Francoise J.-P., Naber G.L., Tsun T.S. (editors) Encyclopedia of Mathematical Physics
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Further Reading
Abers ES and Lee BW (1973) Gauge theories. Physics reports
C9: 1–141.
Bona M, Ciuchina M, and Franco E et al. (2005) The 2004 UTfit
collaboration report on the status of the unitarity triangle in the
standard model. Journal of High Energy Physics 0507: 028–059
(hep-ph/0501199).
CharlesJ,HockerA,LackerHet al. (2005) CP violation and the
CKM matrix: assesing the impact of the asymmetric B factories.
European Physical Journal C 41: 1–131 (hep-ph/0406184).
Erler J and Langacker P (2004) Electroweak Model and constraints
on new physics. In: Eidelman S, Hayes KG, Olive KA et al. (ed.)
Review of Particle Physics, Particle Data Group, Physics Letters
B vol. 592, pp. 1–1109, (http://pdglbl.gov/2005).
Fukugita M and Yanagida T (2003) Physics of Neutrinos and
Applications to Astrophysics. Berlin: Springer.
Gilman FJ, Kleinknecht K, and Renk B The Cabibbo–Kobayashi–
Maskawa quark-mixing matrix. In Eidelman S, Hayes KG,
Olive KA et al. (eds.) PDG Review of Particle Physics, Particle
Data Group, Physics Letters B, vol. 592, pp. 1–1109 (http://
pdg.lbl.gov/2005).
Glashow SL, Iliopoulos J, and Maiani L (1970) Weak interactions with
Lepton–Hadron symmetry. Physical Review D 2: 1285–1292.
Kobayashi M and Maskawa T (1972) CP violation in renormaliz-
able theory of weak interaction. Progress in Theoretical
Physics 49: 652–657.
Salam A (1968) Weak and electromagnetic interactions. In:
Svartholm N (ed.) Elementary Particle Theory,p.367.,
Lerum, Sweden: Almqvist Forlag AB.
Shirai J (2005) Neutrino experiments: review of recent results.
Nuclear Physics B (Proc. Suppl.) 144: 286–296.
Taylor JC (1976) Gauge Theories of Weak Interactions. Cam-
bridge: Cambridge University Press.
Taylor JC (ed.) (2001) Gauge Theories in the Twentieth Century.
London: Imperial College Press.
’t Hooft G (1971) Renormalizable Lagrangians for massive Yang–
Mills fields. Nuclear Physics B35: 167–188.
’t Hooft G and Veltman M (1973) Diagrammar. CERN Yellow
Report 73–9.
Weinberg S (1967) Theory of Leptons. Physical Review Letters
19: 1264–1266.
Elliptic Differential Equations: Linear Theory
C Amrouche, Universite
´
de Pau et des Pays
de l’Adour, Pau, France
M Krbec and S
ˇ
Nec
ˇ
asova´ , Academy of Sciences,
Prague, Czech Republic
B Lucquin-Desreux, Universite
´
P.-M. Curie, Paris VI,
Paris, France
ª 2006 Elsevier Ltd. All rights reserved.
Introduction
Motivation: A Model Problem
Many physical problems can be modeled by partial
differential equations. Let us consider, for example,
the case of an elastic membrane , with fixed
boundary , subject to pressure forces f. The vertical
membrane displacement is represented by a real-
valued function u, which solves the equation
uðxÞ¼f ðxÞ; x ¼ðx
1
; x
2
Þ2 ½1
where the Laplace operator is defined, in two
dimensions, by
u ¼
@
2
u
@x
2
1
þ
@
2
u
@x
2
2
As the membrane is glued to the curve , u satisfies
the condition
uðxÞ¼0; x 2 ½2
The system [1]–[2] is the homogeneous Dirichlet
problem for the Laplac e operator. It enters the more
general framework of (linear) elliptic boundary
value problems, which consist of a (linear) partial
differential equation (in the example above, of order
two: the highest order in the derivatives) insid e an
open set of the whole space R
N
, satisfying some
‘‘elliptic’’ property, completed by (linear ) conditions
on the boundary of , called ‘‘boundary condi-
tions.’’ In the sequel, we only consider the linear
case.
Our aim is to answer the following questions: does
this problem admit a solution? in which space? is this
solution unique? does it depend continuously on the
given data f ?Incaseofpositiveanswers,wesaythat
the problem is ‘ ‘well posed’’ in the Hadamard sense.
But other questions can also be raised, such as the sign
of the solution, for example, or its regularity. We give a
full survey of linear elliptic problems in a bounded or in
an exterior domain with a sufficiently smooth bound-
ary and in the whole space. In the general theory of the
elliptic problem, we consider only smooth coefficients.
We survey the standard theory, which can be found in
the several well-known monographs of the 1960s. The
new trends in the investigation of the elliptic problems
is to consider more general domains with nonsmooth
boundaries and nonsmooth coefficients. On the other
hand, the regularity results for elliptic systems have not
been improved during last 30 years. New trends also
require employment of more general function spaces
and more general functional background.
The number of references (see ‘‘Further reading’’
section) is strictly limited here; we list only some of
the most important publications. The basic facts can
usually be found in more places and sometimes we
do not mention the particular reference. Am ong the
216 Elliptic Differential Equations: Linear Theory
very basic references are Friedman (1969), Gilbarg
and Trudinger (1977), Dautray and Lions (1988),
Ho¨ rmander (1964), Ladyzhenskaya and Uraltseva
(1968), Lions and Magenes (1968), Renardy and
Rogers (1992),andWeinberger (1965); of course,
there are many others.
The Method
To answer the above questions, we generally use, for
such elliptic problems, an approach based on what is
called a ‘‘variational formulation’’ (see the section
‘‘Variationalapproach’’):theboundary-valueproblem
is first transformed into a variational problem of lower
order, which is solved in a Hilbertian frame with help of
the Lax–Milgram theorem (based on the representation
theorem). All questions are then solved (e.g., existence,
uniqueness, continuity in terms of the data, regularity).
