偏微分方程基础理论导引

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"《Partial Differential Equations I Basic Theory》是由Michael E. Taylor撰写的一本关于偏微分方程基础理论的书籍，属于Applied Mathematical Sciences系列的第115卷。该书旨在为具有美国大学基本研究生数学课程背景的读者提供一个学习偏微分方程主要方面的课程，这些课程包括初等实分析、复分析、微分几何和测度论。" 本书的内容深入探讨了偏微分方程（PDE）这一多维度的学科，它最初是为了解释如振动弦和风力吹动等物体的机械行为而发展起来的，现在已经与许多数学分支，如微分几何、复分析和调和分析，以及数学物理中的问题描述和解析紧密相关。作者Michael E. Taylor是北卡罗来纳大学的数学教授，他在书中为读者构建了一个详尽的PDE理论框架。书中涵盖了PDE的基础理论，可能包括以下几个核心知识点： 1. **偏微分方程的基本概念**：介绍偏微分方程的基本形式，如线性和非线性PDE，常微分方程与偏微分方程的区别，以及它们在描述连续系统动态行为中的应用。 2. **解析解与数值解**：讨论PDE的解析解在简单情况下的存在性和唯一性，同时介绍用于复杂问题的数值方法，如有限差分法、有限元方法和谱方法。 3. **线性PDE理论**：包括拉普拉斯方程、热方程和波动方程等基本类型，以及它们的特征线、格林函数和傅里叶变换的应用。 4. **椭圆型PDE**：详述这类PDE的性质，如正则性理论和解的稳定性，它们在几何分析和物理问题中的重要作用。 5. **抛物型PDE**：涉及时间演化问题，如扩散过程，研究其解的性质和行为。 6. **双曲型PDE**：适用于描述物理现象中的传播过程，如声波和光波的传播，讨论初边值问题和特征线方法。 7. **泛函分析和变分法**：在PDE理论中的应用，如Lax-Milgram定理和变分问题，它们在寻找PDE解的方法中起到关键作用。 8. **适定性理论**：包括解的存在性、唯一性和连续依赖性，以及对初边值问题的适定性分析。 9. **偏微分方程的几何应用**：与微分几何的交叉，如曲面的曲率、流形上的PDE等。 10. **实际问题的建模**：展示如何将实际世界的问题转化为PDE模型，如热传导、流体力学和电磁学等领域的问题。此书对于希望深入理解偏微分方程及其在数学物理和工程领域应用的读者来说，是一本宝贵的资源。通过阅读，读者可以建立起对PDE的坚实基础，并为后续高级理论的学习做好准备。此外，书中可能还包括习题和实例，帮助读者巩固所学知识并提升解决实际问题的能力。

xvi Preface

Laplace operator on R

nK; with the Dirichlet boundary condition. In addition to

studying how a given obstacle scatters waves, we consider the inverse problem:

how to determine an obstacle given data on how it scatters waves.

Chapter 10 is devoted to the Atiyah–Singer index theorem. This gives a for-

mula for the index of an elliptic operator D on a compact manifold M; deﬁned by

(7) IndexD D dim ker D dim ker D



We establish this formula, which is an integral over M of a certain differential

form deﬁned by a pair of “curvatures,” when D is a ﬁrst order differential oper-

ator of “Dirac type,” a class that contains many important operators arising from

differential geometry and complex analysis. Special cases of such a formula in-

clude the Chern–Gauss–Bonnet formula and the Riemann–Roch formula. We also

discuss the signiﬁcance of the latter formula in the study of Riemann surfaces.

In Chap. 11 we study Brownian motion, described mathematically by Wiener

measure on the space of continuous paths in R

: This provides a probabilistic

approach to diffusion and it both uses and provides new tools for the analysis of

the heat equation and variants, such as

(8)

Du CV u;

where V is a real-valued function. There is an integral formula for solutions to (8),

known as the Feynman–Kac formula; it is an integral over path space with respect

to Wiener measure, of a fairly explicit integrand. We also derive an analogous

integral formula for solutions to

(9)

Du CXu;

where X is a vector ﬁeld. In this case, another tool is involved in constructing the

integrand, the stochastic integral. We also study stochastic differential equations

and applications to more general diffusion equations.

In Chap. 12 we tackle the

@-Neumann problem, a boundary problem for an el-

liptic operator (essentially the Laplace operator) on a domain   C

,whichis

very important in the theory of functions of several complex variables. From a

technical point of view, it is of particular interest that this boundary problem does

not satisfy the regularity criteria investigated in Chap. 5.If is “strongly pseudo-

convex,” one has instead certain “subelliptic estimates,” which are established in

Chap. 12.

The third and ﬁnal volume of this work contains Chaps. 13–18. It is here that

we study nonlinear PDE.

We prepare the way in Chap. 13 with a further development of function space

and operator theory, for use in nonlinear analysis. This includes the theory of

-Sobolev spaces and H¨older spaces. We derive estimates in these spaces on

nonlinear functions F.u/, known as “Moser estimates,” which are very useful.

