Even more care is necessary in interpreting imbeddings into spaces of type (ii):
wm,p(~-2) ~ W j,q (~'2k)
where f2k is the intersection of f2 with a plane of dimension k < n. Each element of wm'P(~'2)
is,
by T h e o r e m 3.17, a limit in that space of a sequence {ui} of functions in C~(S2). The functions ui have t r a c e s o n f2k (that is, restrictions to S2k) that belong to C~(f2k). The above imbedding signifies that these traces converge in W j'q (f2k) to a function u* that is independent of the choice of {ui}and satisfies
I .* II q
-<K II. IIm
with K independent of u.
4.3 Let us note as a point of interest, though of no particular use to us later, that the imbedding W m'p (~'2) --+ W j'q
(~2)
is equivalent to the simple containment W m'p ( ~ ) C wJ'q(~'2). Certainly the former implies the latter. To verify the converse, suppose W m'p (~'2) C W j'q(~"2),
and let I be the linear operator taking W m,p(~"2)
into W j'q(~'2)
defined by I u = u for u ~ W m'p(~'2).
If uk --+ u in wm'P(~'2) (and hence in L P ( ~ 2 ) ) and I u k ~ v in wJ'q (f2) (and hence in L q ( ~ 2 ) ) , then, passing to a subsequence if necessary, we have by Corollary 2.17 that u k ( x ) ~ u ( x ) a.e. on f2, u ~ ( x ) = I u k ( x ) --+ v ( x ) a.e. on f2. Thus u ( x ) = v ( x ) a.e. on f2, that is, I u = v, and I is continuous by the closed graph theorem of functional analysis.Geometric Properties of Domains
4.4 ( S o m e Definitions) Many properties of Sobolev spaces defined on a do- main f2, and in particular the imbedding properties of these spaces, depend on regularity properties of ~ . Such regularity is normally expressed in terms of geo- metric or analytic conditions that may or may not be satisfied by a given domain.
We specify below several such conditions and consider their relationships. First we make some definitions.
Let v be a nonzero vector in ~n, and for each x ~ 0 let / ( x , v) be the angle between the position vector x and v. For given such v, p > 0, and x satisfying 0 < x _< re, the set
C - {x ~ R n "x - 0 o r 0 <
Ixl
~ p, L(x, v) ~ x / 2 }is called a f i n i t e c o n e of height p, axis direction v and aperture angle x with vertex at the origin. Note that x + C -- {x + y 9 y 6 C} is a finite cone with vertex at x but the same dimensions and axis direction as C and is obtained by parallel translation of C.
82 The Sobolev Imbedding Theorem Given n linearly independent vectors yl . . . Yn ~ IR n , the set
is a p a r a l l e l e p i p e d with one vertex at the origin. Similarly, x + P is a parallel translate of P having one vertex at x. The centre of x + P is the point given by c ( x + P ) = x + ( 1 / 2 ) ( y l + . . . + Yn). Every parallelepiped with a vertex at x is contained in a finite cone with vertex at x and also contains such a cone.
An open cover tY of a set S C R n is said to be locally finite if any compact set in En can intersect at most finitely many members of 6 . Such locally finite collections of sets must be countable, so their elements can be listed in sequence. If S is closed, then any open cover of S by sets with a uniform bound on their diameters possesses a locally finite subcover.
We now specify six regularity properties that a domain S2 C I~ n may possess. We denote by S2~ the set of points in f2 within distance 6 of the boundary of g2:
f2~ = {x ~ S2" dist(x, b d r y ~ ) < 6}.
4.5 (The S e g m e n t Condition) As defined in Paragraph 3.21, a domain g2 satisfies the segment condition if every x E bdry g2 has a neighbourhood Ux and a nonzero vector yx such that if z E f2 M Ux, then z + tyx ~ f2 for 0 < t < 1.
Since the boundary of g2 is necessarily closed, we can replace its open cover by the neighbourhoods Ux with a locally finite subcover { U1, U2 . . . . } with corresponding vectors y l, y2 . . . . such that if x E f2 M Uj for some j , then x + tyj ~ S2 for 0 < t < l .
4.6 (The Cone Condition) f2 satisfies the cone condition if there exists a finite cone C such that each x E f2 is the vertex of a finite cone Cx contained in f2 and congruent to C. Note that Cx need not be obtained from C by parallel translation, but simply by rigid motion.
4.7 (The W e a k Cone Condition) Given x E S2, let R(x) consist of all points y E f2 such that the line segment from x to y lies in f2; thus R ( x ) is a union of rays and line segments emanating from x. Let
F(x) -- {y E R ( x ) ' l y - x l < 1}.
We say that f2 satisfies the w e a k cone condition if there exists 6 > 0 such that /~n(F(X)) ~ 6 for all x E ~ ,
Geometric Properties of Domains 83 where ].l n is the Lebesgue measure in IR ~ . Clearly the cone condition implies the weak cone condition, but there are many domains satisfying the weak cone condition that do not satisfy the cone condition.
4.8 (The Uniform Cone Condition) f2 satisfies the uniform cone condition if there exists a locally finite open cover { Uj } of the boundary of ~2 and a corre- sponding sequence {Cj } of finite cones, each congruent to some fixed finite cone C, such that
(i) There exists M < cx~ such that every Uj has diameter less then M.
(ii) f2~ C Uj~=l uj for some 6 > 0.
(iii) Qj - U x ~ n v ~ (x + cj) c ~ for every j .
(iv) For some finite R, every collection of R + 1 of the sets Qj has empty intersection.
4.9 (The Strong Local Lipschitz Condition) ~ satisfies the strong local Lipschitz condition if there exist positive numbers 6 and M, a locally finite open cover { Uj } of bdry f2, and, for each j a real-valued function ~ of n - 1 variables, such that the following conditions hold:
(i) For some finite R, every collection of R + 1 of the sets Uj has empty intersection.
(ii) For every pair of points x, y E f2~ such that Ix - Y l < 6, there exists j such that
x, y E Vj = {x E Uj "dist(x, b d r y U j ) > 3}.
(iii) Each function j~ satisfies a Lipschitz condition with constant M: that is, if
= (~1 . . . ~n-1) and p = (pl . . . pn-1) are in R ~-1 , then
[ f ( ~ ) - f ( p ) l _< Ml~ - Pl.
(iv) For some Cartesian coordinate system ( ( j , 1 . . .
(j,n)
in Uj, g2 n Uj is represented by the inequality~j,n < OO(~'j,1, " ' ' , ~'j,n-1).
If f2 is bounded, the rather complicated set of conditions above reduce to the simple condition that f2 should have a locally Lipschitz boundary, that is, that each point x on the boundary of f2 should have a neighbourhood Ux whose intersection with bdry f2 should be the graph of a Lipschitz continuous function.
84 The Sobolev Imbedding Theorem
4.10