1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

9. WEAK SOLUTION FORMULATION 375

D^2 f ( x) uniformly on Bp ( !r) \ W 01. This in turn implies that b..9 fc:k ___,. b..9 f

uniformly on Bp ( !r) \ W 01. We compute using b..fc:k S 0,

Since meas (W 01 ) S 51 and b..f E Lfoc by Remark 7.123(i), if we let 51 ___,. O+,

then f w b..f · <pdμ 9 ___,. 0 and

"1

r, 1 b..<pdμ9 s r

1

b..J. <pdμ9 = r, b..J. <pdμ9.

JM JBP(2r) JM

If b..f s k in the support sense, then by Remark 7.123(iii), f satisfies

b..f s k a.e. on M. Combining this with (7.145), we get

JM f b..<pdμ9 s JM <pkdμ9.

(ii) This equivalence is well known. One can find a proof that f satisfies

b..f s k in the viscosity sense if and only if f is a supersolution of b..f s k

in L. Hormander's book [205] on p. 147, Proposition 3.2.10. One can find a

proof in Juutinen, Lindqvist, and Manfredi [226] that f satisfying b..f skin

the viscosity sense is equivalent to b..f s k holding in the weak sense, where

they actually prove the equivalence for div (1\7 flp-^2 \7 f) s k for p > 0.

Actually, the equivalence of the notions of weak and viscosity supersolutions
was first proved by Ishii and Lions [214]. D

9.5. Comparison theory for Riemannian distance d and re-

duced distance £. As a simple application of the results in the previous

subsection we consider the distance function d on a Riemannian manifold

and the reduced distance f of a solution to the backward Ricci flow.