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Partial differential equations.
The book has been completely rewritten for this new edition. While most of the material found in the earlier editions has been retained, though in changed form, there are considerable additions, in which extensive use is made of Fourier transform techniques, Hilbert space, and finite difference methods. A condensed version of the present work was presented in a series of lectures as part of the Tata Institute of Fundamental Research -Indian Insti- tute of Science Mathematics Programme in Bangalore in 1977. I am indebted to Professor K. G. Ramanathan for the opportunity to participate in this excit- ing educational venture, and to Professor K. Balagangadharan for his ever ready help and advice and many stimulating discussions. Very special thanks are due to N. Sivaramakrishnan and R. Mythili, who ably and cheerfully prepared notes of my lectures which I was able to use as the nucleus of the present edition. A word about the choice of material. The constraints imposed by a partial differential equation on its solutions (like those imposed by the environment on a living organism) have an infinite variety of con- sequences, local and global, identities and inequalities. Theories of such equations usually attempt to analyse the structure of individual solutions and of the whole manifold of solutions by testing the compatibility of the differential equation with various types of additional constraints.
- GenresMathematics
Paperback
First published November 30, 1951
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September 19, 2023
Ever since the invention of modern theoretical physics during its formative era in the nineteenth century, partial differential equations have formed the core of the discipline. A sketch of why, going back to the Stoics who were the first to work out a field theory of the continuum (cf. S. Sambursky, Physics of the Stoics, 1987): if matter is to be comprehensible to human experience, it must be that the change over time of material quantities is governed by local spatial gradients. Hence Newtonian ordinary differential equations or Schrödinger’s equation of quantum mechanics on the microscale ought to give rise on the macroscale not to an unanalyzable chaos but to orderly phenomena describable in spatio-temporal terms by a partial differential equation – as in fact happens when dissipative processes smooth out irregularities, which is why partial differential equations have relevance in our world. If it were not for this, they would constitute rather nothing more than an abstract curiosity (as in our day do, say, quantum groups or p-adic quantum mechanics and other strange beasts in the stable of what currently gets called mathematical physics) and their study would not have been cultivated by generations of applied mathematicians. But after two centuries they accordingly fill out a vast reach: where ought the student anxious to acquaint himself with this central field of research go to find his bearings?
Fritz John’s classic Partial Differential Equations (vol. 1 in Springer’s Applied Mathematical Sciences series, originally published in 1971 and now in its fourth edition) is ideally suited to serve as an accessible introduction at the beginning graduate level. What one may hope for would be a grounding in partial differential equations that is mathematically up to par but not so intensely into hard analysis as to make it off-putting for the neophyte (start here, the really difficult results of which there are many can be deferred to later).
As long as one commands enough familiarity with analysis not to be scared off by epsilon-delta arguments, John’s style should be welcome and pleasant – mathematically rigorous and almost always easy to follow. This reviewer encountered only one place where the reasoning became tortuous, in a section on higher-order hyperbolic equations with constant coefficients which strikes him as very confusing owing to the way the different parts of the problem are interwoven, here it was useful to look up some Princeton lecture notes based on John’s text but filling in the details in an orderly fashion (which, upon taking another look, are implicit of course in the original). Elsewhere, John can be admirably clear.
All of the standard material one might expect is represented here. John’s strategy is to explore the main ideas in the simplest possible case and afterwards to sketch how they might extend to a more general setting (non-constant coefficients, or systems of equations). Apart from a few instances in chapter one, the text does not go beyond quasi-linear systems. First-order equations and their solution via the method of characteristics form the subject of the first chapter. In what seems a judicious pedagogical move, John begins the study of second-order equations in chapter two with hyperbolic equations in two independent variables, of which the wave equation would be the canonical example. With the foundations thus laid, one will be ready in chapter three to take up characteristic manifolds and the Cauchy problem, including complete proof of the Cauchy-Kovalevsky theorem and Holmgren’s uniqueness theorem. The standard example of an elliptic equation, namely the Laplace equation, is important enough to occupy its own chapter four (maximum principle, Dirichlet problem, Green’s functions, Poisson’s formula, Perron’s method and Hilbert-space methods). Then chapters five and six cover what more there is to say about second-order hyperbolic equations in higher dimensions, as well as higher order and symmetric systems for which existence of solutions is proved by the illuminating method of finite differences. Chapter six explains how to treat higher-order elliptic equations with constant coefficients along lines similar to what was done for the Laplace equation, with the aid of Green’s functions again and of Fourier transforms. The penultimate chapter seven contains a substantial discussion of linear parabolic equations, mainly the heat equation then the general second-order linear parabolic case.
