
INTRODUCTION TO PARTIAL DIFFERENTIAL EQUATIONS
Second cycle degree in MATHEMATICS
Campus:
PADOVA
Language:
English
Teaching period:
First Semester
Lecturer:
FABIO ANCONA
Number of ECTS credits allocated:
8
Prerequisites:

Differential and integral calculus.
Elementary theory of ordinary differential equations.
Basic theory of complex analysis (functions of complex variables, holomorphic and analytic functions). 
Examination methods:

The exam consists of a final oral examination on the topics treated in class. There will be both theoretical questions and the discussion of some exercise to solve. 
Course unit contents:

Didactic plan:
 First order PDEs: transport equation with constant coefficients, conservation laws (classical and weak solutions, RankineHugoniot conditions, Riemann problem).
 Wave equation: existence of solutions, D'alembert formnula, method of spherical means, Duhamel's principle, uniqueness, finite speed of propagation.
 Laplace equation: fundamental solution, harmonic functions and main properties, mean value formulas, Harnack's inequality, maximum principle. Poisson equation. Green's function and Poisson's representation formula of solutions.
 Heat equation: fundamental solution, existence of solutions for the Cauchy problem and representation formula. Uniqueness and stability of solutions. Mean value formulas, maximum principle, Hopf's maximum principle. 

