With respect to the supremum norm on R 2 that is px y maxlx y we have As fs for

# With respect to the supremum norm on r 2 that is px y

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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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