Mathematics

LUCA MASSIMO ANDREA MARTINAZZI
LUCA MASSIMO ANDREA MARTINAZZI

Knowledge and understanding:The course gives to successful students some advanced tools for the study of various linear and nonlinear PDE's. They will reach a good familiarity with the most recent notions of solutions and their qualitative properties.Skills and attributes:Successful students will able to deal with the advanced study of the solutions to various types of linear and nonlinear PDE's.

Understanding of several methods for partial differential equations.

Mathematical analysis, functional analysis, real analysis.

First order equations. Transport equation and continuity equation.

Conservation laws and Burgers equation.

Constant coefficients second order linear equations (Laplace, heat and wave equations)-

Second order elliptic equations in divergence form (nonconstant coefficients).

L^2 regularity for elliptic equations in divergence form.

Second order elliptic equations in non-divergence form.

Holder regularity.

De Giorgi's theorem and the solution of XIX Hilbert's problem.

Lecture notes di Clement Mouhot (per il teorema di Cauchy-Kovalevskaya, l'equazioni del trasporto e di Burgers)

M. Giaquinta, L. Martinazzi: An introduction to the regularity theory for elliptic systems, harmonic maps and minimal graphs, second edition, Edizioni della Normale (2012)

F. Johns, Partial Differential Equations, Springer

Classroom lectures and homeworks

Warmly encouraged

Oral exam

Statements and proofs of various theorems.