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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

All rights reserved.

ISBN: 978-0-486-15098-7

CHAPTER 1

*The Basic Examples of Linear PDEs and Their Fundamental Solutions*

1

**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* = (*x*1, ..., *x*n) the variables in the Euclidean space **R n**. Usually the

*Laplace operator*is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

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 **R n**, |[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

**R**and

*n**f*any infinitely differentiable function of

*x*, then

*(1.1)*

(Δ*f*)(*Tx*) = Δ{*f(Tx)*}, *x* [member of] **R n**.

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 **R n** 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

*(1.2)*

Δ*h* = 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

*(1.3)*

|[xi]}2 = [xi]21 + ··· [xi]2*n*

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 **R***n*+1 where the variables are denoted by ([xi]1, ..., [xi]*n*, τ). It is essentially the form

*(1.4)*

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

corresponding to the partial differential operator in **R n+1** (where the variables are denoted by

*x*1, ...,

*xn, t*):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

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 **R n+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)

**R**which leave the quadratic form (1.4) invariant. They form a group much used in physics since the advent of relativity: the Lorentz group.

*n*+1The solutions of the wave equation

*(1.5)*

[??]*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 **R n+1**,

*(1.6)*

[partial derivative]/[partial derivative]*t* - Δ*x*,

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

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

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 + *i*τ, replacing [partial derivative]/[partial derivative]*t* by *i*τ 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

*(1.7)*

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

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

*(1.8)*

[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

*(1.9)*

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

We have set

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

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

*(1.10)*

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

*(1.11)*

*i*/2([xi] + *i*η)

(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

*(1.12)*

1/*i* [partial derivative]/[partial derivative]*t* - Δ*x*.

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 *N*1*N*2 linear partial differential operators *Pjk* (*j* = 1, ..., *N*1, *k* = 1, ..., *N*2) and that we consider the *N*1 equations in *N*2 unknown functions *uk*

*(1.13)*

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The system (1.13) is said to be *determined* if *N*1 = *N*2, that is, if there are exactly as many equations as there are unknowns; *overdetermined* if *N*1 >*N*2, 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.

*(Continues...)*

Excerpted fromBasic Linear Partial Differential EquationsbyFrancois Treves. Copyright © 2003 François Trèves. Excerpted by permission of Dover Publications, Inc..

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