But this variational formalism does not necessary allow
to treat all the situations and it is limited to the
Hilbertian case. Other strategies can then be developed,
based on aprioriestimates and duality arguments for
the existence problem, or maximum principle for the
question of unicity. Without forgetting the particular
cases where an explicit Green kernel is computable
(e.g., the Laplacian operator in the whole space case).
Moreover, the study of linear elliptic equations is
directly linked to the background of function spaces. It is
the reason why we first deal with Sobolev spaces – both
of the integer and fractional order and we survey their
basic properties, imbedding and trace theorems. We pay
attention to the Riesz and Bessel potentials and we
define weighted Sobolev spaces important in the context
of unbounded opens. Second, we present the variational
approach and the Lax–Milgram theorem as a key point
to solve a large class of boundary-value problems. We
give examples: the Dirichlet and Neumann problems for
the Poisson equation, the Newton problem for more
general second-order operators; we also investigate
mixed boundary conditions and present an example of
a problem of fourth order. Then, we briefly present the
arguments for studying general elliptic problems and
concentrate on second-order elliptic problems; we recall
the weak and strong maximum principle, formulate the
Fredholm alternative and tackle the regularity questions.
Moreover, we are interested in the existence and
uniqueness of solution of the Laplace equation in the
whole space and in exterior opens. Finally, we present
some particular examples arising from physical pro-
blems, either in fluid mechanics (the Stokes system) or in
elasticity.
Sobolev and Other Types of Spaces
Throughout, R
N
will generally be an open
subset of the N-dimensional Euclidean space R
N
.
A domain will be an open and connected subset of
R
N
. We shall use standard notations for the spaces
L
p
(), C
1
(), etc., and their norms. Let us agree
that C
k,r
(), k 2 N, r 2 (0, 1), denote the space of
functions f in C
k
(), whose derivatives D
f , =
(
1
, ...,
N
) 2 N
N
, of order jj=
P
N
i = 1
i
= k are
all r-Ho¨ lder continuous. In the notations for some of
these spaces, by
we mean that the functions have
the corresponding property on and that they can be
continuously extended to
.
Let us recall several fundamental concepts. The
space D() of the test functions in consists of
all infinitely differentiable ’ with a compact
support in . A locally convex topology can be
introduced here. The elements of the dual space
D
0
() are called the distributions. If f 2 L
1
loc
()
(i.e., f 2 L
1
(K) for all compact subsets K of ),
then f is a regular distribution; the duality is
represented by
R
f (x)’(x)dx.Iff 2D
0
(), we
define the distributional or the weak derivative
D
of f as the distribution ’ 7!(1)
jj
hf , D
’i.
Plainly, if f 2 L
1
loc
has ‘‘classical’’ partial deriva-
tives in L
1
loc
, then it coincides with the correspond-
ing weak derivative.
If =R
N
, it is sometimes more suitable to work
with the tempered distributions. The role of D()is
played by the space S(R
N
)ofC
1
-functions
with finite pseudonorms sup j D
f (x)j(1 þjxj)
k
, jj,
k = 0, 1, 2, .... Recall that the Fourier transform F
maps S(R
N
) into itself and the same is true for the
space of the tempered distributions S
0
(R
N
).
Sobolev Spaces of Positive Order
The Sobolev space W
k, p
(), 1 p 1, k 2 N,is
the space of all f 2 L
p
() whose weak derivatives up
to order k are regular distributions belonging to
L
p
(); in W
k, p
() we introduce the norm
kf k
W
k;p
ðÞ
¼
X
jjk
Z
jD
f ðxÞj
p
dx
0
@
1
A
1=p
½3
when p < 1 and max
jjk
sup ess
x2
jD
f (x)j if
p = 1. The space W
k,p
() is a Banach space,
separable for p < 1 and reflexive for 1 < p < 1;
it is a Hilbert space for p = 2, more simply denoted
H
m
(). In the following, we shall consider only the
range p 2 (1, 1).
The link with the classical derivatives is given by
this well-known fact: a function f belongs to
W
1,p
() if and only if it is a.e. equal to a function
~
u, absolutely continuous on almost all line segments
in parallel to the coordinate axes, whose
(classical) derivatives belong to L
p
() (the Beppo–
Levi theorem).
Elliptic Differential Equations: Linear Theory 217
For 1 < p < 1 and noninteger s > 0 the Sobolev
space W
s,p
() of order s is defined as the space of all
f with the finite norm
kf k
W
s;p
ðÞ
¼
kf k
p
W
½s
p
ðÞ
þ
X
jj¼½s
Z
Z
jD
f ðxÞD
f ðyÞj
p
jx yj
Nþpðs½sÞ
!
1=p
where [s] is the integer part of s (for details, see, e.g.,
Adams and Fournier (2003) and Ziemer (1989)).
Imbedding Theorems
One of the most useful and important features of the
functions in Sobolev spaces is an improvement of
their integrability properties and the compactness of
various imbeddings. Theorems of this type were first
proved by Sobolev and Kondrashev. Let us agree that
the symbols ,! and ,!,! stand for an imbedding and
for a compact imbedding, respectively.
Theorem 1 Let be a Lipschitz open. Then
(i) If sp < N, then W
s,p
() ,!L
p
() with
p
= Np=(N ps)(the Sobolev exponent). If
jj < 1, then the target space is any L
r
()
with 0 < r p
.
If is bounded, then W
s,p
() ,!,!L
q
() for all
1 q < p
.
(ii) If sp > N, then W
jþs,p
(),!C
j
() for j = 0,1,... .
If has the Lipschitz boundary, then W
jþs, p
() ,!C
j,
(
) for j ¼ 0,1, ... and ¼ s N=p:
If sp > N, then W
jþs,p
() ,!,!C
j
(), j = 0,1, ...
and W
jþs,p
() ,!,!W
j
q
() for all 1 q 1. If,
moreover, has the Lipschitz boundary, then the
target space can be replaced by C
j,
(
) provided
sp > N > (s 1)pand0 <<s N = p.
