96729 - Higher Analysis and Pdes

Course Unit Page

  • Teacher Alberto Parmeggiani

  • Credits 6

  • SSD MAT/05

  • Teaching Mode Traditional lectures

  • Language Italian

  • Campus of Bologna

  • Degree Programme Second cycle degree programme (LM) in Mathematics (cod. 5827)

Academic Year 2023/2024

Learning outcomes

At the end of the course, the student will have an introductory knowledge of the modern theory of Partial Differential Equations, and will be able to study fundamental solutions (in the context of distribution theory and/or function spaces), 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 interested in theoretical and applied mathematics. The theory that will be developed is the so-called "modern theory" which is also the basis for the further development in the "geometric analysis of PDEs".

The main topics will be:

  • The Frobenius Theorem for involutive systems of vector fields;
  • Fundamental solutions of ODEs and some classical PDEs; wave (3+1 dimensions), heat equation, Laplace equation, Cauchy-Riemann;
  • Cauchy problem for the wave operator (1+3 dimensions) and for the Schördinger equation;
  • Hypoellipticity and singular support of the fundamental solution; Parametrices of operatorswith constant coefficients whose symbol is a hypoelliptic polynomial;
  • Local solvability in L2 of PDEs with smooth coefficients;
  • Hoermamder's Inequality for partial differential operators with constant coefficients;
  • Periodic distributions and distributions on flat n-tori; summary of the calculus of differential k-forms; the Hodge Theorem on flat n-tori;
  • The Galerkin method for the weak solvability of partial differential equations on a torus. The Friedrichs Lemma and strong periodic solutions locally in L2 of Rn.
  • Time permitting: Sobolev spaces on flat tori and applications.


  1. L. Hörmander: Linear Partial Differential Operators, Springer (1969 Edition).
  2. C. Zuily: Eléments de distributions et d'équations aux dérivées partielles. Dunod.

Teaching methods

The general theory is completed by some problems as well as applications, mainly in the theoretical context.

Assessment methods

The final exam consists of a written and oral exam, to be taken withing the same session. In the written exam, which consists of a written text 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/good/excellent, and a score in thirtieths. In case of the rating "insufficient" the student will have to repeat the written exam, in case of rating at least "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