Preface xvii

We extend the theory of pseudodifferential operators to cases where the symbols

have limited smoothness, and also develop a variant of DO theory, the theory

of “paradifferential operators,” which has had a signiﬁcant impact on nonlinear

PDE since about 1980. We also estimate these operators, acting on the function

spaces mentioned above. Other topics treated in Chap. 13 include Hardy spaces,

compensated compactness, and “fuzzy functions.”

Chapter 14 is devoted to nonlinear elliptic PDE, with an emphasis on second

order equations. There are three successive degrees of nonlinearity: semilinear

equations, such as

(10) u D F.x;u; ru/;

quasi-linear equations, such as

(11)

.x; u; ru/@

u D F.x;u; ru/;

and completely nonlinear equations, of the form

(12) G.x; D

u/ D 0:

Differential geometry provides a rich source of such PDE, and Chap. 14 contains a

number of geometrical applications. For example, to deform conformally a metric

on a surface so its Gauss curvature changes from k.x/ to K.x/; one needs to solve

the semilinear equation

(13) u D k.x/  K.x/e

As another example, the graph of a function y D u.x/ is a minimal submanifold

of Euclidean space provided u solves the quasilinear equation

(14)



1 Cjruj



u C.ru/  H.u/.ru/ D 0;

called the minimal surface equation. Here, H.u/ D .@

u/ is the Hessian matrix

of u: On the other hand, this graph has Gauss curvature K.x/ provided u solves

the completely nonlinear equation

(15) det H.u/ D K.x/



1 Cjruj



.nC2/=2

;

a Monge-Amp`ere equation. Equations (13)–(15) are all scalar, and the maximum

principle plays a useful role in the analysis, together with a number of other tools.

Chapter 14 also treats nonlinear systems. Important physical examples arise in

studies of elastic bodies, as well as in other areas, such as the theory of liquid

crystals. Geometric examples of systems considered in Chap. 14 include equa-

tions for harmonic maps and equations for isometric imbeddings of a Riemannian

manifold in Euclidean space.

xviii Preface

In Chap. 15, we treat nonlinear parabolic equations. Partly echoing Chap. 14,

we progress from a treatment of semilinear equations,

(16)

D Lu C F.x;u; ru/;

where L is a linear operator, such as L D ; to a treatment of quasi-linear equa-

tions, such as

(17)

.t; x; u/@

u C X.u/:

(We do very little with completely nonlinear equations in this chapter.) We study

systems as well as scalar equations. The ﬁrst application of (16) we consider is

to the parabolic equation method of constructing harmonic maps. We also con-

sider “reaction-diffusion” equations, `  ` systems of the form (16), in which

F.x;u; ru/ D X.u/; where X is a vector ﬁeld on R

,andL is a diagonal opera-

tor, with diagonal elements a

; a

 0: These equations arise in mathematical

models in biology and in chemistry. For example, u D .u

;:::;u

/ might repre-

sent the population densities of each of ` species of living creatures, distributed

over an area of land, interacting in a manner described by X and diffusing in a

manner described by a

: If there is a nonlinear (density-dependent) diffusion,

one might have a system of the form (17).

Another problem considered in Chap. 15 models the melting of ice; one has

a linear heat equation in a region (ﬁlled with water) whose boundary (where the

water touches the ice) is moving (as the ice melts). The nonlinearity in the problem

involves the description of the boundary. We conﬁne our analysis to a relatively

simple one-dimensional case.

Nonlinear hyperbolic equations are studied in Chap.16. Here continuum

mechanics is the major source of examples, and most of them are systems, rather

than scalar equations. We establish local existence for solutions to ﬁrst order hy-

perbolic systems, which are either “symmetric” or “symmetrizable.” An example

of the latter class is the following system describing compressible ﬂuid ﬂow:

(18)

v C



grad p D 0;

@

 C  divv D 0;

for a ﬂuid with velocity v; density ; and pressure p; assumed to satisfy a relation

p D p./; called an “equation of state.” Solutions to such nonlinear systems tend

to break down, due to shock formation. We devote a bit of attention to the study

of weak solutions to nonlinear hyperbolic systems, with shocks.

We also study second-order hyperbolic systems, such as systems for a k-

dimensional membrane vibrating in R

; derived in Chap. 2. Another topic covered

in Chap. 16 is the Cauchy–Kowalewsky theorem, in the nonlinear case. We use a

method introduced by P. Garabedian to transform the Cauchy problem for an an-

alytic equation into a symmetric hyperbolic system.

Preface xix

In Chap. 17 we study incompressible ﬂuid ﬂow. This is governed by the

Euler equation

(19)

v Dgrad p; div v D 0;

in the absence of viscosity, and by the Navier–Stokes equation

(20)

v D Lv  grad p; div v D 0;

in the presence of viscosity. Here L is a second-order operator, the Laplace opera-

tor for a ﬂow on ﬂat space; the “viscosity”  is a positive quantity. The (19)shares

some features with quasilinear hyperbolic systems, though there are also signif-

icant differences. Similarly, (20) has a lot in common with semilinear parabolic

systems.