Note that if the imbedding W
s,p
() ,!L
q
()is
compact for some q p, then jj < 1. Moreover,
if lim sup
r !1
j{x 2 ; r jxj < r þ 1}j > 0, then
W
s,p
() ,!L
q
() cannot be compact.
Traces and Sobolev Spaces of Negative Order
Let s > 0andlet be, for simplicity, a bounded open
subset of R
N
with boundary of class C
[s],1
. Then
with the help of local coordinates, we can define
Sobolev spaces W
s,p
() (also denoted H
s
()forp = 2)
on =@ (see, e.g., Nec
ˇ
as (1967) and Adams and
Fournier (2003) for details). If f 2 C(
), then f
j
has
sense. Introducing the space D(
)ofrestrictionsin
of functions in D(R
N
), one can show that if f 2D(
),
we have kf
j
k
W
11=p, p
()
Ckf k
W
1, p
()
so that, in view
of the density of D(
)inW
1, p
(), the restriction
of f to can be uniquely extended to the whole
W
1, p
(). The result is the bounded trace operator
0
: W
1,p
() !W
11=p,p
(). Moreover, every g 2
W
11=p,p
() can be extended to a (nonunique) function
f 2 W
1,p
() and this extension operator is bounded
with respect to the corresponding norms.
More generally, let us suppose is of class C
k1,1
and define the operator Tr
n
for any f 2D(
)by
Tr
n
f = (
0
f ,
1
f , ...,
k1
f ), where
j
f ðxÞ¼
@
j
f
@n
j
ðxÞ
¼
X
jj¼j
j!
!
ð@
f ðxÞ=@x
Þn
; x 2
is the jth-order derivative of f with respect to the
outer normal n at x 2 ; by density, this operator
can be uniquely extended to a continuous linear
mapping defined on the space W
k,p
(); moreover,
0
(W
k,p
()) = W
k1=p,p
().
The kernel of this mapping is the space
W
k,p
()
(denoted by H
k
0
() for p = 2), where
W
s,p
()is
defined as the closure of D()inW
s,p
()(s > 0). For
1 < p < 1, the following holds:
W
s,p
(R
N
) =
W
s,p
(R
N
),
W
s,p
() = W
s,p
() provided 0 < s 1 p.
If s < 0, then the space W
s,p
() is defined as the dual
to W
˚
s,p
0
(), where p
0
= p=(p 1) (see, e.g., Triebel
(1978, 2001)). Observe that, for an arbitrary ,a
function f 2 W
1,p
() has the zero trace if and only if
f (x)=dist(x, ) belongs to L
p
().
For p = 2, we simply denote by H
k
() the dual
space of H
k
0
(). In the case of bounded opens, we recall
the following useful Poincare´–Friedrichs inequality (for
simplicity, we state it here in the Hilbert frame):
Theorem 2 Let be bounded (at least in one
direction of the space). Then there exists a positive
constant C
P
() such that
kvk
L
2
ðÞ
C
P
ðÞkrvk
½L
2
ðÞ
N
for all v 2 H
1
0
ðÞ½4
The Whole-Space Case: Riesz and
Bessel Potentials
The Riesz potentials I
naturally occur when one
defines the formal powers of the Laplace operator .
Namely, if f 2S(R
N
)and>0, then
FðÞ
=2
f
hi
ðÞ= jj
Ff ðÞ:
This can be taken formally as a definition of the
Riesz potential I
on S
0
(R
N
),
I
f ð:Þ¼F
1
jj
Ff ðÞ½ð:Þ
218 Elliptic Differential Equations: Linear Theory
for any 2 R.If0<<N,thenI
f (x) = (I
f )(x),
where I
is the inverse Fourier transform of jj
,
I
ðxÞ¼C
jxj
N
C
¼ ðN Þ=2ðÞ
N=2
2
ð=2Þ
1
where is the Gamma function and I
is the Riesz
kernel. The following formula is also true:
I
ðxÞ¼C
Z
1
0
t
ðNÞ= 2
e
jxj
2
=t
dt
t
Recall that every f 2S(R
N
) can be represented as
the Riesz potential I
g of a suitable function g 2
S(R
N
), namely g = ()
=2
f ; we get the representa-
tion formula
f ðxÞ¼I
gðxÞ
¼ C
Z
R
N
gðyÞ
jx yj
N
dy
The standard density argument implies then an
appropriate statement for functions in W
k,p
(R
N
)
with an integer k and for the Bessel potential spaces
H
,p
(R
N
) – see below for their definition. The
original Sobolev imbedding theorem comes from
the combination of this representation and the basic
continuity property of I
, p < N,
I
: L
p
ðR
N
Þ!L
q
ðR
N
Þ;
1
q
¼
1
p
N
To get an isomorphic representation of a Bessel
potential space (of a Sobolev space with positive
integer smoothness in particular) it is more convenient
to consider the Bessel potentials (of order 2 R),
G
f ðxÞ¼ðG
f ÞðxÞ
¼F
1
½1 þjj
2
=2
Ff ðÞ
ðxÞ
(with a slight abuse of the notations); the following
formula for the Bessel kernel G
is well known:
G
ðxÞ¼c
1
Z
1
0
t
ðNÞ= 2
e
ðjxj
2
=tÞðt=4Þ
dt
t
(cf. the analogous formula for I
), where c
=
(4)
=2
(=2). The kernels G
can alternatively be
expressed with help of Bessel or Macdonald functions.
Now we can define the Bessel potential spaces.
For s 2 R and 1 < p < 1, let H
s,p
(R
N
) be the space
of all f 2S
0
(R
N
) with the finite norm
kf k
H
s;p
ðR
N
Þ
¼
Z
R
N
F
1
ð1 þjj
2
Þ
s=2
Ff ðÞ
p
d
1=p
In other words, the spaces H
s,p
(R
N
) are isomorphic
copies of L
p
(R
N
).
