8 2. PRELIMINARIES CONCERNING MANIFOLDS
1) Mi c 2B, c D, i = 1,2,...
3) YliLi
^2Mi(x)
^ 1
for
a// x G fi
4) diamB, ^ dist(B;,F) ^ diamBz for alii = 1,2...
Hereafter, XE denotes the characteristic function of a measurable set E C X.
Also, 2B stands for the ball of the same center as B but with radius 2 times larger.
2.2. The Sobolev space # ^ ( X , Y )
The various classes of mappings / : X » Y in this paper are defined based on the
classical Sobolev theory of real valued functions. Note that W1^P(X) = W1^P(X, R)
is a Banach space equipped with the norm
(2-2) l / I U
P
= « / I U + I A f l U
We adopt the classical results in Rn to our manifold setting, see for instance [48].
LEMMA 2.2. Smooth functions in ^°°(X) are dense in ^
1 P
(X), 1 ^ p oo.
LEMMA
2.3.
[POINCARE INEQUALITY]
For every set E of a positive measure in
X and f e
W^P(X)
we have
\f-h\p^C^
j
\Df(x)\pdx
The constant CE actually depends only on the measure of E. As usual, the
integral average of / over the set E is denoted by
fE
= r fix) dx —— I fix) dx
JE JEjy }
|E| V
)
The local variant of Poincare inequality reads as:
LEMMA
2.4. For every legitimate ballM = B(a, r), we have
I \f - h\p rp f \Df(x)\pdx, whenever f e Whp(X)
JM JM
As a matter of fact this inequality is true for all geodesic balls in X, but we shall
exploit this inequality only for legitimate balls. Regarding the implied constant, we
must emphasize that it depends neither on / nor on the radius r.
Now given two Riemannian manifolds X and Y, we shall consider the Sobolev
space W1,P(X, Y) of mappings whose tangent linear map (differential)
(2.3) Df(x):TxX^TyY, y = f(x)
is Jzfp-integrable. Our description, and certainly a rigorous definition of
W^P{X, Y), relies on an imbedding Y c l ^ [47].
THEOREM
2.5. (J. Nash) Every ff00 -smooth Riemannian manifold Y can be
^°° -isometrically imbedded in some Euclidean space M,N.
The reader is also referred to M. L. Gromov and V. A. Rohlin [16] for an account
of the imbedding problem. The Nash theorem allows us to consider
W1,P(X,
Y)
as a subclass of a linear space of mappings / : X RN such that f(x) G Y at
almost every x G X. The metric topology in W1,P(X, Y) will be inherited from the
associated norm topology in the linear space W1'P(X, R^). In this way the Sobolev
class W1,P(X, Y) becomes a complete metric space. In what follows we shall tacitly
use the fact that #^'
P
(X, Y) is also closed under weak topology of
W1^P{X, RN).
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