2.3. DIFFERENTIAL FORMS
9
2.3. Differential forms
Throughout this paper we let ^°°(A^X), 0 ^ £ ^ n = dimX, denote the space
of smooth £-forms on X. Two differential operators on forms will be of particular
interest to us. First is the exterior derivative,
(2.4) d : * f °° (A*X) - ^°° ( A m X )
Second is the formal adjoint of d, also called Hodge codifferential,
(2.5) d* = ( - l ) n £ + 1 * d* : S?°°(AmX) -• ^°°(A£X)
where * : ^^(A^X) »
^°°(An~£X)
denotes the Hodge star duality operator. Here,
we conveniently set ^^(A^X) = 0, whenever £ 0 or £ n. Note that ** =
(-l)*(n-€) on
^°°(A£X).
The point-wise scalar product of forms a,/? e ^°°(A^X)
is given by (a, /3) dx = a A */? G
^°°(AnX),
and hence
(2.6) / (a,(3) dx= [ aA*(3
Jx Jx
The duality between d and d* is emphasized in the formula of integration by parts
(2.7) [ (dp,1)= /*,*»
for if G
^°°(A€X)
and ^ G
^oc(A€+1X).
Now a differential form p G
£?p(AeX)
is
said to be closed in the sense of distributions if J
x
((/?, d*ip) = 0 for every test form
i/j G ^ ^ ( A ^ X ) . We write it as dip = 0. Similarly, we establish what it means
for ip to be coclosed, and write it as d*ip = 0. Forms of the type da, with a G
^ ^ ( A ^ X ) , are called exact while those of type d*/3, with (3 G # ^ ( A ^
+ 1
X ) , are
called coexact. It follows from the identities dod 0 and d* od* = 0 that the £-forms
da G
J*?P(A£X)
and d*/? G
J*fp(A£X)
are closed and coclosed, respectively. Finally,
those forms h G
Jt?p(A£X)
which are closed and coclosed will be called harmonic
fields of degree £. We denote by
H(A£X)
the space of all harmonic fields of degree £
and regard it as well known that this space is finite dimensional.
7i(AlX)
consists
of ^°°-smooth forms. Being so, all possible norms on
H(A£X)
are equivalent. For
instance, we shall benefit from the estimate
(2.8) \\h\\ , ^ \\h\\
1
,
and further,
(2.9)
ftU(A£x)
*
(JXWP)P forallP«
In relation to the imbedding _2^eak(A*X) C
5fp(AeX),
with 0 p 1, we record
the following estimate
(2.10) IMU(A,X) ^ ( J ^ ) " ^
\%\^ supy fhtdx t0\
as is easily verified by Tchebyshev inequality, see (4.6).
2.3.1. Sobolev classes of differential forms. Four spaces of differential
forms have a special place in our studies. These spaces are:
- The Sobolev space of closed forms:
(A^X)nkerd
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