Basic Linear Partial Differential Equations

Basic Linear Partial Differential Equations

by Francois Treves

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Focusing on the archetypes of linear partial differential equations, this text for upper-level undergraduates and graduate students employs nontraditional methods to explain classical material. Topics include the Cauchy problem, boundary value problems, and mixed problems and evolution equations. Nearly 400 exercises enable students to reconstruct proofs. 1975 edition… See more details below


Focusing on the archetypes of linear partial differential equations, this text for upper-level undergraduates and graduate students employs nontraditional methods to explain classical material. Topics include the Cauchy problem, boundary value problems, and mixed problems and evolution equations. Nearly 400 exercises enable students to reconstruct proofs. 1975 edition.

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Basic Linear Partial Differential Equations

By Francois Treves

Dover Publications, Inc.

Copyright © 2003 François Trèves
All rights reserved.
ISBN: 978-0-486-15098-7


The Basic Examples of Linear PDEs and Their Fundamental Solutions


The Basic Examples of Linear PDEs

The theory of linear PDEs stems from the intensive study of a few special equations, whose importance was recognized in the eighteenth and nineteenth centuries. These were the basic equations in mathematical physics (gravitation, electromagnetism, sound propagation, heat transfer, and quantum mechanics). After their introduction in applied mathematics, they were shown to play important roles in pure mathematics: For instance, the Laplace equation was first studied as the basic equation in the theory of Newton's potential and in electrostatics; later, suitably reinterpreted, it was used to study the geometry and topology of Riemannian manifolds. Similarly, the heat equation was studied by Fourier in the context of heat transfer. Later it was shown to be related to probability theory. One of the basic examples, which we describe below, does not seem to have originated in applications to physics: the Cauchy-Riemann operator, which is used to define analytic functions of a complex variable. But to my knowledge, all the remaining ones have their origin in applied mathematics. At any rate, the general theory of linear PDEs is an elaboration of the respective theories of these special operators. During the twentieth century it was recognized that many properties which had seemed to be the prerogative of the Laplace equation or of the wave equation could in fact be extended to wide classes of equations. These properties usually center around a question or a problem that only makes sense for one or the other equation: for instance, around the Dirichlet problem, which makes sense for the Laplace equation but not really for the wave equation, or the Cauchy problem, which is well posed for the latter but not for the former. The purpose of this introductory course is to help the student to understand some of these problems and some of their solutions—but always by staying very close to the special equation for which they were originally considered. It is therefore necessary that we have the nature of the basic examples clearly in mind.

1.1 The Laplace Equation in n > 1 Variables

Let us denote by x = (x1, ..., xn) the variables in the Euclidean space Rn. Usually the Laplace operator is


Some people call the Laplace operator that which in our notation would be —Δ. They have very good reasons to do this; it is a pity that historical custom is not on their side, but they are gaining ground. Indeed, —Δ is a positive operator; its Fourier transform is the square of the norm of the variable in Rn, |[xi]|2. The latter remark underlines the close relationship between the Laplace operator and the Euclidean norm, the spheres in the Euclidean space, the orthogonal transformations, and so on. Indeed, Δ is invariant under orthogonal transformations; that is, if T is any such transformation in Rn and f any infinitely differentiable function of x, then


f)(Tx) = Δ{f(Tx)}, x [member of] Rn.

This, of course, is a crucial symmetry property of the Laplace operator and is part of the reason for its role in the description of many phenomena in isotropic media.

As a matter of fact, any linear transformation T of Rn such that (1.1) holds for all C∞ functions f must be orthogonal: The orthogonal transformations are exactly those which leave Δ invariant (i.e., which commute with Δ).

The functions that satisfy the homogeneous Laplace equation


Δh = 0

are called harmonic functions.

1.2 The Wave Equation

For reasons which will become clear when we begin using the Fourier transformation, it is convenient to replace the partial differentiations [partial derivative]/[partial derivative]xj by purely imaginary variables √-1 [xi]j (j = 1, ..., n). Thus the operator —Δ becomes


|[xi]}2 = [xi]21 + ··· [xi]2n

which is a positive-definite quadratic form. Its signature is (n, 0): It has n positive eigenvalues and no nonpositive ones. We may also look at quadratic forms with different signatures. An important case is the form with all eigenvalues strictly positive except one which is strictly negative. For various reasons it is convenient to consider such a form on an (n + 1) -dimensional space Rn+1 where the variables are denoted by ([xi]1, ..., [xi]n, τ). It is essentially the form


|[xi]|2 - τ2 = [xi]21 + ··· + [xi]2n - τ2

corresponding to the partial differential operator in Rn+1 (where the variables are denoted by x1, ..., xn, t):


This is the wave operator (sometimes called the d'Alembertian): The xj's are called the space variables, and t is the time variable. It is the operator used to describe oscillatory phenomena and wave propagation.

If we are interested in those linear transformations of Rn+1 which commute with ?, we will have no trouble in determining what they are. Of course, they are the same as the linear transformations in (the dual space) Rn+1 which leave the quadratic form (1.4) invariant. They form a group much used in physics since the advent of relativity: the Lorentz group.

The solutions of the wave equation


[??]g = 0

have properties that are radically different from those of the Laplace equation, as will become clear when we take a closer look at them.