For k = 0, 1, 2, ..., plainly H
k,2
(R
N
) = W
k,2
(R
N
)
by virtue of the Plancherel theorem. But it is true
also for integer s and general 1 < p < 1 (see, e.g.,
Triebel (1978)).
Remark 3 Much more comprehensive theory of
general Besov and Lizorkin–Triebel spaces in R
N
has
been established in the last decades, relying on the the
Littlewood–Paley theory. Spaces on opens can be
defined as restrictions of functions in the corresponding
space on the whole R
N
, allowing to derive their
properties from those valid for functions on R
N
.The
justification for that are extension theorems. In parti-
cular, there exists a universal extension operator for the
Lipschitz open, working for all the spaces mentioned up
to now. We refer to Triebel (1978, 2001).
Unbounded Opens and Weighted Spaces
The study of the elliptic problems in unbounded
opens is usually carried out with use of suitable
Sobolev weighted space. The Poisson equation
u ¼ f in R
N
; N 2 ½5
is the typical example; the Poincare´ inequality [4] is
not true here and it is suitable to introduce Sobolev
spaces with weights.
Let m 2 N,1< p < 1, 2 R, k = m N=p
if N=p þ 2 {1, ..., m} and k = 1 elsewhere. For
an open R
N
, we define
W
m;p
ðÞ¼
n
v 2D
0
ðÞ; 0 jjk;
mjj
ðlog Þ
1
D
u 2 L
p
ðÞ;
k þ 1 jjm;
mþjj
D
u 2 L
p
ðÞ
o
where (x) = (1 þjxj
2
)
1=2
: Note that W
m,p
is a
reflexive Banach space for the norm k.k
W
m,p
defined by
kuk
p
W
m;p
¼
X
0jjk
k
mþjj
ðlog Þ
1
D
uk
p
L
p
ðÞ
þ
X
kþ1jjm
k
mþjj
D
uk
p
L
p
ðÞ
We also introduce the following seminorm:
juj
W
m;p
¼
X
jj¼m
k
D
uk
p
L
p
ðÞ
0
@
1
A
1=p
Let
W
m;p
ðÞ¼ v 2 W
m;p
;
0
ðvÞ¼
m1
ðvÞ¼0
g
Elliptic Differential Equations: Linear Theory 219
If is a Lipschitz domain, then
W
m,p
()is
the closure of D()inW
m,p
(), while D(
)is
dense in W
m,p
(). We denote by W
m,p
0
() the dual
of
W
m,p
()(p
0
= p=(p 1)). We note that these
spaces also contain polynomials,
P
j
W
m;p
ðÞ
,
j ¼
m
N
p
if
N
p
þ =2Z
j ¼ m
N
p
elsewhere
8
>
>
<
>
>
:
where [s] is the integer part of s and P
[s]
= {0} if
[s] < 0. The fundamental property of functions
belonging to these spaces is that they satisfy the
Poincare´ weighted inequality. An open is an
exterior domain if it is the complement of a closure
of a bounded domain in R
N
.
Theorem 4 Suppose that is an exterior domain
or =R
N
þ
or =R
N
. Then
(i) the seminorm jj
W
m,p
()
is a norm on W
m,p
()=P
j
,
equivalent to the quotient norm with
j
0
= min (m 1, j);
(ii) the seminorm jj
W
m,p
()
is equivalent to the full
norm on
W
m, p
().
Variational Approach
Let us first describe the method on the model problem
[1]–[2], supposing f 2 L
2
()and bounded. We first
suppose that this problem admits a sufficiently smooth
function u.Letv be any arbitrary (smooth) function;
we multiply eqn [1] by v(x) and integrate with respect
to x over ;thisgives
Z
ðuvÞðxÞdx ¼
Z
ðfvÞðxÞdx
Using the following Green’s formula (d(x) denotes
the measure on =@ and @u(x)=@n = ru(x) n(x),
where n(x) is the unit normal at point x of
oriented towards the exterior of ):
Z
ðuvÞðxÞdx ¼
Z
ðru rvÞðxÞdx
þ
Z
@u
@n
v
ðÞd ½6
we get, since vj
= 0: A(u, v) = L(v), where we
have set
Aðu; vÞ¼
Z
ruðxÞrvðxÞdx
LðvÞ¼
Z
f ðxÞv ðxÞdx
½7
The idea is to study in fact this new problem
(showing first its equivalence with the boundary-
value problem), noting that it makes sense for far
less regular functions u, v (and also f ), in fact u, v 2
H
1
0
() (and f 2 H
1
()).
The Lax–Milgram Theorem
The general form of a variational problem is
to find u 2 V such that
Aðu; vÞ¼LðvÞ for all v 2 V ½8
where V is a Hilbert space, A a bilinear continuous
form defined on V V and L a linear continuous form
defined on V. We say, moreover, that A is V-elliptic if
there exists a positive constant such that
Aðu; u Þkuk
2
V
for all u 2 V ½9
The following theorem is due to Lax and Milgram.
Theorem 5 Let V be a Hilbert space. We suppose
that A is a bilinear continuous form on V V which
is V-elliptic and that L is a linear continuous form
on V. Then the variational problem [8] has a unique
solution u on V. Moreover, if A is symmetric, uis
characterized as the minimum value on V of the
quadratic functional E defined by
for all v 2 V; EðvÞ¼
1
2
Aðv; vÞLðv Þ½10
Remark 6
(i) We have the following ‘‘energy estimate’’:
kuk
V
1
kLk
V
0
where V
0
is the dual space to V.In
the particular case of our model problem, this
inequality shows the continuity of the solution u 2
H
1
0
() with respect to the data f 2 L
2
() (that can
be weakened by choosing f 2 H
1
()).
