Introduction of Elliptic PDES (Gilbarg-Trudinger)
1
C L A S S I C S I N M A T H E M A T I C S
David Gilbarg • Neil S. Trudinger
Elliptic
Partial Dierential
Equations
of Second Order
2
Contents
Contents 3
Preface 5
1 Introduction 7
1.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2 Further Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.3 Basic Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2 Laplace’s Equation 19
Appendices 23
3
4 CONTENTS
preface
This revision of the 1983 second edition of “Elliptic Partial Dierential Equations of
Second Order” corresponds to the Russian edition, published in 1989, in which we essen-
tially updated the previous version to 1984. The additional text relates to the boundary
Hölder derivative estimates of Nikolai Krylov, which provided a fundamental component
of the further development of the classical theory of elliptic (and parabolic), fully non-
linear equations in higher dimensions. In our presentation we adapted a simplication
of Krylov’s approach due to Luis Caarelli.
The theory of nonlinear second order elliptic equations has continued to ourish
during the last fteen years and, in a brief epilogue to this volume, we signal some of
the major advances. Although a proper treatment would necessitate at least another
monograph, it is our hope that this book, most of whose text is now more than twenty
years old,can continue to serve as background for these and future developments.
Since our rst edition we have become indebted to numerous colleagues, all over the
globe. It was particularly pleasant in recent years to make and renew friendships with
our Russian colleagues, Olga Ladyzhenskaya, Nina Ural’tseva, Nina Ivochkina, Nikolai
Krylov and Mikhail Safonov, who have contributed so much to this area. Sadly, we
mourn the passing away in 1996 of Ennico De Giorgi, whose brilliant discovery forty
years ago opened the door to the higher-dimensional nonlinear theory.
October 1997 David Gilbarg • Neil S. Trudinger
5
Chapter 0: CONTENTS
6
1. Introduction
1.1 Summary
The principal objective of this work is the systematic development of the general
theory of second order quasilinear elliptic equations and of the linear theory required in
the process. This means we shall be concerned with the solvability of boundary value
problems (primarily the Dirichlet problem) and related general properties of solutions
of linear equations
Lu ≡ a
ij
(x)D
ij
u + b
i
(x)D
i
u + c(x)u = f(x), i, j = 1, 2, . . . , n, (1.1)
and of quasilinear equations
Qu ≡ a
ij
(x, u, Du)D
ij
u + b(x, u, Du) = 0. (1.2)
Here Du = (D
1
u, . . . , D
n
u), where D
i
u = ∂u /∂x
i
, D
ij
u = ∂
2
u/∂x
i
∂x
j
, etc., and the
summation convention is understood. The ellipticity of these equations is expressed by
the fact that the coecient matrix [a
ij
] is (in each case) positive denite in the domain
of the respective arguments. We refer to an equation as uniformly elliptic if the ratio γ
of maximum to minimum eigenvalue of the matrix [a
ij
]is bounded We shall be concerned
with both non-uniformly and uniformly elliptic equations.
The classical prototypes of linear elliptic equations are of course Laplace’s equation
∆u =
D
ii
u = 0
and its inhomogeneous counterpart, Poisson’s equation ∆u = f. Probably the best
known example of a quasilinear elliptic equation is the minimal surface equation
D
i
(D
i
u/(1 + |Du|
2
)
1/2
) = 0,
which arises in the problem of least area. This equation is non-uniformly elliptic with
γ = 1 + |Du|
2
. The properties of the dierential operators in these examples motivate
much of the theory of the general classes of equations discussed in this book.
7
Chapter 1: Introduction
The relevant linear theory is developed in Chapters 2-9 (and in part of Chapter 12).
Although this material has independent interest, the emphasis here is on aspects needed
for application to nonlinear problems. Thus the theory stresses weak hypotheses on the
coecients and passes over many of the important classical and modern results on linear
elliptic equations.
Since we are ultimately interested in classical solutions of equation (1.2), what is
required at some point is an underlying theory of classical solutions for a suciently large
class of linear equations. This is provided by the Schauder theory in Chapter 6, which
is an essentially complete theory for the class of equations (1.1) with Hölder continuous
coecients. Whereas such equations enjoy a denitive existence and regularity theory
for classical solutions, corresponding results cease to be valid for equations in which the
coecients are assumed only continuous.
