- Docente: Eleonora Cinti
- Crediti formativi: 6
- SSD: MAT/05
- Lingua di insegnamento: Inglese
- Modalità didattica: Convenzionale - Lezioni in presenza
- Campus: Bologna
- Corso: Laurea Magistrale in Matematica (cod. 5827)
-
dal 20/02/2025 al 30/05/2025
Conoscenze e abilità da conseguire
At the end of the course, students will be able to study linear PDEs of first and second order, mainly by classical methods. This knowledge is fundamental to all the theoretical and modelling. applications.
Contenuti
Generalities on PDEs
- What is a PDE?
- Examples of PDEs
- Classification of second order PDEs
- Well-posed problems
First-order PDEs
- Linear equations with constant coefficients
- The Method of the Characteristics
- Existence and Uniqueness for the Cauchy Problem
Second order PDEs: Harmonic functions
- The Laplace operator. Harmonic functions
- Harmonic functions in open subsets of R2
- Some integral identities
- Radial harmonic functions in RN, N ≥ 2
- The fundamental solution of the Laplacian and a representation formula
- Mean Value Theorems for harmonic functions
- Surface and solid average operators for continuous functions
- Mean Value properties imply harmonicity
- Some convergence theorems
- The weak Laplacian
- The Harnack inequality
- Monotone sequences of harmonic functions
- Strong maximum principle and boundary estimates for harmonic functions
- Analyticity of the harmonic functions
- Liouville Theorems
- Maximum Principles for linear second order PDOs with nonnegative characteristic form
The Dirichlet problem for the Laplace operator: the Perron method
- Introduction
- Preliminaries: the Green function
- The Green functions for the Euclidean ball
- The Poisson kernel for the Euclidean ball
- The solution of the Dirichlet problem on the Euclidean balls
- Superharmonic functions
- The Perron-Wiener solution of the Dirichlet problem
- Boundary behavior of the Perron-Wiener solution
The Heat operator
- The Heat operator. Caloric functions
- Fundamental solution of H. Solvability of the Cauchy problem
- Green identity for H
- Some representation formulas in terms of H
- Smoothness of caloric functions and some convergence theorems
- Weak caloric functions
- Mean value Theorem for caloric functions
- Reverse of the Mean Value Property
- The caloric strong Maximum Principle
- The weak caloric Harnack inequality
- Monotone sequences of caloric functions
- The caloric Harnack inequality
- The parabolic weak maximum principle for the heat operator
- Uniqueness for the Cauchy problem
- Representation theorems on strips
- Liouville theorems for caloric functions
The wave operator
- The wave operator
- The Cauchy problem for the wave equation in R×]0, ∞[. D’Alembert formula.
- Some properties of the surface average. Darboux formula.
- The Cauchy problem for the wave equation n R3 ×]0, ∞[. Kirchhoff formula.
- Energy estimate and uniqueness for compactly supported data.
Testi/Bibliografia
Lawrence C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, Volume 19 American Mathematical Society.
Metodi didattici
The course consists of lessons describing the fundamental concepts of the program. Lessons are completed with examples illuminating the theoretical content.
Modalità di verifica e valutazione dell'apprendimento
Oral exam
Strumenti a supporto della didattica
Notes of the teacher on virtuale [http://virtuale.unibo.it/]
Orario di ricevimento
Consulta il sito web di Eleonora Cinti