Non Linear Partial Differential Equations
A.Y. 2024/2025
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 can be attended as a single course.
Course syllabus and organization
Single session
Responsible
Lesson period
First semester
The lectures and exercise sessions will be held in person (in the assigned) classroom. If, for reasons of the continuing health emergency it becomes necessary to offer (all or in part) of the lectures and exercise sessions online, they will be offered live on the Zoom platform.
Course syllabus
1. Maximum principles for linear elliptic 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. Differentiability of convex functions:
· Subdifferentials and the first order theory.
· Alexandroff's theoem on the second order differentiability in the Peano sense.
· Second order subdifferentials and superdifferentials, semicontinuous functions and semiconvex functions.
· The Jensen e Slodkowski lemmas.
3. Visocsity solutions of fully nonlinear elliptic equations:
· The notions of viscosity subsolutions and supersolutions for fully nonlinear second order equations.
· Existence of viscosity solutions of the Dirichlet problem via Perron's method (upper envelope of viscosity subsolutions) and Ishii's theorem.
· Validity of the comparison principle by way of the analysis of local extema of semicontinuous functions.
· Construction of suitable upper and lower solutions.
Time permitting, some or all of the following arguments will be treated:
· The Harvey-Lawson nonlinear potential theory approach and the notion of Krylov of viscosity solutions with admissibility constraints.
· Hölder regularity of viscosity solutions.
· The notion of generalized principle eigenvalue for homogeneous nonlinear operators.
· 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. Differentiability of convex functions:
· Subdifferentials and the first order theory.
· Alexandroff's theoem on the second order differentiability in the Peano sense.
· Second order subdifferentials and superdifferentials, semicontinuous functions and semiconvex functions.
· The Jensen e Slodkowski lemmas.
3. Visocsity solutions of fully nonlinear elliptic equations:
· The notions of viscosity subsolutions and supersolutions for fully nonlinear second order equations.
· Existence of viscosity solutions of the Dirichlet problem via Perron's method (upper envelope of viscosity subsolutions) and Ishii's theorem.
· Validity of the comparison principle by way of the analysis of local extema of semicontinuous functions.
· Construction of suitable upper and lower solutions.
Time permitting, some or all of the following arguments will be treated:
· The Harvey-Lawson nonlinear potential theory approach and the notion of Krylov of viscosity solutions with admissibility constraints.
· Hölder regularity of viscosity solutions.
· The notion of generalized principle eigenvalue for homogeneous nonlinear operators.
Prerequisites for admission
Real Analysis and Partial Differential Equations.
Teaching methods
Traditional blackboard lectures. Attendance strongly suggested.
Teaching Resources
In addition to numerous articles (which will be cited during the course), the following monographs are useful:
-- 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 of an oral exam to be conducted in one of the two following mechanisms (chosen by the candidate):
Traditional oral examination on the content of the syllabus. During the oral exam, the candidate will be asked to present various results covered in the syllabus. Candidates must demonstrate mastery of the details of the proofs, capacity to place the results in their proper context and illustrate their intrinsic importance as well as relevant applications.
Oral examination in the form of a seminar on an individual topic related to the syllabus and agreed upon with each candidate and agreed upon in time to allow for the preparation of the seminar. During the presentation, candidates must demonstrate mastery of the topic and ability to place the topic in its proper context within the syllabus. In addition, candidates must respond adequately to questions concerning clarifications and interpretations of the results presented.
The final examination is passed if the oral exam is successfully passed. Final marks are given using the numerical range 0-30, and will be communicated immediately after the oral examination.
Traditional oral examination on the content of the syllabus. During the oral exam, the candidate will be asked to present various results covered in the syllabus. Candidates must demonstrate mastery of the details of the proofs, capacity to place the results in their proper context and illustrate their intrinsic importance as well as relevant applications.
Oral examination in the form of a seminar on an individual topic related to the syllabus and agreed upon with each candidate and agreed upon in time to allow for the preparation of the seminar. During the presentation, candidates must demonstrate mastery of the topic and ability to place the topic in its proper context within the syllabus. In addition, candidates must respond adequately to questions concerning clarifications and interpretations of the results presented.
The final examination is passed if the oral exam is successfully passed. Final marks are given using the numerical range 0-30, and will be communicated immediately after the oral examination.
MAT/05 - MATHEMATICAL ANALYSIS - University credits: 6
Lessons: 42 hours
Professor:
Payne Kevin Ray
Shifts:
Turno
Professor:
Payne Kevin RayEducational website(s)
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