A natural starting point for the study of classical solutions is the theory of Laplace’s
and Poisson’s equations. This is the content of Chapters 2 and 4. In anticipation
of later developments the Dirichlet problem for harmonic functions with continuous
boundary values is approached through the Perron method of subharmonic functions.
This emphasizes the maximum principle, and with it the barrier concept for studying
boundary behavior, in arguments that are readily extended to more general situations in
later chapters. In Chapter 4 we derive the basic Hölder estimates for Poisson’s equation
from an analysis of the Newtonian potential. The principal result here (see Theorems
4.6, 4.8) states that all C
2
(Ω) solutions of Poisson’s equation, ∆u = f , in a domain Ω
of R
n
satisfy a uniform estimate in any subset Ω
′
⊂⊂ Ω
∥u∥
C
2,α
(
¯
Ω
′
)
⩽ C(sup
Ω
|u| + ∥f∥
C
α
(
¯
Ω)
), (1.3)
where C is a constant depending only on α(0 < α < 1), the dimension n and dist
(Ω
′
, ∂Ω); (for notation see Section 4.1). This interior estimate (interior since Ω
′
⊂⊂ Ω)
can be extended to a global estimate for solutions with suciently smooth boundary
values provided the boundary ∂Ω is also suciently smooth. In Chapter 4 estimates up
to the boundary are established only for hyperplane and spherical boundaries, but these
suce for the later applications.
The climax of the theory of classical solutions of linear second order elliptic equations
is achieved in the Schauder theory, which is developed in modied and expanded form
in Chapter 6. Essentially, this theory extends the results of potential theory to the
class of equations (1.1) having Hölder continuous coecients. This is accomplished by
the simple but fundamental device of regarding the equation locally as a perturbation
of the constant coecient equation obtained by xing the leading coecients at their
values at a single point. A careful calculation based on the above mentioned estimates
for Poisson’s equation yields the same inequality (1.3) for any C
2,α
solution of (1.1),
where the constant C now depends also on the bounds and Hölder constants of the
8
Section 1.1: Summary
coecients and in addition on the minimum and maximum eigenvalues of the coecient
matrix [a
ij
] in Ω. These results are stated as interior estimates in terms of weighted
interior norms (Theorem 6.2) and, in the case of suciently smooth boundary data, as
global estimates in terms of global norms (Theorem 6.6). Here we meet the important
and recurring concept of an apriori estimate; namely, an estimate(in terms of given
data) valid for all possible solutions of a class of problems even if the hypotheses do
not guarantee the existence of such solutions. A major part of this book is devoted to
the establishment of apriori bounds for various problems.(We have taken the liberty of
replacing the latin a priori with the single word apriori, which will be used throughout.)
The importance of such apriori estimates is visible in several applications in Chapter
6, among them in establishing the solvability of the Dirichlet problem by the method
of continuity (Theorem 6.8) and in proving the higher order regularity of C
2
solutions
under appropriate smoothness hypotheses (Theorems 6.17, 6.19). In both cases the
estimates provide the necessary compactness properties for certain classes of solutions,
from which the desired results are easily inferred
We remark on several additional features of Chapter 6, which are not needed for
the later developments but which broaden the scope of the basic Schauder theory. In
Section 6.5 it is seen that for continuous boundary values and a suitably wide class
of domains the proof of solvability of the Dirichlet problem for (1.1) can be achieved
entirely with interior estimates, thereby simplifying the structure of the theory. The
results of Section 6.6 extend the existence theory for the Dirichlet problem to certain
classes of non-uniformly elliptic equations. Here we see how relations between geometric
properties of the boundary and the degenerate ellipticity at the boundary determine
the continuous assumption of boundary values. The methods are based on barrier
arguments that foreshadow analogous (but deeper) results for nonlinear equations in
Part II. In Section 6.7 we extend the theory of (1.1) to the regular oblique derivative
problem. The method is basically an extrapolation to these boundary conditions of
the earlier treatment of Poisson’s equation and the Schauder theory (without barrier
arguments, however).
