1. PRELIMINARIES
3
all bounded subsets of £1. For continuous functions u we have |w|0(Q) =
Hwll^Q). For an integer nonnegative k we define
\
{/P
\uhJa)
= [ ]T
\\Dau\\pp(Q)X
\a\k
The space L (Q) is the set of all functions with the finite norms || || (Q).
The space
Hk,p(Q)
is the set of all functions having on Q (generalized)
partial derivatives of order k with the finite norm || \\k (Q). We as-
sume 1 p +oo.
Hk,p(Q.)
is the subspace of H
,p(£l)
consisting of
the limits of sequences of functions from C™(Q) with respect to the norm
|| \\k p(Q.). The spaces C (Q.) and H
,/7(£2)
are Banach spaces with the
norms introduced above. The space H^(Q) = H ' (Q) is even a Hilbert
space that has the scalar product
(u,v){k)(0)= ^2 [
DauD^dx.
We often write
Ck
and
Hk'p
instead of
Ck(Rn)
and
Hk'p(Rn)
and
sometimes omit the integration domain in integrals over R" .
We summarize necessary properties of H
,p
-spaces as follows.
THEOREM
1.1.2. Let Q be a bounded domain with the Lipschitz boundary.
i) If 1 p, then there is a constant C such that
\\u\\p{dO) C\\u\\lp(Q) for all u e
Cl(U).
ii) If n kp, A k - n/p, then there is C such that
\u\x(0) C\\u\\kp(Q) for all u e
Hk'p(Q).
iii) If u e
Hk'p{£l)
and
Dau
= 0 on 9ft for \a\ k - 1, then u e
HS'P(Q).
iv) (Extension). There is a linear operator Lext mapping H
,P(Q)
into
H0
jP(Rn)
with the finite operator norm such that Lu(x) = u(x) for I G Q .
v) (Interpolation ) . There is a constant C = C(Q, k) such that
\\u\\(k)(Q)
C\\u\\k(^(a)\\u\\l-k/m(n)
for all u e H{m)(Q), 0 k m.
Proofs of these results are to be found in the book of Morrey [100], Chapter
3.
We note as well that an analytic function means a function on a domain
in R" that can be locally represented as the sum of its Taylor series, and a
complex analytic function is a function on a domain in
Cn
satisfying the
Cauchy-Riemann equations.
In many situations the following formula is useful:
(1.1.1) / D.udx= / un.dT (integration by parts).
JQ
J
Jon
J
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