1.3 The Heat Equation

The examples given in §1.1 and §1.2 are both homogeneous second-order differential operators, that is, differential operators which involve second-order partial differentiations and none of order ≠2. The heat operator in Rn+1,


[partial derivative]/[partial derivative]t - Δx,

is not of this type. It is used to describe various transfer phenomena, like the transfer of heat in isotropic media. At first glance the heat and the wave equations look alike, and indeed they have some properties in common. But there are also very deep differences. No wave propagation phenomena are associated with the solutions of the heat equation; phenomena of the diffusion type are. As a matter of fact, there is some similarity with the Laplace equation. It should not come as a surprise: The leading terms in the heat equation, that is, the second-order partial derivatives, are the same as in the Laplace equation in space variables.

We have just seen what are probably the most important examples of linear partial differential equations. The Laplace equation is the archetype of a large class of equations, called the elliptic PDEs. The reason for this is obvious: If we look at the quadratic form


it is equal, up to a change of scale, to the symbol (1.3) of the Laplace operator in two variables. It is also the function in R2 whose level curves are ellipses.

Similarly, the wave equation is the archetype of the hyperbolic PDEs: The level curves of the function [xi]2 - τ2 in R2 are the standard hyperbola. The heat equation is the archetype of the parabolic PDEs: Its symbol can be defined as being the function [xi]2 - τ in R2 whose level curves are the standard parabola. As a matter of fact, again in view of our use of the Fourier transformation, we prefer to define its symbol as |[xi]|2 + iτ, replacing [partial derivative]/[partial derivative]t by iτ rather than by —τ.

This has been the classical way of categorizing partial differential equations, when only those of first and second order were studied by mathematicians. It is quite inadequate to classify systems of PDEs, higher order equations, or equations with complex coefficients. It turns out that some of the essential properties of the Laplace equation follow from the fact that its symbol (1.3) only vanishes at the origin—and not from the fact that it is a positive-definite quadratic form. In other words, these properties subsist in other equations which partake of the former characteristic but not of the latter. This is the case of the equation we study next.

1.4 The Cauchy—Riemann Equation

Let x, y denote the variables in the plane R2. The homogeneous Cauchy-Riemann equation reads



Here f = u + iv is a complex-valued differentiable function (u, v are real). Equation (1.7) is equivalent to the system


[partial derivative]u/[partial derivative]x = [partial derivative]v/[partial derivative]y, [partial derivative]v/[partial derivative]x = -[partial derivative]u/[partial derivative]y.

Let us set z = x + iy, z = x - iy, or, equivalently, x = 1/2(z + [bar.z]), y = (1/2i)(z - z). Thus any function such as f(x, y) in a subset of R2 can also be viewed as a function of (z, [bar.z]). Equation (1.7) can then be rewritten (by the chain rule of differentiation) as


[partial derivative]f/[partial derivative]z = 0.

We have set


Roughly speaking, (1.9) tells us that f is "independent of [bar.z]"; more precisely, it states that f (supposed to be sufficiently smooth) is an analytic function of z, i.e., has a complex derivative [at every point where (1.9) holds]. It is convenient to introduce also the "anti-Cauchy—Riemann" operator

[partial derivative]/[partial derivative]z = 1/2([partial derivative]/[partial derivative]x - √ -1 [partial derivative]/[partial derivative]y).

Note that


4 [partial derivative]2/[partial derivative]z]partial derivative][bar.z] = Δ,

the Laplace operator in two variables. The identity (1.10) points to strong relations between the Laplace and the Cauchy-Riemann equations. These will be confirmed when we study them. The symbol of [partial derivative]/[partial derivative][bar.z] is


i/2([xi] + iη)

(we have denoted by [xi], η the variables in the dual plane R2). Note that, like the symbol of the Laplace operator, it only vanishes at the origin. This property will have important consequences. Because of it, in the modern terminology, the Cauchy-Riemann operator is also said to be elliptic.

1.5 The Schrödinger Equation

In the study of partial differential equations one is quickly taught to expect important and deep implications to follow from merely formal differences. This is confirmed by everything that follows and is well exemplified by the theories of the heat and of the Schrödinger equations. The Schrödinger operator with constant coefficients in n-space variables is


1/i [partial derivative]/[partial derivative]t - Δx.

The only difference with the heat operator is the presence of the factor i—1 in front of [partial derivative]/[partial derivative]t. Yet the solutions of the two equations exhibit very different kinds of behavior, as will be seen later. The Schrödinger equation was originally introduced to describe the behavior of the electron and other elementary particles. It has the defect of not being Lorentz-invariant and therefore of not fitting in the relativistic formulation of quantum mechanics. It is still used as an approximation, but in a more rigorous setup, it has been replaced by Dirac's equations.

So far we have only looked at examples of a single, or scalar, linear partial differential equation. But there are many important (for mathematics and for physics) examples of systems of equations. This means that we are given N1N2 linear partial differential operators Pjk (j = 1, ..., N1, k = 1, ..., N2) and that we consider the N1 equations in N2 unknown functions uk



The system (1.13) is said to be determined if N1 = N2, that is, if there are exactly as many equations as there are unknowns; overdetermined if N1 >N2, that is, if there are strictly more equations than unknowns; and underdetermined if there are strictly fewer equations than unknowns. The theory of systems is more difficult than the theory of single equations, especially the theory of overdetermined systems. At this stage we shall content ourselves with some examples. The Maxwell equations, on which classical electromagnetism is based, constitute an example of a determined system, as are the Dirac equations, alluded to above. Both are hyperbolic systems. Without getting into the technicalities of the definition, let us say that hyperbolic systems have formal and nonformal properties closely related to those of the wave equation. We next give some examples of systems of linear PDEs which are not determined.


Excerpted from Basic Linear Partial Differential Equations by Francois Treves. Copyright © 2003 François Trèves. Excerpted by permission of Dover Publications, Inc..
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