Sobolev Spaces
We are going to define Sobolev spaces of integer order on a Riemannian manifold. First we shall be concerned with density problems. Then we shall prove the Sobolev imbedding theorem and the Kondrakov theorem. After that we shall introduce the notion of be
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Sobolev Spaces
§1. First Definitions 2.1 We are going to define Sobolev spaces of integer order on a Riemannian manifold. First we shall be concerned with density problems. Then we shall prove the Sobolev imbedding theorem and the Kondrakov theorem. After that we shall introduce the notion of best constant in the Sobolev imbedding theorem. Finally, we shall study the exceptional case of this theorem (i.e., H'i on n-dimensional manifolds). For Sobolev spaces on the open sets in n-dimensional, real Euclidean space [R", we recommend the very complete book of Adams [1]. 2.2 Definitions. Let (M", g) be a smooth Riemannian manifold of dimension n (smooth means COO). For a real function cp belonging to Ck(M,,) (k ~ 0 an integer), we define:
In particular, IVocpl = Icpl, IVlcpl2 = kth covariant derivative of cpo
IVcpl2
= V"cpVvCP. Vkcp will mean any
Let us consider the vector space (f~ of C7) functions cp, such that ~ t ~ k, where k and t are integers and p ~ 1 is a real number.
IV(cpl E LiM,,), for all t with 0
2.3 Definitions. The Sobolev space respect to the norm
IlcpllH/:
H~(M II)
is the completion of
(f~
with
k
=
L IlVlcpllp. (=0
Ht(M II) is the closure of P)(M II) in Ht(M II). P)(MII) is the space of COO functions with compact support in M" and Hg = Lp.
T. Aubin, Some Nonlinear Problems in Riemannian Geometry © Springer-Verlag Berlin Heidelberg 1998
33
§2. Density Problems
It is possible to consider some other norms which are equivalent; for instance, we could use
H;
When p = 2, is a Hilbert space, and this norm comes from the inner product. For simplicity we will write Hi; for the Hilbert space H;.
§2. Density Problems 2.4 Theorem.
~(IR")
is dense in Ht(IR").
Proof Letf(t) be a Coo decreasing function on IR, such thatf(t) = 1 for t ~ 0 andf(t) = 0 for t ~ 1. It is sufficient to prove that a function rp E Coo(IR") n Hr(IR") can be approxi-
mated in Ht(IR") by functions of ~(IR"). We claim that the sequence of functions rp}{x) = rp(x).f(llxll - j), of ~(IR"), converges to qI(x) in H~(IR"). Let us verify this for the functions and the first derivatives, that is, in the case of H1(IR"). Whenj -+ :C, rp}{x) -+ qI(x) everywhere and IqI}{x) I ~ IqI(x)l, which belongs to L". So by the Lebesgue dominated convergence theorem IIrpj - qllI" -+ O. Moreover, whenj -+ 00, IVrp}{x)l-+ IVrp(x) I everywhere, and IVrp}{x) I ~ IVqI(x) I + IqI(x) I SUPte[O.1l I f'(t) I which belongs to L". Thus
IIV(rpj - rp)lI" -+ O.
This proves the density assertion for Leibnitz's formula.
H~(IR").
For k > 1, we have to use •
2.S Remark. The preceding theorem is no~ true for a bounded open set 0 in Euclidean space. Indeed, let us verify that Hf(O) is strictly included in Hf(O). For this purpose consider the inner product
For
y, E Coo(O) n
Hf(O) and rp (qI,
If Y, ¥:- 0 satisfies
y, , ilUn).
E ~(O),
i t aiiy,)rp
y,> = (y, -
y, = ~j= I auY"
n
1=1
dx.
then for all rp E ~(O), (rp,
.
y,> = 0, so that
Such a function y, exists on a bounded open set 0; for instance, '" = sinh Xl (X I the first coordinate of x),
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