# 34676 - Advanced Analysis 2

## Learning outcomes

At the end of the course, the student will have an introductory knowledge of the theory of Partial Differential Equations, and will be able to study fundamental solutions (in the context of distribution theory), parametrices and a priori estimate techniques, for studying solvability of the equation and qualitative properties of the solutions.

## Course contents

The course is an introduction to the theory of Partial Differential Equations (PDEs) for students majoring in both pure and applied mathematics.

The main topics will be:

• Fundamental solutions of some classical PDEs; wave, heat equation, Laplace equation, Cauchy-Riemann and Schördinger equation;
• Paramterices of elliptic operators;
• A priori estimate techniques for the solvability of PDEs;
• The method of characteristics for the solvability of first order PDEs;
• Rudiments on the Cauchy problem (time permitting).

1. L. Hörmander: Linear Partial Differential Operators, Springer (1969 Edition).
2. F. Treves: Basic Linear Differential Equations, Dover.
3. J. Chazarain - A. Piriou: Introduction to the Theory of Linear Partial Differential Equations, North Holland.
4. C. Zuily: Eléments de distributions et d'équations aux dérivées partielles. Dunod.

## Teaching methods

The general theory is completed by a number of problems as well as applications. This should provide the students with an applied math major of a sufficient background in their discipline.

## Assessment methods

1. Homeworks.
2. Problems solved in class by the teacher as well as by the studen
3. The final exam consists of a written and oral exam, to be taken withing the same session. In the written exam, related to the arguments developed during the course (2 hours; no notes or electronic devices are allowed) the student will receive an evaluation: insufficient/sufficient. In case of the rating "insufficient" the student will have to repeat the written exam, in case of the rating "sufficient", the student will be able to proceed to the oral exam. The latter always starts from the exposition of some (relevant) topic chosen by the student. A sufficient written exam will be held valid within the session.

## Office hours

See the website of Alberto Parmeggiani