(ii) Theorem 5 can be extended to sesquilinear
continuous forms A defined on V V; such a form
is called V-elliptic if there exists a positive constant
such that
Re Aðu; uÞkuk
2
V
for all u 2 V ½11
(iii) Denoting by A the linear operator defined on
the space V by A(u, v) = hAu, vi
V
0
, V
, for all v 2 V,the
Lax–Milgram theorem shows that A is an isomorph-
ism from V onto its dual space V
0
, and the problem [8]
is equivalent to solving the equation Au = L.
(iv) Let us make some remarks concerning the
numerical aspects. First, this variational formulation is
the starting point of the well-known finite element
method: the idea is to compute a solution of an
approximate variational problem stated on a finite
subspace of V (leading to the resolution of a linear
220 Elliptic Differential Equations: Linear Theory
system), with a precise control of the error with the exact
solution u. Second, the equivalence with a minimization
problem allows the use of other numerical algorithms.
Let us now present some classical examples of
second-order elliptic problems than can be solved
with help of the variational theory.
The Dirichlet Problem for the Poisson Equation
We consider the problem on a bounded Lipschitz
open R
N
,
u ¼ f
u ¼ u
0
on ¼ @
½12
with u
0
2 H
1=2
(), so that there exists U
0
2 H
1
()
satisfying
0
(U
0
) = u
0
. The variational formulation
of problem [12] is
to find u 2 U
0
þ H
1
0
ðÞ such that
for all v 2 H
1
0
ðÞ; Aðu; vÞ¼LðvÞ½13
with A given by [7] and a more general L with f 2
H
1
(), defined by
LðvÞ¼hf ; vi
H
1
ðÞ;H
1
0
ðÞ
½14
The existence and uniqueness of a solution of [13]
follows from Theorem 5 (and Poincare´ inequality [4]).
Conversely, thanks to the density of D()inH
1
0
(), we
can show that u satisfies [12]. More precisely, we get:
Theorem 7 Let us suppose f 2 H
1
() and u
0
2
H
1=2
(); let U
0
2 H
1
() satisfy
0
(U
0
) = u
0
. Then the
boundary-value problem [12] has a unique solution u
such that u U
0
2 H
1
0
(). This is also the unique
solution of the variational problem [13]. Moreover,
there exists a positive constant C = C() such that
kuk
H
1
ðÞ
C kf k
H
1
ðÞ
þku
0
k
H
1=2
ðÞ
½15
which shows that u depends conti nuously on the
data f and u
0
.
Moreover, using techniques of Nirenberg’s differ-
ential quotients, we have the following regularity
result (see, e.g., Grisvard (1980)):
Theorem 8 Let us suppose that is a bounded
open subset of R
N
with a boundary of class C
1, 1
and
let f 2 L
2
(), u
0
2 H
3=2
(). Then u 2 H
2
() and
each equation in [12] is satisfied almost everywhere
(on for the first one and on for the boundary
condition). Mo reover, there exists a positive con-
stant C = C() such that
kuk
H
2
ðÞ
C½kf k
L
2
ðÞ
þkgk
H
3=2
ðÞ
½16
By induction, if the data are more regular, that is,
f 2 H
k
() and u
0
2 H
kþ3=2
() (with k 2 N), and if
is of class C
kþ1, 1
, we get u 2 H
kþ2
().
Remark 9 Let us point out the importance of the
open geometry. For example, if is a bounded
plane polygon, one can find u 2 H
1
0
() with u 2
C
1
(
), such that u =2H
1þ=w
(), where w is the
biggest value of the interior angles of the polygon. In
particular, if the polygon is not convex, the solution
of the Dirichlet problem [12] cannot be in H
2
().
The Neumann Problem for the Poisson Equation
We consider the problem (n is the unit outer normal
on )
u ¼ f in
@u
@n
¼ h on
½17
Setting E() = {v 2 H
1
(); v 2 L
2
()}, the space
D(
) is a dense subspace, and we have the following
Green formula for all u 2 E() and v 2 H
1
():
Z
uðxÞvðxÞdx
¼
Z
ruðxÞrv ðxÞdx þ
@u
@n
;
0
v
H
1=2
ðÞ;H
1=2
ðÞ
If u 2 H
1
() satisfies [17] with f 2 L
2
()andh 2
H
1=2
(), then for any function v 2 H
1
(), we have,
by virtue of the above Green formula,
Aðu; vÞ¼
~
LðvÞ
~
Lv ¼
Z
ðfvÞðxÞdx þhh;
0
vi
H
1=2
ðÞ;H
1=2
ðÞ
But, here the form A is not H
1
()-elliptic; in fact,
one can check that, if problem [17] has a solution,
then we have necessarily (take v = 1 above)
Z
f ðxÞdx þhh; 1i
H
1=2
ðÞ;H
1=2
ðÞ
¼ 0 ½18
Moreover, we note that if u is a solution, then
u þ C, where C is an arbitrary constant, is also a
solution. So the variational problem is not well
posed on H
1
(). It can, however, be solved in the
quotient space H
1
()=R, which is a Hilbert space
for the quotient norm
k
_
vk
H
1
ðÞ=R
¼ inf
k2R
kv þ kk
H
1
ðÞ
½19
but also for the seminorm v 7!jvj
H
1
()
=
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
A(v, v)
p
,
which is an equivalent norm on this quotient space
(see Nec
ˇ
as (1967)).
Elliptic Differential Equations: Linear Theory 221
Then, supposing that the data f and h satisfy the
‘‘compatibility condition’’ [18], we can apply the
Lax–Milgram theorem to the variational problem
to find
_
u 2 V such that
Að
_
u;
_
vÞ¼
~
Lð
_
vÞ for all
_
v 2 V ½20
with V = H
1
()=R. We get the following result (see,
e.g., Nec
ˇ
as (1967):
Theorem 10 Let us suppose that is connected
and that the data f 2 L
2
() and h 2 H
1=2
() satisfy
[18]. Then the variational problem [20] has a unique
solution
_
u in the space H
1
()=R and this solution is
continuous with respect to the data, that is, there
exists a positive constant C = C() such that
juj
H
1
ðÞ
C kf k
L
2
ðÞ
þkhk
H
1=2
ðÞ
for all u 2
_
u
Moreover, if is of class C
1,1
and if the data
satisfy f 2 L
2
(), g 2 H
1=2
(), then every u 2
_
u is
such that u 2 H
2
() and it satisfies each equation in
[17] almost everywhere.