In the preceding considerations, especially in the existence theory and barrier argu-
ments, the maximum principle for the operator L (when c ⩽ 0) plays an essential part.
This is a special feature of second order elliptic equations that simplifes and strengthens
the theory. The basic facts concerning the maximum principle, as well as illustrative
applications of comparison methods, are contained in Chapter 3. The maximum prin-
ciple provides the earliest and simplest apriori estimates of the general theory. It is of
considerable interest that all the estimates of Chapters 4 and 6 can be derived entirely
from comparison arguments based on the maximum principle, without any mention of
the Newtonian potential or integrals.
An alternative and more general approach to linear problems, without potential
9
Chapter 1: Introduction
theory, can be achieved by Hilbert space methods based on generalized or weak solutions,
as in Chapter 8. To be more specic, let L
′
be a second order dierential operator, with
principal part of divergence form, dened by
L
′
u ≡ D
i
(a
ij
(x)D
j
u + b
i
(x)u) + c
i
(x)D
i
u + d(x)u.
If the coecients are suciently smooth, then clearly this operator falls within the class
discussed in Chapter 6. However, even if the coecients are in a much wider class and
u is only weakly dierentiable (in the sense of Chapter 7), one can still dene weak or
generalized solutions of L
′
u = g in appropriate function classes. In particular, if the
coecients a
ιj
, b
i
, c
i
are bounded and measurable in Ω and g is an integrable function in
Ω, let us call ua weak or generalized solution of L
′
u = g in Ω if u ∈ W
1,2
(Ω) (as dened
in Chapter 7) and
Ω
(a
ij
D
j
u + b
i
u)D
i
v − (c
i
D
i
u + du)v
dx = −
Ω
gv dx (1.4)
for all test functions v ∈ C
1
0
(Ω). It is clear that if the coecients and g are suciently
smooth and u ∈ C
2
(Ω), then u is also a classical solution.
We can now speak also of weak solutions u of the generalized Dirichlet problem,
L
′
u = g in Ω, u = φ on ∂Ω,
if u is a weak solution satisfying u − φ ∈ W
1,2
0
(Ω). where φ ∈ W
1,2
(Ω). Assuming that
the minimum eigenvalue of [a
ij
] is bounded away from zero in Ω, that
D
i
b
i
+ d ⩽ 0 (1.5)
in the weak sense, and that also g ∈ L
2
(Ω). we nd in Theorem 8.3 that the generalized
Dirichlet problem has a unique solution u ∈ W
1,2
(Ω). Condition (1.5), which is the
analogue of c ⩽ 0 in (1.1), assures a maximum principle for weak solutions of L
′
u ⩾ 0(⩽
0) (Theorem 8.1) and hence uniqueness for the generalized Dirichlet problem. Existence
of a solution then follows from the Fredholm alternative for the operator L
′
(Theorem
8.6), which is proved by an application of the Riesz representation theorem in the Hilbert
space W
1
,
2
0
(Ω).
The major part of Chapter 8 is taken up with the regularity theory for weak solu-
tions. Additional regularity of the coecients in (1.4) implies that the solutions belong
to higher W
k,2
spaces (Theorems 8.8, 8.10). It follows from the Sobolev imbedding
theorems in Chapter 7 that weak solutions are in fact classical solutions provided the
coecients are suciently regular. Global regularity of these solutions is inferred by
extending interior regularity to the boundary when the boundary data are suciently
10
Section 1.1: Summary
smooth (Theorems 8.13, 8.14).
The regularity theory of weak solutions and the associated pointwise estimates are
fundamental to the nonlinear theory. These results provide the starting point for the
“bootstrap”arguments that are typical of nonlinear problems. Briey, the idea here is
to start with weak solutions of a quasilinear equation, regarding them as weak solutions
of related linear equations obtained by inserting them into the coecients, and then to
proceed by establishing improved regularity of these solutions. Starting anew with the
latter solutions and repeating the process, still further regularity is assured, and so on,
until the original weak solutions are nally proved to be suitably smooth. This is the
essence of the regularity proofs for the older variational problems and is implicit in the
nonlinear theory presented here.
