Professur Analysis






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Analysis Partieller Differentialgleichungen

Vorlesung im Sommersemester 2007 an der TU Chemnitz.

Übungen bei Carsten Schubert

Daten:
Di 11:30-13:00 in SR8
Mi 19:00-21:00 in SR6

Allgemeines:

Viele Probleme außerhalb (Naturwissenschaften, Wirtschaftswissenschaften, ...) und innerhalb der Mathematik führen auf partielle Differentialgleichungen. Für die Lösung solcher Gleichungen stehen keine allgemeinen Existenz- oder Eindeutigkeitsaussagen zur Verfügung. Stattdessen betrachtet man wichtige Typen von Differentialgleichungen.

Content:

Introduction
1. The Laplace equation
  1.1 Introduction
  1.2 Harmonic functions - the maximum principle
  1.3 A first look at convolution
  1.4 The Fundamental solution for the Laplacian
  1.5 Green's functions for the ball
  1.6 The Harnack inequality
2. Energy methods and weak solutions
  2.1 The Dirichlet principle
  2.2 Weak derivatives
  2.3 Weak Formulation of the Dirichlet problem
  2.4 Dirichlet problem for elliptic second order pde 3. The wave equation I: 1-d and uniqueness
  3.1 The 1-d wave equation
  3.2 Energy methods and uniqueness
4. The Fourier transform
  4.1 Definition and first properties
  4.2 Fourier transform and Sobolev spaces
  4.3. The wave equation II: existence via Fourier transform
5. The heat equation and Cauchy problems
  5.1. The heat equation
  5.2. Introduction to strongly continuous semigroups
6. The Schrödinger equation, Stone's theorem and selfadjoint operators
  6.1 The Schrödinger equation
  6.2 Stone's theorem
  6.3 Perturbation theory and the Coulomb potential
7. A glimpse at variational methods for nonlinear PDE and Brouwer's fixed point theorem.

To be continued ...

Noch Fragen? Hier ist die Antwort ... here

Literatur:

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

J.Jost: Partial differential equations. 2nd ed.
(English) Graduate Texts in Mathematics 214. Springer, New York, 2007