Problem with Mixed Boundary Conditions
Here we consider more general boundary condi-
tions: the Dirichlet conditions on a closed subset
1
of =@, and the Neumann, or more generally the
‘‘Robin’’, conditions on the other part
2
=
1
.
We seek u such that (f 2 L
2
(), h 2 L
2
(
2
),
a 2 L
1
(
2
))
u ¼ f in
u ¼ 0on
1
au þ
@u
@n
¼ h on
2
½21
Let V = {v 2 H
1
();
0
v = 0on
1
}. Then [8] is the
variational formulation of this problem with
1. A(u,v) =
R
ru(x) rv(x)dx þ
R
2
(a
0
u
0
v)()d;
2. L(v) =
R
f (x)v(x)dx þ
R
2
(h
0
v)()d.
Supposing, for example, a 0, we get a unique
solution u 2 V for this variational problem by virtue
of the Lax–Milgram theorem. Moreover, if u 2
H
2
(), then u is the unique solution in H
2
() \ V
of the problem [21].
The Newton Problem for More General Operators
Let be a bounded open subset of R
n
. We now
consider more general second-order operators of the
form v 7!r.(Mrv) þ b rv þ cv, where b 2
[W
1,1
()]
N
, c 2 L
1
(), M is an N N square
matrix with entries M
ij
,andr(Mrv) stands for
X
N
i;j¼1
@
@x
i
M
ij
@u
@x
j
We also assume that there is a positive constant
M
such that
X
N
i;j¼1
M
ij
ðxÞ
i
j
M
X
N
i¼1
2
i
for a.e. x 2 and ¼ð
1
; ...;
N
Þ2R
N
For given data f 2 L
2
(), h 2 L
2
(), we look for a
solution u of the problem
r ðMruÞþb ru þ cu ¼ f in
au þ n ðMruÞ¼h on
½22
We assume that a 2 L
1
(). The variational formu-
lation of this problem is still [8], with V = H
1
()
and
Aðu; vÞ¼
Z
Mru rvdx
þ
Z
½b ru þcuvdx þ
Z
a
0
u
0
vd ½23
LðvÞ¼
Z
f ðxÞvðxÞdx þ
Z
ðh
0
vÞðÞd ½24
If the conditions
c
1
2
rb C
0
0 a.e. on
a þ
1
2
b C
1
0 a.e. on
are fulfilled, with (C
0
, C
1
) 6¼ (0, 0), then the bilinear
form A is V-elliptic and the Lax–Milgram theorem
applies.
A Biharmonic Problem
We consider the Dirichlet problem for the operator
of fourth order: (c 2 L
1
()):
2
u þ cu ¼ f in ½25
u ¼ u
0
on ;
@u
@n
¼ h on ½ 26
Theorem 11 Let us suppose that has a boundary
of class C
1,1
and that the data satisfy f 2 H
2
(), u
0
2
H
3=2
(), h 2 H
1=2
(). Let U
0
2 H
2
() be such that
0
(U
0
) = u
0
,
1
(U
0
) = h. Then, if c 0 a.e. in , the
boundary value problem [25]–[26] has a unique
222 Elliptic Differential Equations: Linear Theory
solution u such that u U
0
2 H
2
0
(), and u is also the
unique solution of the variational problem
to find u 2 U
0
þ H
2
0
ðÞ such that
Aðu; vÞ¼lðvÞ for all v 2 H
2
0
ðÞ½27
where l(v) = hf , vi
H
2
(), H
2
0
()
and
Aðu; vÞ¼
Z
uðxÞvðxÞdx þ
Z
ðcuvÞðxÞdx ½28
Moreover, there exists a positive constant C = C()
such that
kuk
H
2
ðÞ
C ½kf k
H
2
ðÞ
þku
0
k
H
3=2
ðÞ
þkhk
H
1=2
ðÞ
½29
which shows that u depends continuously upon the
data f, u
0
, and h.
Remark 12 The Hilbert space choice V is of crucial
importance for the V-ellipticity. In fact, let us
consider for example the problem [25], with
u ¼ 0on;
@u
@n
¼ 0on ½30
In fact, the associated bilinear form is not V-elliptic
for V = H
2
() but it is V-elliptic for V = {v 2 L
2
();
v 2 L
2
)}.
General Elliptic Problems
Here will be a bounded and sufficiently regular
open subset of R
N
. Let us consider a general linear
differential operator of the form
Aðx; DÞu ¼
X
jjl
a
ðxÞD
u; a
ðxÞ2C ½31
Setting A
0
(x, ) =
P
jj= l
a
(x)
, we say that the
operator A is elliptic at a point x if A
0
(x, ) 6¼ 0 for
all 2 R
N
{0}. One can show that, if N 3, l is
even, that is, l = 2m; the same result holds for N = 2
if the coefficients a
are real. Moreover, for N 3,
every elliptic operator is properly elliptic, in the
following sense: for any independent vectors ,
0
in
R
N
, the polynomial 7!A
0
(., þ
0
) has m roots
with positive imaginary part.
The aim here is to study boundary-value problems
of the following type:
Au ¼ f in ½32
B
j
u ¼ g
j
on ; j ¼ 0; ...; m 1; ½33
where A is properly elliptic on
, with sufficiently
regular coefficients, and the operators B
j
are bound-
ary operators, of order m
j
2m 1, that must
satisfy some compatibility conditions with respect to
the operator A (see Renardy and Rogers (1992) for
details; these conditions were introduced by Agmon,
Douglis, and Nirenberg). For example, A = (1)
m
m
and B
j
= @
j
=@n
j
is a convenient choice.