The Hölder estimates for weak solutions that are so vital for the nonlinear theory are
derived in Chapter 8 from Harnack inequalities based on the Moser iteration technique
(Theorems 8.17, 8.18, 8.20, 8.24). These results generalize the basic apriori Hölder esti-
mate of De Giorgi, which provided the initial breakthrough in the theory of quasilinear
equations in more than two independent variables. The arguments rest on integral esti-
mates for weak solutions u derived from judicious choice of test functions v in (1.4). The
test function technique is the dominant theme in the derivation of estimates throughout
most of this work.
In this edition we have added new material to Chapter 8 covering the Wiener criterion
for regular boundary points, eigenvalues and eigenfunctions, and Hölder estimates for
rst derivatives of solutions of linear divergence structure equations.
We conclude Part I of the present edition with a new chapter, Chapter 9, concerning
strong solutions of linear elliptic equations. These are solutions which possess second
derivatives, at least in a weak sense, and satisfy (1.1) almost everywhere. Two strands
are interwoven in this chapter. First we derive a maximum principle of Aleksandrov,
and a related apriori bound (Theorem 9.1) for solutions in the Sobolev space W
2,n
(Ω),
thereby extending certain basic results from Chapter 3 to nonclassical solutions. Later in
the chapter, these results are applied to establish various pointwise estimates, including
the recent Hölder and Harnack estimates of Krylov and Safonov (Theorems 9.20,9.22;
Corollaries 9.24, 9.25). The other strand in this chapter is the L
p
theory of linear second-
order elliptic equations that is analogous to the Schauder theory of Chapter 6. The basic
estimate for Poisson’s equation, namely the Calderon-Zygmund inequality is derived
through the Marcinkiewicz interpolation theorem, although without the use of Fourier
transform methods. Interior and global estimates in the Sobolev spaces W
2,p
(Ω), 1 <
p < ∞, are established in Theorems 9.11, 9.13 and applied to the Dirichlet problem for
strong solutions, in Theorem 9.15 and Corollary 9.18.
Part II of this book is devoted largely to the Dirichlet problem and related estimates
for quasilinear equations. The results concern in part the general operator (1.2) while
11
Chapter 1: Introduction
others apply especially to operators of divergence form
Qu ≡ div A(x, u, Du) + B(x, u, Du) (1.6)
where A(x, z, p) and B(x, z, p) are respectively vector and scalar functions dened on Ω×
R × R
n
.
Chapter 10 extends maximum and comparison principles (analogous to results in
Chapter 3) to solutions and subsolutions of quasilinear equations. We mention in par-
ticular apriori bounds for solutions of Qu ⩾ 0 (= 0), where Q is a divergence form
operator satisfying certain structure conditions more general than ellipticity (Theorem
10.9).
Chapter 11 provides the basic framework for the solution of the Dirichlet problem
in the following chapters. We are concerned principally with classical solutions, and the
equations may be uniformly or non-uniformly elliptic. Under suitable general hypotheses
any globally smooth solution u of the boundary value problem for Qu = 0 in a domain
Ω with smooth boundary can be viewed as a xed point, u = T u, of a compact operator
T from C
1,α
(
¯
Ω) to C
1,α
(
¯
Ω) for any α ∈ (0, 1). In the applications the function dened
by T u,for any u ∈ C
1,α
(
¯
Ω), is the unique solution of the linear problem obtained by
inserting u into the coecients of Q. The Leray-Schauder xed point theorem (proved
in Chapter 11) then implies the existence of a solution of the boundary value problem
provided an apriori bound in C
1,α
(
¯
Ω), can be established for the solutions of a related
continuous family of equations u = T (u; σ), 0 ⩽ σ ⩽ 1, where T (u; 1) = T u (Theorems
11.4, 11.6). The establishment of such bounds for certain broad classes of Dirichlet
problems is the object of Chapters 13-15.
The general procedure for obtaining the required apriori bound for possible solutions
u is a four-step process involving successive estimation of sup
Ω
|u|, sup
∂Ω
|Du|, sup
Ω
|Du| and
∥u∥
C
1,α
(
¯
Ω)
for some α > 0. Each of these estimates presupposes the preceding ones and
the fnal bound on ∥u∥
C
1,α
(
¯
Ω)
completes the existence proof based on the Leray-Schauder
theorem.
