An Introduction to Second Order Partial Differential Equations

The following sections are included:

• Dedication
• Preface
• Contents
• List of Symbols

The first chapter contains generalities about partial differential equations (PDEs) as well as a list of the most popular PDEs in the literature, starting with the classical Laplace equation, the heat and wave equations, and ending with the Maxwell equations. A general existence and uniqueness theorem is given for analytic data…

We have seen in Chapter 1 that the Laplace, heat, and wave equations are among the most important partial differential equations. It turns out that they are the representative equations for the three major types of PDEs: elliptic, parabolic, and hyperbolic equations, respectively. As we shall see in the next chapters, there are many results (existence, uniqueness, qualitative properties) concerning these three types of equations. These results are specific for each class of equations. This is why when given a PDE, it is important to know which type it is. The aim of this chapter is to give the rules to answer this question…

The following sections are included:

• The Laplace equation
  • The Laplacian in polar coordinates
• Harmonic functions by the method of separation of variables
  • The two-dimensional case
  • The three-dimensional case
• The Dirichlet problem in special geometries
  • The solution for the unit disc
  • The solution of the exterior Dirichlet problem in ℝ³
• Green formulas and related properties of harmonic functions
  • Green formulas
  • Green representation theorem and consequences
• Green functions and the Laplace equation
  • Properties of Green functions and Poisson integral
  • The Neumann problem in the unit disc in ℝ²
• General elliptic equations

The most well-known parabolic equation is the heat equation

The wave equation

In the previous chapters, we have solved well-posed PDEs which admit classical solutions, that is, solutions that are smooth functions possessing continuous derivatives. We have also seen in Section 3.6 that some PDEs do not have solutions. In particular, Example 3.21 shows that the problem −Δu = f does not always have a solution when N ≥ 2 and f is only continuous. Such equations model physical phenomena, hence a solution is expected to exist, but maybe not in the classical sense. In this case, the idea is to consider suitable functional spaces where a solution can exist in a “weaker” sense. Sobolev spaces are the appropriate ones in this setting. We introduce them in the next chapter…

As mentioned in the introduction to Chapter 6, Sobolev spaces are the appropriate functional settings for finding “weak” solutions of a partial differential equation when a classical solution does not exist. In these spaces, differentiation has to be understood in the “weak” (or distributional) sense. The aim of this chapter is precisely to introduce these notions…

The Sobolev Embedding Theorems state one of the most interesting properties of the Sobolev spaces. It is precisely the embedding properties of these spaces which make them suitable spaces in the study of solutions of partial differential equations. They show that in any dimension N, functions in W^{1, p} have actually better integrability properties depending on p and N. This is due to the fact that these functions and their derivatives are in L^p…

We are now ready to study solutions, in the weak sense, of some second order linear elliptic partial differential equations in the divergence form. We consider the equation

We now consider time-dependent partial differential equations of second order. We shall prove existence and uniqueness results for two types of linear evolution problems in divergence form, a parabolic problem and a hyperbolic one. To begin with, we shall show how the notions of variational formulation and weak solution can be extended to evolution problems. For simplicity, we only treat the case of Dirichlet boundary conditions. The other boundary conditions, as those considered in Chapter 9 for the elliptic case, can be treated by similar arguments. For general results concerning evolution equations, we refer the reader to [24], [31], [41], and [44] (see also [6], [8], and [16]).

The following sections are included:

• Bibliography
• Index