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
(In case of multiple editions, please check the period, as it may vary)
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
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.