GENERAL NATURE OF PDE

It is no exaggeration to state that partial differential equations (PDE) haveplayed a vital role in the development of science and technology, primarilysince the beginning of the twentieth century. In the earlier stage, PDE weremainly used to describe physical phenomena, like vibrations of strings, heatconduction in solids, transport phenomena, to mention a few. Later, with theadvantage of mathematical modelling, the scope of using PDE for thedescription of phenomena occurring in biology, economics and even sociologybecame prominent.

Since the days of Newton or even earlier, many have attempted to describephysical processes using mathematics. Such a mathematical description oftenleads to linear differential, integral and even integro-differentialequations. Thus, a large number of PDE naturally come from mathematicalphysics. The initial developments in PDE, though, were mainly geared towardsobtaining solutions to a particular physical or engineering problem, it wassoon realized that many of the problems will have common features andsimilarities. This naturally led to the grouping of PDE that can be tackledin a single framework. This automatically leads to the abstraction of thesubject and the theoretical analysis that follows, hence, becomes moreimportant. This is one of the features we try to follow in the present book.Indeed, unlike ordinary differential equations (ODE), all PDE including thelinear ones cannot be treated in a single theoretical framework, leading tothe necessity of a classification. In fact, due to the diverse nature ofphysical phenomena, we remark that we cannot classify all the PDE.Nevertheless, a fairly good classification is available for the second-orderequations and interestingly a large number of physical and other problemslead to second-order equations. Also, for the three important classes ofequations, namely elliptic, hyperbolic and parabolic, general theories havebeen developed.

As mentioned above, a wide class of physical problems is described bysecond-order linear differential equations of the form Here the variablex varies in an open set in the physical spaceℝn; n = 1;2; 3 and the coefficients aij;bi and c are known from thephysical process; u is the unknown function andf denotes an external quantity, if any, influencing thephysical process.

INTRODUCTION

In this chapter, we discuss the classification of partial differentialequations (PDE), concentrating mostly on linear equations or equations withlinear principal part. In the modern theory of the subject, especially sincethe middle of the last century, the question of classification has takenaltogether new directions, which we only mention very briefly in the sectionon Notes. Also, the classification of PDEs is in general not complete. Thesecond-order equations in two variables have a fairly completeclassification, which is our main topic of discussion in this chapter. Webegin by a discussion on a Cauchy problem for a second-order linear PDE andhighlighting certain subtle differences with a Cauchy problem for anordinary differential equation (ODE). This naturally motivates towards thestudy of the classification of PDE. The general references for this chapterare Courant and Hilbert (1989), Rubinstein and Rubinstein (1998), Koshlyakovet al. (1964), John (1978), Mikhailov (1978), Ladyzhenskaya (1985), McOwen(2005), Renardy and Rogers (2004), Prasad and Ravindran (1996), Vladimirov(1979, 1984), and Hörmander (1976, 1984), among many others.

CAUCHY PROBLEM

We begin by describing a Cauchy problem or aninitial value problem (IVP) for a linear PDE andcomparing it with an IVP associated to a linear ODE. For simplicity of theexposition, we consider the second-order linear equation in a regionΩ in ℝn:

where the coefficients aij; ai; a andthe function f are given real (smooth) valued functionsdefined in Ω. The Cauchy problem for a second-order linear ODE

where the coefficients b; c and the functiong are smooth functions defined in an interval inℝ, consists in finding a solution u satisfying theinitial conditions and for somex0 and arbitraryu0, u1.

A Cauchy problem (or an IVP) associated with the PDE (6.1), in which initialconditions for u and its normal derivative are assigned onan (n − 1)-dimensional surface Γ in Ω.We will see that, in general, this may lead to a lack of existence oruniqueness depending on the nature of Γ.

INTRODUCTION

The reasons for studying Laplace and Poisson equations are twofold.Primarily, these equations arise in a wide variety of physical contexts.Secondly, as mentioned earlier, Laplace operator is a prototype of a verygeneral class of linear elliptic operators. In fact, the Laplace operatorpossesses many features of the general class of elliptic operators. Recallthat the most general form of second-order linear partial differentialequations (PDE) in n variables is given by

where x ∈ Ω; an open set inℝn,aij = aji. Theoperator L is said to be uniformlyelliptic if there exists an a > 0 suchthat for all. Recall that is the characteristic form associated with theoperator L. The ellipticity condition here implies that thecharacteristic variety is an empty set at all points in Ω.

Thus, the condition requires the uniform positive definiteness of thesymmetric matrix [aij(x)]. Ifwe take bi = ci =d = 0 and

there results the Laplace operator Δ. The ultimate interest is tostudy the classical solutions of the equationLu=f for a given dataf. The study of Laplace equationΔu=0 and Poisson equationΔu = f (potentialtheory) gives a starting point for the general theory ofLu = f. The Schaudertheory provides a general theory for Lu =f when the coefficients are smooth, that is,Hölder continuous and it is essentially an extension of the potentialtheory. The crucial result is the derivation of estimates (known asa-priori estimates), say, of the form: Anyu ∈ C2(Ω), asolution of Δu = f in adomainΩ ⊂ℝn satisfies a uniform estimate:

where Ω′ ⊂⊂Ω; 0 < a < 1 andC is a constant depending only on a.Here is the standard Ḧolder space.

Such estimates eventually (with delicate analysis) lead to the solvability ofthe equation. The above estimate is an interior estimateand we also require boundary/global estimates that depend on the smoothnessof the boundary as well.