This semester, I will be teaching a course on Partial Differential Equations (PDE), based on L. C. Evans’ “Partial Differential Equations”. The course is intended for third-year undergraduate and master’s students.

The course is structured around the study of the three most important differential equations: the Laplace, heat and wave equations. These three equations have been the source of numerous mathematical developments, both pure and applied. We will commence by learning the classical methods for deriving properties of their solutions. Subsequently, we will delve into a more advanced approach, studying distributions, Sobolev spaces and weak solutions. These theoretical instruments will enable us to introduce new methods for investigating PDE.

Weekly schedule: Monday and Tuesday, 13:15 - 15:00, in PER 23, Room 0.05

Codes: SMA.03554 and SMA.04554

For more information, please write me at the address sebastiano.nicolussigolo@unifr.ch

Material

I give you access to my notes. It is a git repository: you can clone or fork the repository, or you can download the source files. Danger: These are NOT lecture notes: these are my personal notes, on which I work to prepare the lectures, and they will keep changing quite a lot. You should have a similar document for yourself. It is not necessary to use LaTeX, or a computer: paper and pen can work equally well. However, it is necessary that you write all proofs yourself: add and subtract details, polish your text until every single step is justified. Dig into your uncertainities!

Diary

Week 1

Monday 17/02

  • Intro to the course.
  • Intermezzo of calculus: Dominated convergence theorem; Integration with parameter.
  • The homogeneous and the nonhomogeneous transport equation
  • Intermezzo of calculus: Characterization of open sets with \(C^k\) boundary.

Tuesday 18/02

  • Definition of (outer) unit normal
  • Surface measure on submanifolds of \(\mathbb R^n\)
  • Coarea formula and integration in polar coordinates
  • LaHa: Introduction to Laplacian, Laplace equation and Poisson equation
  • LaHa: Definition of Harmonic function
  • LaHa: Examples of harmonic functions: affine functions, harmonic polynomials, real and imaginary parts of holomorphic functions
  • LaHa: Symmetries of Laplacian
  • LaHa: Finding spherically symmetric harmonic functions on \(\mathbb R^n\setminus\{0\}\).
  • LaHa: Definition of fundamental solution of the Laplace equation

Week 2

Monday 24/02

  • LaHa: Fundamental solution: properties
  • LaHa: Fundamental solution: solution to the Poisson equation equation
  • Intermezzo: Spherical and ball averages (in abstract)

Tuesday 25/02

  • LaHa: Mean-value formulas
  • LaHa: Strong Maximum principle for harmonic functions
  • LaHa: No: This is an exercise: Positivity of harmonic functions
  • LaHa: Uniqueness for Poisson equation
  • LaHa: Smoothness of harmonic functions
  • LaHa: Estimates on derivatives of harmonic functions (only stated for now)
  • LaHa: Analyticity of harmonic functions (proof not completed yet)

Week 3

Monday 03/03

  • LaHa: Analyticity of harmonic functions (end of the proof)
  • LaHa: Estimates on derivatives of harmonic functions (the proof)
  • LaHa: Liouville’s theorem
  • LaHa: Representation formula for Poisson equation on bdd domain
  • LaHa: Harnack’s inequality
  • LaHa: Representation formula using Green’s function

Tuesday 04/03

  • LaHa: Symmetry of Green’s functions
  • LaHa: Green’s function of the ball
  • LaHa: Uniqueness by energy methods
  • LaHa: Dirichlet’s principle
  • Heat: Intro: homogeneous and nonhomogeneous heat equations
  • Heat: Symmetries
  • Heat: Examples
  • Heat: Fundamental solution: def
  • Heat: Fundamental solution: properties
  • Heat: Solution to the Cauchy problem (first part)

Week 4

Monday 10/03

  • Heat: Solution to the Cauchy problem (second part: approximation of the identity)
  • Heat: Solution to the nonhomogeneous Cauchy problem
  • Heat: Mean-value formula (the statement only)

Tuesday 11/03

  • Heat: Strong Max principle (with proof from Folland)
  • Heat: Uniqueness on bounded domains
  • Heat: Maximum principle for unbounded Cauchy problem
  • Heat: Uniqueness for unbounded domains
  • Heat: Uniqueness by energy methods

Week 5

Monday 17/03

  • Heat: Backward uniqueness by energy methods
  • Wave: Intro
  • Wave: Symmetries
  • Wave: Examples
  • Wave: Solutions in n=1: d’Alambert’s formula
  • Wave: Reflection method: solution for pinched string

Tuesday 18/03

  • Wave: Finite propagation speed, by energy methods
  • Wave: Uniqueness of solution, by energy methods
  • Wave: Euler-Poisson-Darboux equation
  • Wave: Solutions in n=3: Kirchhoff Formula