Non Linear Partial Differential Equations
A.Y. 2019/2020
Learning objectives
Deepen a modern study of the field of partial differential equations in a nonlinear context by way of techniques based on maximum principles for obtaining pointwise information.
Expected learning outcomes
Using techniques based on maximum principles, being able to treat questions of existence, uniqueness and qualitative properties for nonlinear partial differential equations of interest for geometric problems such as equations of prescribed curvature and minimal surfaces and for physical problems such as potential flows and optimal transport. Acquisition of the ability to read and present modern literature in the field.
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.Maximum principles for linear equations and applications to nonlinear equations.
The Hopf theory for uniformly elliptic equations: weak and strong maximum principles, the Hopf lemma, Serrin's comparison principle, the generalized maximum principle for narrow domains. A priori estimates for semilinear equations. The Alexandroff theory for elliptic equations: the Alexandroff estimate and the corresponding maximum principle, the comparison principle for domains with small volume. A priori estimates for quasilinear and fully nonlinear equations.
2. Qualitative properties of solutions: The methods of moving planes, moving spheres and the sliding method. Applications: symmetry, monotonia, nonexistence and classification of solutions for nonlinear equations. Comparison between the results and the techniques.
3. Fully nonlinear equations: The concepts of sub and super solutions for viscosity solutions to fully nonlinear equations. Degenerate elliptic and proper operators with examples. Ishii's Theorem on existence and uniqueness for the Dirichlet problem. The comparison principle and construction of barrier functions. Elements of convex analysis and multivalued functions. Time permitting, the recent theory of Harvey and Lawson will be treated.
The Hopf theory for uniformly elliptic equations: weak and strong maximum principles, the Hopf lemma, Serrin's comparison principle, the generalized maximum principle for narrow domains. A priori estimates for semilinear equations. The Alexandroff theory for elliptic equations: the Alexandroff estimate and the corresponding maximum principle, the comparison principle for domains with small volume. A priori estimates for quasilinear and fully nonlinear equations.
2. Qualitative properties of solutions: The methods of moving planes, moving spheres and the sliding method. Applications: symmetry, monotonia, nonexistence and classification of solutions for nonlinear equations. Comparison between the results and the techniques.
3. Fully nonlinear equations: The concepts of sub and super solutions for viscosity solutions to fully nonlinear equations. Degenerate elliptic and proper operators with examples. Ishii's Theorem on existence and uniqueness for the Dirichlet problem. The comparison principle and construction of barrier functions. Elements of convex analysis and multivalued functions. Time permitting, the recent theory of Harvey and Lawson will be treated.
Prerequisites for admission
Knowledge of partial differential equations including questions of well posedness, differential calculus in several variable and measure theory.
Teaching methods
In order to facilitate the learning process, classroom lectures will be offered to the students, who are strongly encouraged to participate.
Teaching Resources
In addition to numerous mathematical articles whose references will be given durng the course, the following textbooks are recommended:
L. Caffarelli e X. Cabrè - Fully Nonlinear Elliptic Equations, Colloquium Publications, Vol. 43, American Mathematical Society, Providence, 1995.
D. Gilbarg e N.S. Trudinger - Elliptic Partial Differential Equations of Second Order, Classics in Mathematics Series, Springer-Verlag, New York, 2001.
Q. Han e F. Lin - Elliptic Partial Differential Equations, Courant Lecture Notes Series Vol. 1, American Mathematical Society, Providence, 1997.
N. V. Krylov - Lectures on Fully Nonlinear Second Order Elliptic Equations, Rudolph-Lipschitz-Vorlesung, No. 29, Vorleungsreihe, Rheinische Friedrich-Wilhelms Universität
Bonn, 1994.
L. Caffarelli e X. Cabrè - Fully Nonlinear Elliptic Equations, Colloquium Publications, Vol. 43, American Mathematical Society, Providence, 1995.
D. Gilbarg e N.S. Trudinger - Elliptic Partial Differential Equations of Second Order, Classics in Mathematics Series, Springer-Verlag, New York, 2001.
Q. Han e F. Lin - Elliptic Partial Differential Equations, Courant Lecture Notes Series Vol. 1, American Mathematical Society, Providence, 1997.
N. V. Krylov - Lectures on Fully Nonlinear Second Order Elliptic Equations, Rudolph-Lipschitz-Vorlesung, No. 29, Vorleungsreihe, Rheinische Friedrich-Wilhelms Universität
Bonn, 1994.
Assessment methods and Criteria
The final examination consists in an oral exam. The student may choose to take a standard oral exam on the entire syllabus (presnting and illustrating some of the major results presented during the course) or to give a seminar on a topic suggested by the material covered in the course. In this second case, the final evaluation will include a judgement on the clarity and completeness of the presentation an on the abitity to harmoniously integrate the presentation in the context of the material presented in the course.
Final marks are given using the numerical range 18-30, and will be communicated immediately after the oral examination.
Final marks are given using the numerical range 18-30, and will be communicated immediately after the oral examination.
MAT/05 - MATHEMATICAL ANALYSIS - University credits: 6
Lessons: 42 hours
Professor:
Payne Kevin Ray
Shifts:
-
Professor:
Payne Kevin RayProfessor(s)