Programma del Corso di Analisi Funzionale Anno Accademico 2016/2017

Introduction: What Are Partial Differential Equations? . . . . . . . . . . . . . .
The Laplace Equation as the Prototype of an Elliptic Partial Differential Equation of Second Order..........................

  • 2.1  Harmonic Functions: Representation Formula
    for the Solution of the Dirichlet Problem on the Ball
    (Existence Techniques 0) ............................................
  • 2.2  Mean Value Properties of Harmonic Functions.
    Subharmonic Functions. The Maximum Principle .................
The Maximum Principle ...................................................
  • 3.1  The Maximum Principle of E. Hopf.................................
  • 3.2  The Maximum Principle of Alexandrov and Bakelman ............
  • 3.3  Maximum Principles for Nonlinear Differential Equations . . . . . . . .
Existence Techniques I: Methods Based on the Maximum Principle .....................................................................
  • 4.1  Difference Methods: Discretization of Differential Equations . . . . .
  • 4.2  The Perron Method...................................................
  • 4.3  The Alternating Method of H.A. Schwarz ..........................
  • 4.4  Boundary Regularity .................................................
The Dirichlet Principle. Variational Methods for the Solution
of PDEs (Existence Techniques II).......................................
  • 5.1  Dirichlet’s Principle ..................................................
  • 5.2  The Sobolev Space W 1;2 .............................................
  • 5.3  Weak Solutions of the Poisson Equation ............................
  • 5.4  Quadratic Variational Problems .....................................
  • 5.5  Abstract Hilbert Space Formulation of the Variational
    Problem. ...............................
  • 5.6  Convex Variational Problems ........................................
Sobolev Spaces and L2 Regularity Theory ..............................
  • 5.1  General Sobolev Spaces. Embedding Theorems
    of Sobolev, Morrey, and John–Nirenberg ...........................
  • 5.2  L2-Regularity Theory: Interior Regularity of Weak
    Solutions of the Poisson Equation ...................................
  • 5.3  Boundary Regularity and Regularity Results
    for Solutions of General Linear Elliptic Equations .................
  • 5.4  Extensions of Sobolev Functions and Natural
    Boundary Conditions.................................................
  • 5.5  Eigenvalues of Elliptic Operators....................................