Partial Differential Equations
A.Y. 2026/2027
Learning objectives
The course presents the basic concepts of the modern theory of Partial Differential Equations.
Expected learning outcomes
Acquisition of the basic notions and the techniques for solving partial differential equations. Study of the relations with the theory of function spaces, and of various fundamental properties such as maximum principle, weak solutions and regularity theory.
Lesson period: Second 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
Second semester
In case of sanitary or other emergencies, lectures and recitations will be organized in streaming onlone, as well as written and orale exams
Course syllabus
1. Introduction to PDE: basic notions, problems and challenges.
2. Representation formulas for solutions: transport equations, the equations of Laplace/Poisson, heat equation and wave equation. Method of characteristics, fundamental solutions, Green's functions, mean value formulas, maximum principles and applications. Energy methods and Dirichlet's principle as motivation for Sobolev spaces.
3. Sobolev spaces: weak derivatives and theor properties. Sobolev spaces: elementary properties. Sobolev spaces as function spaces (completeness, separability, reflexivity). Strong derivatives and approximation: theorems of local and global approximation with applications to the chain rule in Sobolev spaces. Boundary traces: motivation, the trace operator and the Poincaré inequality. Extension operators. Weak formulation of the Dirichlet problem by way of Dirichlet's principle (revisited). (rivisitato). Sobolev inequalities and immersions: thew inequalities of Gagliardo-Nirenberg-Sobolev, Morrey and associated immersions, compactness and the Rellich-Kondrachov theorem.
4. Linear elliptic equations of second order. Introduction: notions of ellipticity, equations in divergence form and not, Gårding's inequality and the Fredholm alternative. Characterization of Dirichlet eigenvalues. Regularity of weak solutions: difference quotients and weak derivatives, interior and boundary regularity Maximum principles: Hopf's lemma and weak and strong maximum principles for classical solutions and weak solutions, sub and super solutions. Eigenvalues and eigenfunctions for the Dirichlet problem, molteplicicty of real eigenvlaues, orthogoal bases, and the notion of principal eigenvalue. Variartional characterization.
5. Parabolic and hyperbolic equations of second order: Galerkin's method for the Cauchy-Dirichlet problem.
2. Representation formulas for solutions: transport equations, the equations of Laplace/Poisson, heat equation and wave equation. Method of characteristics, fundamental solutions, Green's functions, mean value formulas, maximum principles and applications. Energy methods and Dirichlet's principle as motivation for Sobolev spaces.
3. Sobolev spaces: weak derivatives and theor properties. Sobolev spaces: elementary properties. Sobolev spaces as function spaces (completeness, separability, reflexivity). Strong derivatives and approximation: theorems of local and global approximation with applications to the chain rule in Sobolev spaces. Boundary traces: motivation, the trace operator and the Poincaré inequality. Extension operators. Weak formulation of the Dirichlet problem by way of Dirichlet's principle (revisited). (rivisitato). Sobolev inequalities and immersions: thew inequalities of Gagliardo-Nirenberg-Sobolev, Morrey and associated immersions, compactness and the Rellich-Kondrachov theorem.
4. Linear elliptic equations of second order. Introduction: notions of ellipticity, equations in divergence form and not, Gårding's inequality and the Fredholm alternative. Characterization of Dirichlet eigenvalues. Regularity of weak solutions: difference quotients and weak derivatives, interior and boundary regularity Maximum principles: Hopf's lemma and weak and strong maximum principles for classical solutions and weak solutions, sub and super solutions. Eigenvalues and eigenfunctions for the Dirichlet problem, molteplicicty of real eigenvlaues, orthogoal bases, and the notion of principal eigenvalue. Variartional characterization.
5. Parabolic and hyperbolic equations of second order: Galerkin's method for the Cauchy-Dirichlet problem.
Prerequisites for admission
Elements of real and functional analysis as seen in the courses Mathematical Analysis 1 -5 and Real Analysis: differential and integral calculus in onbe and several variables, the fundamental theorems of integral calculus in severval variables, Lebesgue spaces and Hilbert spaces, spectral theorem for compact operators on Hilbert spaces, Lebesgue's differentiation theorem.
Teaching methods
Lectures and problem sessions (esercitazioni) will be presented in class and the partecipation is strongly recommended.
Teaching Resources
Evans, L.C. - Partial Differential Equations, Graduate Studies in Mathematics, Vol. 19, Amer. Math. Soc., Providence, RI, 1998. (Also available in second edition, 2010).
Gilbarg, D and Trudinger, N. - Elliptic Partial Differential Equations of the Second Order, Second Edition, Springer-Verlag, 1983. (Also available din paperback in the series "Classics in Mathematics", 2001).
Giusti, E. - Metodi Diretti nel Calcolo delle Variazioni - Unione Matematica Italiana, Bologna, 1994. (Adesso disponibile anche in inglese e pubblicato da World Scientific Press, 2003)
Payne, K.R. - Equazioni alle Derivate Paziali: Appunti del Corso, disponibile in rete al sito http://www.mat.unimi.it/users/payne/PDE_AA15_16.pdf
Gilbarg, D and Trudinger, N. - Elliptic Partial Differential Equations of the Second Order, Second Edition, Springer-Verlag, 1983. (Also available din paperback in the series "Classics in Mathematics", 2001).
Giusti, E. - Metodi Diretti nel Calcolo delle Variazioni - Unione Matematica Italiana, Bologna, 1994. (Adesso disponibile anche in inglese e pubblicato da World Scientific Press, 2003)
Payne, K.R. - Equazioni alle Derivate Paziali: Appunti del Corso, disponibile in rete al sito http://www.mat.unimi.it/users/payne/PDE_AA15_16.pdf
Assessment methods and Criteria
The final exam consists in problems to solve at home (homework) that takes the palce of a written exam and a comprehensive oral exam.
For the component of homework, exercizes to solve at home will be assigned in the form of open ended responses which should
d be detailed, yet synthetic, for the purpose of testing whether the basic mechanical aspects and applications of the theory have been digested.
Only students who have handed in their homeworks will be admitted to the final exam. During the oral exam, candidates will be asked to illustrate the most important results presented in the course as well as to solve some specific problems related to the main themes of the course, in order to evaluate the comprehension of the arguments treated in the course,, as well as the ability to apply them.
The duration of the oral exam depends on the speed and clarity of the candidate, but should last about 45 minutes.
For the component of homework, exercizes to solve at home will be assigned in the form of open ended responses which should
d be detailed, yet synthetic, for the purpose of testing whether the basic mechanical aspects and applications of the theory have been digested.
Only students who have handed in their homeworks will be admitted to the final exam. During the oral exam, candidates will be asked to illustrate the most important results presented in the course as well as to solve some specific problems related to the main themes of the course, in order to evaluate the comprehension of the arguments treated in the course,, as well as the ability to apply them.
The duration of the oral exam depends on the speed and clarity of the candidate, but should last about 45 minutes.
MATH-03/A - Mathematical Analysis - University credits: 9
Exercises: 24 hours
Lessons: 49 hours
Lessons: 49 hours
Professors:
Bucur Claudia Dalia, Payne Kevin Ray
Professor(s)