.Y. 2024/2025
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.
Lesson period: First semester
Assessment methods: Esame
Assessment result: voto verbalizzato in trentesimi
Single course
This course cannot be attended as a single course. Please check our list of single courses to find the ones available for enrolment.
Course syllabus and organization
Single session
Responsible
Lesson period
First semester
Course syllabus
1. PDE examples
2. The basic model: harmonic functions.
3. Linear equations: Harnack's inequality, maximum principles and Schuder's estimates.
4. Geometrical properties of solutions to elliptic PDEs via the maximum principle.
5. Nonlinear variational problems: regularity and De Giorgi-Nash-Moser theory.
6. Some recent results and open problems.
2. The basic model: harmonic functions.
3. Linear equations: Harnack's inequality, maximum principles and Schuder's estimates.
4. Geometrical properties of solutions to elliptic PDEs via the maximum principle.
5. Nonlinear variational problems: regularity and De Giorgi-Nash-Moser theory.
6. 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).
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.
3. Xavier Fernandez-Real and Xavier Ros-Oton, Regularity Theory for Elliptic PDE, available online.
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.
3. Xavier Fernandez-Real and Xavier Ros-Oton, Regularity Theory for Elliptic PDE, available online.
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
Shifts:
Turno
Professor:
Ciraolo GiulioProfessor(s)