As already observed, bounds on sup
Ω
|u| are discussed in Chapter 10. In the later
chapters this bound is either assumed in the hypotheses or is implied by properties of
the equation.
Equations in two variables (Chapter 12) occupy a special place in the theory This is
due in part to the distinctive methods that have been developed for them and also to
the results, some of which have no counterpart for equations in more than two variables.
The method of quasiconformal mappings and arguments based on divergence structure
equations (cf. Chapter 11) are both applicable to equations in two variables and yield
relatively easily the desired
C
1,α
apriori estimates, from which a solution of the Dirichlet
problem follows readily.
12
Section 1.1: Summary
Of particular interest is the fact that solutions of uniformly elliptic linear equations in
two variables satisfy an apriori C
1,α
estimate depending only on the ellipticity constants
and bounds on the coecients, without any regularity assumptions (Theorem 12.4).
Such a C
1,α
estimate, or even the existence of a gradient bound under the same general
conditions is unknown for equations in more than two variables. Another special feature
of the two-dimensional theory is the existence ofan apriori C
1
bound |Du| ⩽ K for
solutions of arbitrary elliptic equations
au
xx
+ 2bu
xy
+ cu
yy
= 0, (1.7)
where u is continuous on the closure of a bounded convex domain Ω and has boundary
value φ on ∂Ω satisfying a bounded slope(or three-point) condition with constant K.
This classical result, usually based on a theorem of Radó on saddle surfaces, is given
an elementary proof in Lemma 12.6. The stated gradient bound, which is valid for all
solutions u of the general quasilinear equation (1.7) in which a = a(x, y, u, u
x
, u
y
), etc.,
and such that u = φ on ∂Ω, reduces this Dirichlet problem to the case of uniformly
elliptic equations treated in Theorem 12.5. In Theorem 12.7 we obtain a solution of the
general Dirichlet problem for (1.7), assuming local Hölder continuity of the coecients
and a bounded slope condition for the boundary data (without further smoothness
restrictions on the data).
Chapters 13, 14 and 15 are devoted to the derivation of the gradient estimates
involved in the existence procedure described above. In Chapter 13, we prove the fun-
damental results of Ladyzhenskaya and Ural’tseva on Hölder estimates of derivatives of
elliptic quasilinear equations. In Chapter 14 we study the estimation of the gradient
of solutions of elliptic quasilinear equations on the boundary. After considering gen-
eral and convex domains, we give an account of the theory of Serrin which associates
generalized boundary curvature conditions with the solvability of the Dirichlet problem.
In particular, we are able to conclude from the results of Chapters 11, 13 and 14 the
Jenkins and Serrin criterion for solvability of the Dirichlet problem for the minimal sur-
face equation, namely, that this problem is solvable for smooth domains and arbitrary
smooth boundary values if and only if the mean curvature of the boundary (with respect
to the inner normal) is nonnegative at every point (Theorem 14.14).
Global and interior gradient bounds for solutions u of quasilinear equations are estab-
lished in Chapter 15. Following a renement of an old procedure of Bernstein we derive
estimates for sup
Ω
|Du| in terms of sup
∂Ω
|Du| for classes of equations that include both
uniformly elliptic equations satisfying natural growth conditions and equations sharing
common structural properties with the prescribed mean curvature equation (Theorem
15.2). A variant of our approach yields interior gradient estimates for a more restricted
class of equations (Theorem 15.3). We also consider uniformly and non-uniformly el-
13
Chapter 1: Introduction
liptic equations in divergence form (Theorems 15.6, 15.7 and 15.8), in which cases, by
appropriate test function arguments, we deduce gradient estimates under dierent types
of coecient conditions than in the general case. We conclude Chapter 15 with a selec-
tion of existence theorems, chosen to illustrate the scope of the theory. These theorems
are all obtained by various combinations of the apriori estimates in Chapters 10, 14 and
15 and a judicious choice of a related family of problems to which Theorem ll.8 can be
applied.