The 173 homework exercises scattered throughout the text are all of manageable difficulty if one avails oneself of the hints. To speed things up it will be convenient to resort to Mathematica (cf. our comments in this regard in our earlier review of Earl A. Coddington’s fine textbook on ordinary differential equations, here). In a few cases, this reviewer found it helpful to restate them in terms of differential forms and to use Stokes’ theorem even though it goes beyond the official scope of the text. The few problems on distributions and Hilbert-space theory are easy because conceptual. In the later chapters, John includes several extended exercises supplied with a sketch of the argument; these are not too bad if one follows the hints provided but it is unlikely one could figure out overly technical steps like those appealing to Gronwall’s lemma on one’s own!
To close this review let us pose a couple questions:
1) How far is a good knowledge of ordinary differential equations indispensable to understand partial differential equations? The two disciplines have an entirely different flavor, even though in a formal sense an ordinary differential equation represents a trivial case of a partial differential equation in one independent variable. Nevertheless, one ought to know something about ordinary differential equations before tackling this text since symmetry considerations often will reduce a problem in several variables to one in just one variable. Besides this, the powerful method of characteristics for first-order partial differential equations makes essential use of auxiliary ordinary differential equations. Thus, many of the problems in chapter one will test one’s mettle in the theory of ordinary differential equations (which must be acquired elsewhere); otherwise – in later chapters – skill at them turns out not actually to be that necessary.
2) Does learning about partial differential equations improve one’s intuition for functional analysis? It helps in view of the fact that the typical problems and associated function spaces are concrete, often special methods apply that enable more precise deductions as with the Laplace equation, where Hilbert-space techniques can be employed to solve the Dirichlet problem. Thus, a working knowledge of elementary functional analysis proves useful to a perusal of this text. Another point: the great David Hilbert was always keen to insist that a full specification of the problem involves not just the partial differential equation itself, but also the pertinent boundary conditions. In the present text, John will frequently consider the same equation subject to different criteria, say, an initial-value condition or mixed conditions in which one imposes prescribed values on the boundary of the domain (as would happen in the laboratory, if one holds certain surfaces at a given potential or regulates the flux of reactants entering a continuously stirred tank). For instance, in chapter seven, the uniqueness of solutions to a parabolic equation may depend on what conditions one is willing to impose on the initial data (uniqueness does not in fact hold in general, but can be arranged if one requires growth conditions or positivity of the solution). Going forward, one may declare as what should be a truism, but these days oftener ignored, that the practicing mathematical physicist will want to have at his disposal a store of good examples from the theory of partial differential equations in order to render the functional analysis less otherworldly and abstract. Perhaps the failure on the part of most theorists in our generation to comply with such a rule explains why there has been so little progress in quantizing gravity.
In answer to both questions, working the exercises is imperative! The present text by Fritz John would make an excellent choice for someone who may have had some exposure to partial differential equations at the undergraduate level and who now wishes to step up to the next level: mathematically sophisticated but not pretentious about it, supplied with a good number of but not too many homework exercises, makes enough of a foray to give a sense of the subject without overwhelming the beginner with hard analysis.
Fritz John’s classic Partial Differential Equations (vol. 1 in Springer’s Applied Mathematical Sciences series, originally published in 1971 and now in its fourth edition) is ideally suited to serve as an accessible introduction at the beginning graduate level. What one may hope for would be a grounding in partial differential equations that is mathematically up to par but not so intensely into hard analysis as to make it off-putting for the neophyte (start here, the really difficult results of which there are many can be deferred to later).
As long as one commands enough familiarity with analysis not to be scared off by epsilon-delta arguments, John’s style should be welcome and pleasant – mathematically rigorous and almost always easy to follow. This reviewer encountered only one place where the reasoning became tortuous, in a section on higher-order hyperbolic equations with constant coefficients which strikes him as very confusing owing to the way the different parts of the problem are interwoven, here it was useful to look up some Princeton lecture notes based on John’s text but filling in the details in an orderly fashion (which, upon taking another look, are implicit of course in the original). Elsewhere, John can be admirably clear.