In order to show that problem [32]–[33] has a
solution u 2 H
2mþr
()(r 2 N), the idea is to show
that the operator P defined by u 7!P(u) =
(Au, B
0
u, ..., B
m1
u) is an index operator from
H
2mþr
() into G = H
r
()
m1
j = 0
H
2mþrm
j
1=2
()
and to express the compatibility conditions through
the adjoint problem.
We recall that a linear continuous operator P is
an index operator if
(1) dim Ker P < 1,andImP closed;
(2) codim Im P < 1.
Then the index (P) is given by (P) =
dim Ker P – codim Im P. We recall the following
Peetre’s theorem:
Theorem 13 Let E, F, and G be three reflexive
Banach spaces such that E ,!,!F, and P a linear
continuous operator from E to G. Then condition
(1) is equivalent to:‘‘there exis ts C 0, such that
for all u 2 E, we have kuk
E
C (kPuk
G
þkuk
F
).’’
Applying this theorem to our problem [32]–[33],
condition (1) results from a priori estimates of the
following type:
kuk
H
2mþr
ðÞ
C kPuk
G
þkuk
H
2mþr1
ðÞ
and condition (2) by similar a priori estimates for
the dual problem.
Second-Order Elliptic Problems
We consider a second-order differential operator of
the ‘‘divergence form’’
Au ¼
X
N
i;j¼1
ða
ij
ðxÞu
x
i
Þ
x
j
þ
X
N
i¼1
b
i
ðxÞu
x
i
þ cðxÞ u ½34
with given coefficient functions a
ij
, b
i
, c (i, j =
1, ..., N), and where we have used the notation
u
x
i
=
@u
@x
i
. Such operators are said uniformly strongly
elliptic in if there exists >0 such that
X
jij¼jjj¼1
a
ij
ðxÞ
i
j
jj
2
for all x 2 ;2 R
N
Remark 14 There exist elliptic problems for which
the associated variational problem does not necessa-
rily satisfy the ellipticity condition. Let us consider
Elliptic Differential Equations: Linear Theory 223
the following example, due to Seeley: let ={(r, ) 2
(,2) [0, 2]} and
A ¼ e
i
@
@
2
e
2i
1 þ
@
2
@r
2
One can check that, for all 2 C, the problem Au þ
u = f in and u = 0on admits nonzero solutions u
whicharegivenby(with such that
2
= )
u = sin r cos (e
i
)andu = sin r sin (e
i
)for 6¼ 0;
u = sin r and u = sin e
i
for = 0.
Most of the results concerning existence, unicity,
and regularity for second-order elliptic problems can
be established thanks to a maximum principle.
There exist different types of maximum principles,
which we now present.
Maximum Principle
Theorem 15 (Weak maximum principle). Let A be
a uniformly strongly elliptic operator of the form
[34] in a bounded open R
N
, with a
ij
, b
i
, c 2
L
1
() and c 0. Let u 2 C
2
() \ C(
) and
Au 0 ½resp: Au 0 in
Then
inf
u inf
@
u
½resp: sup
u sup
@
u
þ
where u
þ
= max (u,0) and u
= min (u, 0). If c = 0
in , one can replace u
[resp. u
þ
] by u.
Theorem 16 (Strong principle maximum). Under
the assumptions of the above theorem, if u is not a
constant function in C
2
() \ C(
) such that Au 0
[resp. Au 0], then inf
u < u (x)[resp . sup
u >
u(x)], for all x 2 .
Remark 17 These two maximum principles can be
adapted to elliptic operators in nondivergence form,
that is,
Au ¼
X
N
i;j¼1
a
ij
ðxÞu
x
i
x
j
þ
X
N
i¼1
b
i
ðxÞu
x
i
þ cðxÞ u ½35
Fredholm Alternative
We now present some existence results which are
based on the Fredholm alternative rather than on the
variational method.
Let us consider two Hilbert spaces V and H,
where V is a dense subspace of H and V ,!,!H.
Denoting by V
0
the dual space of V, and identifying
H with its dual space, we have the following
imbeddings: V ,!H ,!V
0
. Let A be a sesquilinear
form on V V, V-coercive with respect to H, that
is, there exist
0
2 R and >0 such that
ReðAðv; vÞÞ þ
0
kvk
2
H
kvk
2
V
for all v 2 V
Denoting by A the operator associated with the
bilinear form A (see Remark 6(iii)), the equation
Au = f is equivalent to u
0
Tu = g, with T = (A þ
0
Id)
1
and g = Tf . Note that T is an isomorphism
from H onto D(A) = {u 2 H; Au 2 H}).
The operator T : H !H is compact and, thanks to
the Fredholm alternative, there are two situations:
1. either Ker A = 0 and A is an isomorphism from
D(A) onto H;
2. or Ker A 6¼ 0; then Ker A is of finite dimension,
and the problem Au = f with f 2 H admits a
solution if and only if f 2 Im A = [Ker(A
)]
?
.
We now give another example in a non-Hilbertian
frame. Let us consider the problem (Grisvard 1980):
Au = f in and Bu = g on , where is of class
C
1, 1
, A, which is defined by [34], is uniformly
strongly elliptic with a
ij
= a
ji
2 C
0, 1
(
), b
i
, c 2 L
1
(),
and Bu =
0
(u)orBu =
1
(u). One can show that
the operator u 7!(Au, Bu) is a Fredholm operator of
index zero from W
2,p
()inL
p
() W
2d1=p , p
()
(with d = 0ifBu =
0
(u)andd = 1ifBu =
1
(u)).
Regularity
Assume that is a bounded open. Suppose that u 2
H
1
0
() is a weak solution of the equation
Au ¼ f in
u ¼ 0on
½36
where A has the divergence form [34]. We now
address the question whether u is in fact smooth:
this is the regularity problem for weak solutions.
