Sobolev Spaces

By Robert A. Adams and John J. F. Fournier

Sobolev Spaces presents an introduction to the theory of Sobolev Spaces and other related spaces of function, also to the imbedding characteristics of these spaces. This theory is widely used in pure and Applied Mathematics and in the Physical Sciences.

This second edition of Adam's 'classic' reference text contains many additions and much modernizing and refining of material. The basic premise of the book remains unchanged: Sobolev Spaces is intended to provide a solid foundation in these spaces for graduate students and researchers alike.

• Self-contained and accessible for readers in other disciplines
• Written at elementary level making it accessible to graduate students

• Language: English
• Publisher: Academic Press
• Release date: Jun 26, 2003
• ISBN: 9780080541297

Book preview

1

Preliminaries

1.1 Introduction

Sobolev spaces are vector spaces whose elements are functions defined on domains in n−dimensional Euclidean space Rn and whose partial derivatives satisfy certain integrability conditions. In order to develop and elucidate the properties of these spaces and mappings between them we require some of the machinery of general topology and real and functional analysis. We assume that readers are familiar with the concept of a vector space over the real or complex scalar field, and with the related notions of dimension, subspace, linear transformation, and convex set. We also expect the reader will have some familiarity with the concept of topology on a set, at least to the extent of understanding the concepts of an open set and continuity of a function.

In this chapter we outline, mainly without any proofs, those aspects of the theories of topological vector spaces, continuity, the Lebesgue measure and integral, and Schwartz distributions that will be needed in the rest of the book. For a reader familiar with the basics of these subjects, a superficial reading to settle notations and review the main results will likely suffice.

Notation

1.2

Throughout this monograph the term domain and the symbol Ω will be reserved for a nonempty open set in n-dimensional real Euclidean space Rn. For c and two functions u and v, the scalar multiple cu, the sum u + v, and the product uv are always defined pointwise:

at all points x where the right sides make sense.

A typical point in Rn is denoted by x = (x1, …, xn); its norm is given by|x)¹/². The inner product of two points x and y in Rn is x · y xjyj.

If α = (α1, …, αn) is an n-tuple of nonnegative integers αj, we call α a multi-index and denote by xα αj. Similarly, if Dj = ∂/∂xj, then

denotes a differential operator of order |α|. Note that D(0, …, 0)u = u.

If α and β are two multi-indices, we say that β ≤ α provided βj ≤ αj for 1 ≤ j ≤ n. In this case α − β is also a multi-index, and |α − β| + |β| = |α|. We also denote

and if β ≤ α,

The reader may wish to verify the Leibniz formula

valid for functions u and v that are |α| times continuously differentiable near x.

1.3

If G ⊂ Rn the closure of G in Rn. We shall write G is a compact (that is, closed and bounded) subset of Rn. If u is a function defined on G, we define the support of u to be the set

We say that u has compact support in Ω if supp (uΩ. We denote by "bdry G" the boundary of G in Rnc, where Gc is the complement of G in Rn; Gc = Rn − G = {x ∈ Rn : x ∈ G}.

If x ∈ Rn and G ⊂ Rn, we denote by "dist(x, G)" the distance from x to G, that is, the number infy∈G|x − y|. Similarly, if F, G ⊂ Rn are both nonempty,

Topological Vector Spaces

1.4 Topological Spaces

If X is any set, a topology on X is a collection O of subsets of X which contains

(i) the whole set X and the empty set ø,

(ii) the union of any collection of its elements, and

(iii) the intersection of any finite collection of its elements.

The pair (X, O) is called a topological space and the elements of O are the open sets of that space. An open set containing a point x in X is called a neighbourhood of x. The complement X − U = {x ∈ X : x ∉ U} of any open set U is called a closed of any subset S ⊂ X is the smallest closed subset of X that contains S.

Let O1 and O2 be two topologies on the same set X. If O1 ⊂ O2, we say that O2 is stronger than O1, or that O1 is weaker than O2.

A topological space (X, O) is called a Hausdorff space if every pair of distinct points x and y in X have disjoint neighbourhoods.

The topological product of two topological spaces (X, OX) and (Y, OY) is the topological space (X × Y, O), where X × Y = {(x, y) : x ∈ X, y ∈ Y} is the Cartesian product of the sets X and Y, and O consists of arbitrary unions of sets of the form {OX × OY : OX ∈ OX, OY ∈ OY}.