In Chapter l6. we concentrate on the prescribed mean curvature equation and derive
an interior gradient bound (Theorem 16.5) thereby enabling us to deduce existence
theorems for the Dirichlet problem when only continuous boundary values are assigned
(Theorems 16.8. 16.10). We also consider a family of equations in two variables. which
in a certain sense bear the same relationship to the prescribed mean curvature equation
as the uniformly elliptic equations of Chapter 12 bear to Laplace’s equation. Indeed. by
means of a generalized notion of quasiconformal mapping. we derive interior estimates
for frst and second derivatives. The second derivative estimates provide a generalization
of a well known curvature estimate of Heinz for solutions of the minimal surface equation
(Theorem 16.20) and moreover, imply an extension of the famous result of Bernstein
that entire solutions of the minimal surface equation in two variables must be linear
(Corollary 16.19). However, perhaps the striking feature of Theorems 16.5 and 16.20 is
the approach. Rather than working in the domain Ω, we work on the hypersurface S
given by the graph of the solution u and exploit various relations between the tangential
gradient and Laplacian operators on
S
and the mean curvature of
S
. We have also added
to the present edition a new nal chapter. Chapter 17 is devoted to fully nonlinear
elliptic equations, which incorporates recent work on equations of Monge-Ampère and
Bellman-Pucci type. These are equations of the general form
F [u] = F (x, u, Du, D
2
u) = 0 (1.8)
and include linear and quasilinear equations of the forms (1.1) and (1.2) as special
cases. The function F is dened for (x, z, p, r) ∈ Ω × R × R
n
× R
n
where R
n×n
denotes
the linear space of real symmetric n × n matrices. Equation (1.8) is elliptic when
the derivative F
r
is a positive denite matrix. The method of continuity (Theorem
17.8) reduces the solvability of the Dirichlet problem for (1.8) to the establishment of
C
2,α
(
¯
Ω) estimates, for some α > 0; that is, in addition to the rst derivative estimation
required for the quasilinear case, we need second derivative estimates for fully nonlinear
equations. Such estimates are established for equations in two variables (Theorems 17.9,
17.10), uniformly elliptic equations (Theorems 17.14, 17.15) and equations of Monge-
Ampère type (Theorem 17.19, 17.20, 17.26), yielding, in particular, recent results on the
solvability of the Dirichlet problem for uniformly elliptic equations by Evans, Krylov and
Lions (in Theorem 17.17, 17.18), and for equations of Monge-Ampère type by Krylov,
14
Section 1.2: Further Remarks
and Caarelli, Nirenberg and Spruck (in Theorem 17.23).
We conclude this summary with some guides to the reader. The material is not in
strict logical order. Thus the theory of Poisson’s equation (Chapter 4) would normally
follow Laplace’s equation (Chapter 2). However, the elementary character of the results
on the maximum principle(Chapter 3)and the opportunity for the reader to meet early
some general problems with variable coecients recommends its insertion after Chapter
2. In fact, the general maximum principle is not used until the existence theory of
Chapter 6. The basic material on functional analysis (Chapter 5) is needed in only a
minor way for the Schauder theory: the contraction mapping principle and the basic
concepts of Banach spaces suce, except for the proof of the alternative in Theorem
6.15. For applications to nonlinear problems in Part II it is sucient to know the results
of Section 1-3 of Chapter 6. Depending on the reader’s interests, it may be preferable to
study the linear theory by starting directly with L
2
theory in Chapter 8; this assumes
the preliminary material on functional analysis (Chapter 5)and on the calculus of weakly
dierentiable functiòns (Chapter 7).The Harnack inequalities and Hölder estimates in
the regularity theory of Chapter 8 are not applied until Chapter 13.
The theory of quasilinear equations in two variables (Chapter 12) is essentially inde-
pendent of Chapters 7-ll and can be read following Chapter 6 provided one assumes the
Schauder fxed point theorem (Theorem ll.l). The method of quasiconformal mappings
is met again in Chapter I6 but otherwise the remaining chapters are independent of
Chapter l’2. Accordingly, after the basic outline of the nonlinear theory in Chapter ll
the reader can proceed directly to the n-variable theory in Chapters 13-17. Chapter 16
is largely independent of Chapters 13-15. Chapters 6 and 9 are sucient preparation
for most of Chapter 17.