Theorem 18 (H
2
-regularity). Let be open, of
class C
1,1
, a
ij
2 C
1
(
), b
i
, c 2 L
1
(), f 2 L
2
(). Sup-
pose, furthermore, that u 2 H
1
() is a weak solution
of [36]. Then u 2 H
2
() and we have the estimate
kuk
H
2
ðÞ
C ðkf k
L
2
ðÞ
þkuk
L
2
ðÞ
Þ
where the constant C depends only on and on the
coefficients of A.
Theorem 19 (Higher regularity). Let m be a non-
negative integer, be open, of class C
mþ1, 1
and assume
that a
ij
2 C
mþ1
(
), b
i
, c 2 C
mþ1
(
), f 2 H
m
(). Sup-
pose, furthermore, that u 2 H
1
() is a weak solution of
[36]. Then u 2 H
mþ2
() and
kuk
H
mþ2
ðÞ
Cðkf k
H
m
ðÞ
þkuk
L
2
ðÞ
Þ
224 Elliptic Differential Equations: Linear Theory
where the constant C depends only on and on
the coeff icients of A. In particular, if m > N=2, then
u 2 C
2
(
). Moreover, if is of C
1
class and f 2
C
1
(
), a
ij
2 C
1
(
), b
i
, c 2 C
1
(
), then u 2 C
1
(
).
Remark 20
(i) If u 2 H
1
0
() is the unique solution of [36], one
can omit the L
2
-norm of u in the right-hand side
of the above estimate.
(ii) Moreover, let us suppose the coefficients a
ij
, b
i
and c are all C
1
and f 2 C
1
(); then, if u 2
H
1
()satisfiesAu = f , u 2 C
1
(); this is due to
the ‘‘hypoellipticity’’ property satisfied by the
operator A.
We have a similar result in the L
p
frame (Grisvard
1980):
Theorem 21 (W
2, p
-regularity). Let be open, of
class C
1,1
, a
ij
2 C
1
(
), b
i
, c 2 L
1
(). Suppose,
furthermore, that b
i
= 0, 1 i N and c 0 a.e.
Then for every f 2 L
p
() there exists a unique
solution u 2 W
2,p
() of [36].
Unbounded Open
The Whole Space
Note in passing that we shall work with the weighted
Sobolev spaces W
m,p
() defined in the subsection
‘‘Unboundedopensandweightedspaces.’’
Theorem 22 The following claims hold true:
(i) Let f 2 W
1, p
0
(R
N
) satisfy the compatibility
condition
hf ; 1i
W
1;p
0
ðR
N
ÞW
1;p
0
0
ðR
N
Þ
¼ 0 if p
0
N
Then the problem [5] has a solution u 2
W
1,p
0
(R
N
), which is unique up to an element in
P
[1N=p]
and satisfies the estimate
kuk
W
1;p
0
ðR
N
Þ=P
½1N=p
Ckf k
W
1;p
0
ðR
N
Þ
Moreover, if 1 < p < N, then u = E f .
(ii) If f 2 L
p
(R
N
), then the problem [5] has a
solution u 2 W
2,p
0
(R
N
), which is unique up to
an element in P
[2N= p ]
and if 1 < p < N=2, then
u = E f .
The Caldero´n–Zygmund inequality
@
2
’
@x
i
@x
j
L
p
ðR
N
Þ
CðN; pÞk’k
L
p
ðR
N
Þ
’ 2DðR
N
Þ
and Theorem 4 are crucial for establishing Theorem 22.
Further, point (i) means that the Riesz potential of
second order satisfies
I
2
: W
1;p
0
ðR
N
Þ?P
½1N=p
0
! W
1;p
0
ðR
N
Þ=P
½1
N
p
0
(where the initial space is the orthogonal comple-
ment of P
[1N= p
0
]
in W
1, p
0
(R
N
)) and it is an
isomorphism.
Note that here
W
1;p
0
ðR
N
Þ¼fv 2 L
p
ðR
N
Þ; rv 2 L
p
ðR
N
Þg
for 1 < p < N and 1=p
= 1=p 1=N.Andfor
1 < r < N=2, we also have the continuity property
I
2
: L
r
ðR
N
Þ!L
q
ðR
N
Þ; for
1
q
¼
1
r
2
N
Remark 23 The problem
u u ¼ f in R
N
½37
is of a completely different nature than the problem
[5]. The class of function spaces appropriate for the
problem [37] are the classical Sobolev spaces. With
the help of the Caldero´n–Zygmund theory, one can
prove that if f 2 L
p
(R
N
), then the unique solution of
[37] belongs to W
2,p
(R
N
) and can be represented as
the Bessel potential of second order (see Stein
(1970)): u = G f , where G is the appropriate Bessel
kernel, that is, G, for which
b
G() (1 þjj
2
)
1=2
.
Recall that in particular G(x) jxj
1
e
jxj
for N = 3.
In the Hilbert case, f 2 L
2
(R
N
), we get
ð1 þjj
2
Þ
b
u 2 L
2
ðR
N
Þ
which, by Plancherel’s theorem, implies that u 2
H
2
(R
N
). For f 2 W
1, p
(R
N
), the problem [37] has a
unique solution u 2 W
1, p
(R
N
) satisfying the
estimate
kuk
W
1;p
ðR
N
Þ
Cðp; nÞkf k
W
1;p
ðR
N
Þ
Exterior Domain
We consider the problem in an exterior domain with
the Dirichlet boundary condition
u ¼ f in
u ¼ g on ¼ @
½38
where f 2 W
1, p
0
() and g 2 W
11=p, p
(@). Invoking
the results for R
N
and bounded domains, one can
prove the existence of a solution u 2 W
1, p
0
() which
is unique up to an element of the kernel A
p
0
() = {z 2
Elliptic Differential Equations: Linear Theory 225