Let (X, OX) and (Y, OY) be two topological spaces. A function f from X into Y is said to be continuous if the preimage f−1(O) = {x ∈ X : f (x) ∈ O} belongs to OX for every O ∈ OY. Evidently the stronger the topology on X or the weaker the topology on Y, the more such continuous functions f there will be.

1.5 Topological Vector Spaces

We assume throughout this monograph that all vectors spaces referred to are taken over the complex field unless the contrary is explicitly stated.

A topological vector space, hereafter abbreviated TVS, is a Hausdorff topological space that is also a vector space for which the vector space operations of addition and scalar multiplication are continuous. That is, if X is a TVS, then the mappings

from the topological product spaces X × X × X, respectively, into X has its usual topology induced by the Euclidean metric.)

X is a locally convex TVS if each neighbourhood of the origin in X contains a convex neighbourhood of the origin.

We outline below those aspects of the theory of topologicaland normed vector spaces that play a significant role in the studyof Sobolev spaces. For a more thorough discussion of these topics the reader is referred to standard textbooks on functional analysis, for example [Ru1] or [Y].

1.6 Functionals

A scalar-valued function defined on a vector space X is called a functional. The functional f is linear provided

If X is a TVS, a functional on X is continuous if it is continuous from X has its usual topology induced by the Euclidean metric.

The set of all continuous, linear functionals on a TVS X is called the dual of X and is denoted by X′. Under pointwise addition and scalar multiplication X′ is itself a vector space:

X′ will be a TVS provided a suitable topology is specified for it. One such topology is the weak-star topology, the weakest topology with respect to which the functional Fx, defined on X′ by Fx(f) = f(x) for each f ∈ X′, is continuous for each x ∈ X. This topology is used, for instance, in the space of Schwartz distributions introduced in Paragraph 1.57. The dual of a normed vector space can be given a stronger topology with respect to which it is itself a normed space. (See Paragraph 1.11.)

Normed Spaces

1.7 Norms

A norm on a vector space X is a real-valued function f on X satisfying the following conditions:

(i) f (x) ≥ 0 for all x ∈ X and f (x) = 0 if and only if x = 0,

(ii) f (cx) = |c| f (x) for every x ∈ X and c ,

(iii) f (x + y) ≤ f (x) + f (y) for every x, y ∈ X.

A normed space is a vector space X provided with a norm. The norm will be denoted ||·; X|| except where other notations are introduced.

If r > 0, the set

is called the open ball of radius r with center at x ∈ X. Any subset A ⊂ X is called open if for every x ∈ A there exists r > 0 such that Br(x) ⊂ A. The open sets thus defined constitute a topology for X with respect to which X is a TVS. This topology is the norm topology on X. The closure of Br (x) in this topology is

A TVS X is normable if its topology coincides with the topology induced by some norm on X. Two different norms on a vector space X are equivalent if they induce the same topology on X. This is the case if and only if there exist two positive constants a and b such that,

for all x ∈ X, where ||x||1 and ||x||2 are the two norms.

Let X and Y be two normed spaces. If there exists a one-to-one linear operator L mapping X onto Y having the property ||L(x); Y|| = ||x; X|| for every x ∈ X, then we call L an isometric isomorphism between X and Y, and we say that X and Y are isometrically isomorphic. Such spaces are often identified since they have identical structures and only differ in the nature of their elements.

1.8

A sequence {xn} in a normed space X is convergent to the limit x0 if and only if limn→∞, ||xn − x0; X|| = 0 in R. The norm topology of X is completely determined by the sequences it renders convergent.

A subset S of a normed space X is said to be dense in X if each x ∈ X is the limit of a sequence of elements of S. The normed space X is called separable if it has a countable dense subset.

