- Docente: Ermanno Lanconelli
- Credits: 6
- SSD: MAT/05
- Language: Italian
- Moduli: Ermanno Lanconelli (Modulo 1) Giovanni Cupini (Modulo 2)
- Teaching Mode: In-person learning (entirely or partially) (Modulo 1); In-person learning (entirely or partially) (Modulo 2)
- Campus: Bologna
- Corso: Second cycle degree programme (LM) in Mathematics (cod. 8208)
Learning outcomes
At the end of the course the student knows basic notions from
the theory od Elliptic, Parabolic and Hyperbolic Partial
Differentia Equations. He is able to study and
understand elementary
differential models in applied sciences. He is also able to
face high level topics from the general
theory of Partial Dirrential Equations.
Course contents
1) Harmonic functions: mean value properties, Harnack inequality,
Liouville-type Theorems, sequences of harmonic functions. Poisson
formula. Subharmonic functions and Perron method for
the Dirichlet problem. The weak Laplacian.
Dirichlet Principle.
2) Caloric functions: mean value property, Harnack inequality,
Liouville-type Theorems, sequences of caloric functions. A
Poisson-type formula for caloric functions. Subcaloric functions
and Perron method for the first boundary value problem. Cauchy
problem.
Caloric functions in the weak sense.
3) The wave equation. D'Alembert formula. The Energy Integral.
Spherical averages and solution to the Cauchy problem. Weak
solutions. Front wave.
4) Linear Partial Differential Equations with nonnegative
characteristic form: Picone's Maximum Principle.
Readings/Bibliography
Ater notes of the lectures.
Assessment methods
1) Homework: solution of some problems form a list given by the
lecturer.
2) Oral exams on some arguments, chosen by the student, from the
main
chapters of the course.
Office hours
See the website of Ermanno Lanconelli
See the website of Giovanni Cupini