Peter Stollmann - Professur Analysis
Professur Analysis

Partial Differential Equations

Course in the summer term 2003 at the TU Chemnitz.

Informationen im Vorlesungsverzeichnis

Exercise sheets

as pdf- and ps-files:

Sheet 1: .ps   .pdf    Sheet 2: .ps   .pdf    Sheet 3: .ps   .pdf    Sheet 4: .ps   .pdf
Sheet 5: .ps   .pdf    Sheet 6: .ps   .pdf    Sheet 7: .ps   .pdf    Sheet 8: .ps   .pdf
Sheet 9: .ps   .pdf    Sheet 10: .ps   .pdf    Sheet 11: .ps   .pdf   

Generalities:

Many important problems in various fields outside (natural sciences, economics,...) and inside mathematics lead to partial differential equations. The study of such equations is a nontrivial yet rewarding enterprise and has prompted a lot of research in mathematics. We give here an overview of the most important equations and the techniques that have been developed for a treatment. If time allows it we present a choice of functional analytic methods.

Contents:

Introduction

I. Some linear partial differential equations
  1. The transport equation
  2. Laplace's equation and harmonic functions
    2.1. The equation and what it means
    2.2. Harmonic functions
    2.3. A first look at convolution
    2.4. Fundamental solution for the Laplace operator
    2.5. Solution of the Dirichlet problem for the ball
    2.6. Harnack's inequality
    2.7. Outlook: the Dirichlet principle
  3. Function spaces
    3.1. Integration theory
      3.1.1 Sigma-fields and measures
      3.1.2 The Lebesgue measure
      3.1.3 Integrable functions
      3.1.4 Limit theorems
      3.1.5 Product measures, the Theorems of Fubini and Tonelli
      3.1.6 The Lp-spaces
    3.2. Distributional derivatives
    3.3. The Dirichlet problem-revisited
  4. The wave equation
    4.1 The one dimensional wave equation
    4.2 Energy methods and uniqueness
  5. The Fourier transform
  6. The Fourier transform and PDE
    6.1 Bessel potentials
    6.2 Sobolev spaces and the Fourier transform
    6.3 The wave equation
    6.4 The telegraph equation
  7. The heat equation and Cauchy problems
    7.1 The heat equation
    7.2 Introduction to the theory of strongly continuous semigroups
    7.3 Groups and semigroups in Hilbert spaces

To be continued ...

Any questions? You can find the answer ... here

Literature:

L.C. Evans: Partial Differential Equations
Graduate Studies in Mathematics, Vol 19, AMS, Providence, 1999