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....................................