Partial Differential Equations

A.Y. 2020/2021
Overall hours
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.
Course syllabus and organization

Single session

Lesson period
Second semester
Course syllabus
Laplace equation: fundamental solution, Green's function, representation formulas for solution on particular domains.
Sobolev spaces: weak derivatives, Sobolev inequalities and continuous embeddings into Lebesgue spaces L^p.
Compactnes: the theorem of compact embeddings of Rellich-Kondrachov.
Elliptic equations of second order: existence and uniqueness of weak solutions esistenza for the Dirichlet problem, regularity of weak solutions, characterization of the eigenvalues of the Laplacian, maximum principle.
Heat equation: the fundamental solution, representation formula for the Cauchy problem, the Duhamel principle.
parabolic equations: definition of weak solutions, Galerkin approximation, existence and uniqueness of weak solutions.
Wave equation: the transport equation, representation formula.
Hyperbolic equations: definition, existence and uniqueness of weak solutions.
Prerequisites for admission
Real Analysis: L^p-spaces of Lebesgue, dual space, Banach and Hilbert spaces, weak convergence
recommended: Functional analysis
Teaching methods
Lectures in traditional mode, at the blackboard.
Teaching Resources
Evans, L.C. - Partial Differential Equations, Graduate Studies in Mathematics, Vol. 19, Amer. Math. Soc., Providence, RI, 1998, second edition, 2010
Assessment methods and Criteria
The exam consists of a single oral exam (30 minutes) which serves to verify the theoretical knowledge acquired during the course and the ability to solve exercises similar to the one which were proposed during the course.
MAT/05 - MATHEMATICAL ANALYSIS - University credits: 6
Lessons: 42 hours
Professor: Ruf Bernhard