Chapter 18, the last chapter in this work, is devoted to Einstein’s gravitational

equations:

(21) G

D 8T

Here G

is the Einstein tensor, given by G

D Ric

 .1=2/Sg

; where

Ric

is the Ricci tensor and S the scalar curvature, of a Lorentz manifold (or

“spacetime”) with metric tensor g

: On the right side of (21), T

is the stress-

energy tensor of the matter in the spacetime, and  is a positive constant, which

can be identiﬁed with the gravitational constant of the Newtonian theory of grav-

ity. In local coordinates, G

has a nonlinear expression in terms of g

and its

second order derivatives. In the empty-space case, where T

D0; (21) is a quasi-

linear second order system for g

: The freedom to change coordinates provides

an obstruction to this equation being hyperbolic, but one can impose the use of

“harmonic” coordinates as a constraint and transform (21) into a hyperbolic sys-

tem. In the presence of matter one couples (21) to other systems, obtaining more

elaborate PDE. We treat this in two cases, in the presence of an electromagnetic

ﬁeld, and in the presence of a relativistic ﬂuid.

In addition to the 18 chapters just described, there are three appendices,

already mentioned above. Appendix A gives deﬁnitions and basic properties of

Banach and Hilbert spaces (of which L

-spaces and Sobolev spaces are exam-

ples), Fr´echet spaces (such as C

/), and other locally convex spaces (such as

spaces of distributions). It discusses some basic facts about bounded linear oper-

ators, including some special properties of compact operators, and also considers

certain classes of unbounded linear operators. This functional analytic material

plays a major role in the development of PDE from Chap. 3 onward.

Appendix B gives deﬁnitions and basic properties of manifolds and vector

bundles. It also discusses some elementary properties of Lie groups, including

a little representation theory, useful in Chap. 8, on spectral theory, as well as in

the Chern–Weil construction.

xx Preface

Appendix C, Connections and Curvature, contains material of a differential

geometric nature, crucial for understanding many things done in Chaps. 10–18.

We consider connections on general vector bundles, and their curvature. We dis-

cuss in detail special properties of the primary case: the Levi–Civita connection

and Riemann curvature tensor on a Riemannian manifold. We discuss basic prop-

erties of the geometry of submanifolds, relating the second fundamental form to

curvature via the Gauss–Codazzi equations. We describe how vector bundles arise

from principal bundles, which themselves carry various connections and curvature

forms. We then discuss the Chern–Weil construction, yielding certain closed dif-

ferential forms associated to curvatures of connections on principal bundles. We

give several proofs of the classical Gauss–Bonnet theorem and some related re-

sults on two-dimensional surfaces, which are useful particularly in Chaps. 10 and

14. We also give a geometrical proof of the Chern–Gauss–Bonnet theorem, which

can be contrasted with the proof in Chap. 10, as a consequence of the Atiyah–

Singer index theorem.

We mention that, in addition to these “global” appendices, there are appendices

to some chapters. For example, Chap.3 has an appendix on the gamma function.

Chapter 6 has two appendices; Appendix A has some results on Banach spaces

of harmonic functions useful for the proof of the linear Cauchy–Kowalewsky

theorem, and Appendix B deals with the stationary phase formula, useful for the

study of geometrical optics in Chap. 6 and also for results later, in Chap. 9. There

are other chapters with such “local” appendices. Furthermore, there are two sec-

tions, both in Chap.14, with appendices. Section 6, on minimal surfaces, has a

companion, 6B, on the second variation of area and consequences, and 12, on

nonlinear elliptic systems, has a companion, 12B, with complementary material.

Having described the scope of this work, we ﬁnd it necessary to mention a

number of topics in PDE that are not covered here, or are touched on only very

brieﬂy.

For example, we devote little attention to the real analytic theory of PDE. We

note that harmonic functions on domains in R

are real analytic, but we do not

discuss analyticity of solutions to more general elliptic equations. We do prove

the Cauchy–Kowalewsky theorem, on analytic PDE with analytic Cauchy data.

We derive some simple results on unique continuation from these few analyticity

results, but there is a large body of lore on unique continuation, for solutions to

nonanalytic PDE, neglected here.

There is little material on numerical methods. There are a few references to

applications of the FFT and of “splitting methods.” Difference schemes for PDE

are mentioned just once, in a set of exercises on scalar conservation laws. Finite

element methods are neglected, as are many other numerical techiques.

There is a large body of work on free boundary problems, but the only one

considered here is a simple one space dimensional problem, in Chap.15.

While we have considered a variety of equations arising from classical

physics and from relativity, we have devoted relatively little attention to quan-

tum mechanics. We have considered one quantum mechanical operator, given

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