Sobolev Spaces on Riemannian Manifolds
Several books deal with Sobolev spaces on open subsets of R (n), but none yet with Sobolev spaces on Riemannian manifolds, despite the fact that the theory of Sobolev spaces on Riemannian manifolds already goes back about 20 years. The book of Emmanuel He
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Let ( M , g ) be a Riemannian manifold. For k an integer and u E C~(M), Vku denotes the kth covariant derivative of u (with the convention V~
= u). As
an example, the components of Vu in local coordinates are given by (Vu)i = Oiu, while the components of V2u in local coordinates are given by
By definition one has that
iv%12 = # + . . . #~jk (v%)il...i~ (V~u)~,.. jk For k an integer and p > 1 real, we denote by g~(M) the space of smooth functions u E C~(M) such that [VJu I E LP(M) for any j = 0 , . . . , k. Hence,
where, in local coordinates, dr(g) = ~ ) d x ,
and where dx stands for the
Lebesgue's volume element of R '~, n = dimM. If M is compact, one has that
C~(M) = C~(M) for all k and p > 1. D e f i n i t i o n 2.1:
The Sobolev space H~(M) is the completion of g~(M) with
respect to the norm k
Noting that a Cauchy sequence in C~(M) is also a Cauchy sequence in LP(M), and that a Cauchy sequence in C~(M) with converges to 0 in LP(M) converges to 0 in Q~(M), the Sobolev spaces H~(M) can be seen as subspaces of
2.1 First definitions
11
LV(M). This is the point of vue we adopt in the sequel. More precisely, one can look at H~(M) as the space of functions u in LV(M) which are limit in LP(M) of
IlullH~ as above where IvJul, 0 _ j < k, is now the limit in LV(M) of IWu,~l. One checks without any
a Cauchy sequence (urn) in C~(M), and define the norm
difficulty that these spaces are Banach spaces. The following results are then easy to prove. P r o p o s i t i o n 2.2: If p = 2, H~(M) is a Hilbert space when equipped with the
equivalent norm
Ilull =
IVJul2dv(g)
The scalar product (., .) associated to I1.11 is defined by k
(-,vl= E__o/M(,'"'...,'-'-
,o
,.),v(,)
P r o p o s i t i o n 2.3: If M zs compact, HP(M) does not depend on the Riemannian
metric. P r o p o s i t i o n 2.4: If p > 1, H~(M) is reflexive. Of course, proposition 2.3 is not anymore true for non-compact manifolds. Think for instance to some non-compact manifold M endowed with two metrics, the volume of M for one of these two metrics being finite, the volume of M for the other one being infinite. Then the constant function u = 1 belongs to the Sobolev spaces associated to the metric of finite volume, while it does not belong to the Sobolev spaces associated to the metric of infinite volume. Independently, let us now mention that as an easy consequence of Meyers-Serrin's theorem [Ad, theorem 3.16], one has the following. (When we refer to lipschitzian functions we refer to global lipschitzian functions). 2.5: Let (M,g) be a Riemannian manifold and u : M --+ R a lipschiizian funclion on M which equals zero outside a compact subset of M . Then u c H~(M) for any p >_ 1. Lemma
P r o o f o f l e m m a 2.5: Let u : M --~ R be a lipschitzian function on M which equals zero outside a compact subset K of M. Let (~k,r
..... N be a finite
N number of charts such that K C t2k=l~k and such that for any k = 1 , . . . , N,
r
_ g~j < R6ij as bilinear forms = B~(1) and ~15 ij
1 is some real number, B~(1) is th
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