1.2 Further Remarks
Beyond the assumption of basic real analysis and linear algebra the material in this
work is almost entirely self-contained. Thus, much of the preliminary development of po-
tential theory and functional analysis, as well as results on Sobolev spaces and xed point
theorems, will be familiar to many readers, although the proof of the Leray-Schauder
theorem without topological degree in Theorem 11.6 is probably not so well known. A
number of well established auxiliary results, such as the interpolation inequalities and
extension lemmas of Chapter 6, are proved for the sake of completeness.
There is substantial overlap with the monographs of Ladyzhenskaya and Uraltseva
[LU 4] and Morrey [MY 5]. This book diers from the former in some of the analytical
techniques and in the emphasis on non-uniformly elliptic equations in the nonlinear
theory; it diers from the latter in not being directly concerned with variational problems
and methods. The present work also includes material developed since the publication
15
Chapter 1: Introduction
of those books. On the other hand, it is much more limited in various ways. Among
the topics not included are systems of equations, semilinear equations, the theory of
monotone operators, and aspects of the subject based on geometric measure theory.
In a subject that is often quite technical we have not always strived for the greatest
generality, especially with respect to the modulus of continuity, estimates, integral con-
ditions, and the like. We have instead conned ourselves to conditions determined by
power functions: for example, Hölder continuity rather than Dini continuity, L
p
spaces
in Chapter 8 rather than Orlicz spaces, structure conditions in terms of powers of |p|
rather than more general functions of |p|, etc. By suitable modication of the proofs the
reader will usually be able to supply the appropriate generalizations.
Historical material and bibliographical references are discussed primarily in the Notes
at the end of the chapters. These are not intended to be complete but rather to sup-
plement the text and place it in better perspective. A much more extensive survey of
the literature until 1968 is contained in Miranda [MR 2]. The problems attached to the
chapters are also intended to supplement the text; hopefully they will be useful exercises
for the reader.
1.3 Basic Notation
R
n
: Euclidean n-space, n ⩾ 2, with points x = (x
1
, . . . , x
n
), x
i
∈ R (real numbers);
|x| = (
x
2
i
)
1/2
; if b = (b
1
, . . . , b
n
) is an ordered n-tuple, then |b| = (
b
2
i
)
1/2
.
R
n
+
: half-space in R
n
= {x ∈ R
n
|x
n
> 0}.
∂S: boundary of the point set S;
¯
S = closure of S = S ∪ ∂S.
S − S
′
: {x ∈ S|x /∈ S
′
}.
S
′
⊂⊂ S: S
′
has compact closure in S; S
′
is strictly contained in S.
Ω: a proper open subset of R
n
, not necessarily bounded; Ω is a domain if it is also
connected; |Ω| = volume of Ω.
B(y): a ball in R
n
with center y; B
r
(y) is the open ball of radius r centered at y.
ω
n
: volume of unit ball in R
n
=
2π
n/2
nΓ(n/2)
.
D
i
u = ∂u/∂x
i
, D
ij
u = ∂
2
u/∂x
i
∂x
j
, etc.; Du = (D
1
u, . . . , D
n
u) = gradient of u;
D
2
u = [D
ij
u] = Hessian matrix of second derivatives D
ij
u, i, j = 1, 2, . . . , n.
β = (β
1
, . . . , β
n
), β
i
= integer ⩾ 0, with |β| =
β
i
, is a multi-index; we dene
D
β
u =
∂
|β|
u
∂x
β
1
1
···∂x
β
n
n
C
0
(Ω) (C
0
(
¯
Ω)): the set of continuous functions on Ω (
¯
Ω).
C
k
(Ω): the set of functions having all derivatives of order ⩽ k continuous in Ω (k =
integer ⩾ 0 or k = ∞).
C
k
(
¯
Ω): the set of functions in C
k
(Ω) all of whose derivatives of order ⩽ k have
continuous extensions to
¯
Ω.
supp u: the support of u, the closure of the set on which u = 0.
16
Section 1.3: Basic Notation
C
k
0
(Ω): the set of functions in C
k
(Ω) with compact support in Ω.
C = C(∗, . . . , ∗) denotes a constant depending only on the quantities appearing in
parentheses. In a given context, the same letter C will (generally) be used to denote
dierent constants depending on the same set of arguments.
17