All of the standard material one might expect is represented here. John’s strategy is to explore the main ideas in the simplest possible case and afterwards to sketch how they might extend to a more general setting (non-constant coefficients, or systems of equations). Apart from a few instances in chapter one, the text does not go beyond quasi-linear systems. First-order equations and their solution via the method of characteristics form the subject of the first chapter. In what seems a judicious pedagogical move, John begins the study of second-order equations in chapter two with hyperbolic equations in two independent variables, of which the wave equation would be the canonical example. With the foundations thus laid, one will be ready in chapter three to take up characteristic manifolds and the Cauchy problem, including complete proof of the Cauchy-Kovalevsky theorem and Holmgren’s uniqueness theorem. The standard example of an elliptic equation, namely the Laplace equation, is important enough to occupy its own chapter four (maximum principle, Dirichlet problem, Green’s functions, Poisson’s formula, Perron’s method and Hilbert-space methods). Then chapters five and six cover what more there is to say about second-order hyperbolic equations in higher dimensions, as well as higher order and symmetric systems for which existence of solutions is proved by the illuminating method of finite differences. Chapter six explains how to treat higher-order elliptic equations with constant coefficients along lines similar to what was done for the Laplace equation, with the aid of Green’s functions again and of Fourier transforms. The penultimate chapter seven contains a substantial discussion of linear parabolic equations, mainly the heat equation then the general second-order linear parabolic case.
The 173 homework exercises scattered throughout the text are all of manageable difficulty if one avails oneself of the hints. To speed things up it will be convenient to resort to Mathematica (cf. our comments in this regard in our earlier review of Earl A. Coddington’s fine textbook on ordinary differential equations, here). In a few cases, this reviewer found it helpful to restate them in terms of differential forms and to use Stokes’ theorem even though it goes beyond the official scope of the text. The few problems on distributions and Hilbert-space theory are easy because conceptual. In the later chapters, John includes several extended exercises supplied with a sketch of the argument; these are not too bad if one follows the hints provided but it is unlikely one could figure out overly technical steps like those appealing to Gronwall’s lemma on one’s own!
To close this review let us pose a couple questions:
1) How far is a good knowledge of ordinary differential equations indispensable to understand partial differential equations? The two disciplines have an entirely different flavor, even though in a formal sense an ordinary differential equation represents a trivial case of a partial differential equation in one independent variable. Nevertheless, one ought to know something about ordinary differential equations before tackling this text since symmetry considerations often will reduce a problem in several variables to one in just one variable. Besides this, the powerful method of characteristics for first-order partial differential equations makes essential use of auxiliary ordinary differential equations. Thus, many of the problems in chapter one will test one’s mettle in the theory of ordinary differential equations (which must be acquired elsewhere); otherwise – in later chapters – skill at them turns out not actually to be that necessary.
2) Does learning about partial differential equations improve one’s intuition for functional analysis? It helps in view of the fact that the typical problems and associated function spaces are concrete, often special methods apply that enable more precise deductions as with the Laplace equation, where Hilbert-space techniques can be employed to solve the Dirichlet problem. Thus, a working knowledge of elementary functional analysis proves useful to a perusal of this text. Another point: the great David Hilbert was always keen to insist that a full specification of the problem involves not just the partial differential equation itself, but also the pertinent boundary conditions. In the present text, John will frequently consider the same equation subject to different criteria, say, an initial-value condition or mixed conditions in which one imposes prescribed values on the boundary of the domain (as would happen in the laboratory, if one holds certain surfaces at a given potential or regulates the flux of reactants entering a continuously stirred tank). For instance, in chapter seven, the uniqueness of solutions to a parabolic equation may depend on what conditions one is willing to impose on the initial data (uniqueness does not in fact hold in general, but can be arranged if one requires growth conditions or positivity of the solution). Going forward, one may declare as what should be a truism, but these days oftener ignored, that the practicing mathematical physicist will want to have at his disposal a store of good examples from the theory of partial differential equations in order to render the functional analysis less otherworldly and abstract. Perhaps the failure on the part of most theorists in our generation to comply with such a rule explains why there has been so little progress in quantizing gravity.
In answer to both questions, working the exercises is imperative! The present text by Fritz John would make an excellent choice for someone who may have had some exposure to partial differential equations at the undergraduate level and who now wishes to step up to the next level: mathematically sophisticated but not pretentious about it, supplied with a good number of but not too many homework exercises, makes enough of a foray to give a sense of the subject without overwhelming the beginner with hard analysis.
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March 2, 20245.2 c III posed problems. an example of hadamard
October 7, 2008
His treatment of "enveloppes" leaves much to be said. as well as his derivation for the solution to second order linear partial differential equations (id est the wave equation in one spacial dimension) through fourier series. other than that, his writing style is easy to follow, while his proofs remain rigorous.
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