With respect to the supremum norm on R 2, that is
p(x, y) --
max{lx[, [y[}, we
have
As = f(s)
for sufficiently large s.
Hence f2 satisfies conclusion (b) of
Theorem 6.45 (and, since f is monotonic, conclusion (a) as well) if and only if
f(s +E)
lim
= 0
(28)
s~
f(s)
holds for every e > 0. For example, f (x) = exp(-x 2) satisfies this condition but
f (x) = exp(-x) does not. We shall show in Example 6.53 that the imbedding
W m'p (~) ~
L p
(~)
(29)
is compact if (28) holds. Thus (28) is necessary and sufficient for compactness of
the above imbedding for domains of this type. |
y = f(x)
Fig. 5
6.49
EXAMPLE
Let f be as in the previous example, and assume also that
f'(0) -- 0. Let g be a positive, nonincreasing function in C 1 ([0, ~)) satisfying
1
(i) g(O) -- ~ f (0), and g'(O) -- O,
(ii)
g(x) < f(x)
for all x _ O,
(iii) g (x) is constant on infinitely many disjoint intervals of unit length.
Let h (x) -- f (x) - g (x) and consider the domain (Figure 6)
--{(x,y)~I~ 2 O<y<g(x)
ifx>_O, O<y<h(-x)
ifx <0}.

Unbounded Domains ~
Compact Imbeddings of
W m'p
(~-2)
195
Again we have
As -- f(s)
for sufficiently large s, so f2 satisfies the conclusions
of Theorem 6.45 if (28) holds.
1 having edges
If, however, T is a tessellation of R 2 by squares of edge length
parallel to the coordinate axes, and if one of the squares in T has centre at the
origin, then T has infinitely many 1-fat squares with centres on the positive x-axis.
By Theorem 6.38 the imbedding (29) cannot be compact for the domain (2. |
Y
S
S
S
S
S
S
S
I
S
#
#
I
S S
S
SSSSS
SSSSS
,
y=f(x)
t
i
y = g(xi",
1
k
5r
Fig. 6
Unbounded Domains--Compact Imbeddings of
Wm,P(~)
6.50
(Flows)
The above examples suggest that any sufficient condition for the
compactness of the imbedding
W m'p (~) -+ L p (~)
for unbounded domains S2 must involve the rapid decay of volume locally in each
branch of S2r as r tends to infinity. A convenient way to express such local decay
is in terms of flows on f2.
By a
flow
on f2 we mean a continuously differentiable map
: U --+ f2 where
U is an open set in f2 x I~ containing ~ x {0}, and where ~(x, 0) = x for every
xEf2.
For fixed x E f2 the curve t --+
(x, t) is called a
streamline
of the flow. For fixed
t the map ~t : x --+ ~(x, t) sends a subset of f2 into f2. We shall be concerned
with the Jacobian of this map:
det ~t (x) --
0 (xl
.....
Xn)
(x,t)

196
Compact Imbeddings of Sobolev Spaces
It is sometimes required of a flow
that ~s+t
=
r
0 r
but we do not need this
property and so do not assume it.
6.51
EXAMPLE
Let g2 be the domain of Example 6.48. Define the flow
r
y, t) = (x _ t, f (x - t)
)
f(x)
Y
'
0<t<x.
The direction of the flow is towards the line x = 0 and the streamlines (some
of which are shown in Figure 7) diverge as the domain widens.
~t is a local
magnification for t > 0:
f(x
-- t)
det ~t
(X,
y) =
f(x)
Note that limx~
det
~'t
(x, y) = cx~ if f satisfies (28).
For N -- 1 2,
let f2*
....
s = {(x, y) ~ f2 0 < x < N}. Since f2~v is bounded and
satisfies the cone condition, the imbedding
WI'p (~'~*N) ~
LP(~"~*N)
is compact. This compactness, together with properties of the flow r
are sufficient
to force the compactness of
W m'p (~'2) ~
L p
(~) as we now show.

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