- WEAK SOLUTION FORMULATION 363
Now we compute that the evolution of the £-Jacobian along /'Vis given
by= ~ d~'T=T logdet((Ei,Ej) (r)) + ~ d~'T=T <let (Af) + ~ d~'T=T <let (A])
1 n d I
= 2 tt dr T=?' (Ei,Ei) (r)< - _!~ 2 £ (^17) v IT~ ., L.J H (x ' JJ;.) i dr +! 2 ~ L.J IEi -(r)l2
VT 0 i=l i=l T
The last inequality is due to (7.129). Here the Ei (r) are the vector fields
along /'V satisfying
!xEi - =-Re (-) Ei +^1 -
2 TEi,
where Ei(r) = Ei(r) and H (x,Ei) (r) is the matrix Harnack quadratic
given by (7.63). By (7.72), we have (Ei,Ej) (r) = ¥<5ij, and by (7.74), we
have n
~H ( X,Ei) (r) = ~H (X) (r).
i=l
So
(d
d log.CJ) (f) :s; - -~/ 2 {
7
r^312 H (X) dr + n_T 2T lo 2T
1 n
(7.141) = - 2f 3 / 2 K + u·If equality in (7.139) holds, then we have equality .in (7.129) for each
Y (r) = Ei (r), i = 1, · · · ,n. By (7.130), we have
(
- ) 1 ( - - ) I Bi( f) 12
2Rc Ei(r), Ei(r) +VT (HessL) Ei(r), Ei(r) = f
for each i. Since Ei (r) = Ei (f) can be chosen arbitrarily, this implies
(7.140). D
9. Weak solution formulation
The purpose of this section is to prove the integration by parts inequality
(7.148) for the reduced distance Rand to give the inequalities we proved for
R a weak interpretation. We first recall some of the well-known results in
real analysis which we shall need. An excellent reference for properties of