1.9 Banach Spaces

A sequence {xn} in a normed space X is called a Cauchy sequence if and only if for every ∈ > 0 there exists an integer N such that ||Xm − xn; X|| < ∈ holds whenever m, n >N. We say that X is complete and a Banach space if every Cauchy sequence in X converges to a limit in X. Every normed space X is either a Banach space or a dense subset of a Banach space Y called the completion of X whose norm satisfies

1.10 Inner Product Spaces and Hilbert Spaces

If X is a vector space, a functional (·, ·)X defined on X × X is called an inner product on X provided that for every x, y ∈ X and a, b

(i) (x, y)X )Xdenotes the complex conjugate of c )

(ii) (ax + by, z)X = a(x, z)X + b(y, z)X,

(iii) (x, x)X = 0 if and only if x = 0,

Equipped with such a functional, X is called an inner product space, and the functional

(1)

is, in fact, a norm on X If X is complete (i.e. a Banach space) under this norm, it is called a Hilbert space. Whenever the norm on a vector space X is obtained from an inner product via (1), it satisfies the parallelogram law

(2)

Conversely, if the norm on X satisfies (2) then it comes from an inner product as in (1).

1.11 The Normed Dual

A norm on the dual X′ of a normed space X can be defined by setting

for each X′ ∈ Xis complete, with the topology induced by this norm X′ is a Banach space (whether or not X is) and it is called the normed dual of X. If X is infinite dimensional, the norm topology of X′ is stronger (has more open sets) than the weak-star topology defined in Paragraph 1.6.

The following theorem shows that if X is a Hilbert space, it can be identified with its normed dual.

1.12 Theorem (The Riesz Representation Theorem)

Let X be a Hilbert space. A linear functional X′ on X belongs to X′ if and only if there exists x ∈ X such that for every y ∈ X we have

and in this case ||X′; X′|| = ||X′; X||. Moreover, x is uniquely determined by X′ ∈ X.

A vector subspace M of a normed space X is itself a normed space under the norm of X, and so normed is called a subspace of X. A closed subspace of a Banach space is itself a Banach space.

1.13 Theorem (The Hahn-Banach Extension Theorem)

Let M be a subspace of the normed space X. If m′ ∈ M′, then there exists X′ ∈ X′ such that ||X′; X′|| = ||m′; M′|| and X′(m) = m′ (m) for every m ∈ M.

1.14 Reflexive Spaces

A natural linear injection of a normed space X into its second dual space X′ = (X′)′ is provided by the mapping J whose value J x at x ∈ X is given by

Since |J x(X′)| ≤ ||X′; X′|| ||x; X||, we have

However, the Hahn-Banach Extension Theorem assures us that for any x ∈ X we can find X′ ∈ X′ such that ||X′; X′|| = 1 and X′(x) = ||x; X||. Therefore J is an isometric isomorphism of X into X″.

If the range of the isomorphism J is the entire space X′, we say that the normed space X is reflexive. A reflexive space must be complete, and hence a Banach space.

1.15 Theorem

Let X be a normed space. X is reflexive if and only if X′ is reflexive. X is separable if X′ is separable. Hence if X is separable and reflexive, so is X′.

1.16 Weak Topologies and Weak Convergence

The weak topology on a normed space X is the weakest topology on X that still renders continuous each X′ in the normed dual X′ of X. Unless X is finite dimensional, the weak topology is weaker than the norm topology on X. It is a consequence of the Hahn-Banach Theorem that a closed, convex set in a normed space is also closed in the weak topology of that space.

A sequence convergent with respect to the weak topology on X is said to converge weakly. Thus xn converges weakly to x in X provided X′(xn) → X′(xfor every X′ ∈ X′. We denote norm convergence of a sequence {xn} to x in X by xn → x, and we denote weak convergence by xn x. Since we have |X′(xn − x)| ≤ ||X′; X′|| ||xn − x; X||, we see that xn → x implies xn x. The converse is generally not true (unless X is finite dimensional).

1.17 Compact Sets

A subset A of a normed space X is called compact if every sequence of points in A has a subsequence converging in X to an element of A. (This definition is equivalent in normed spaces to the definition of compactness in a general topological space; A is compact if whenever A is a subset of the union of a collection of open sets, it is a subset of the union of a finite subcollection of those sets.) Compact sets are closed and bounded, but closed and bounded sets need not be compact unless X is finite dimensional. A is called precompact in X in the norm topology of X is compact. A is called weakly sequentially compact if every sequence in A has a subsequence converging weakly in X to a point in A. The reflexivity of a Banach space can be characterized in terms of this