- WEAK SOLUTION FORMULATION 373
(ii) Let q E B (p, r) and let coordinates x be the same as in (i), where
\J\Jf (q) = D^2 f (0) exists. For any c > 0 we define the c= function on
TqM,lpq,c .(x) = f (0) +DJ (0) · X + ~XT • \J\Jf (q) · x + ~XT · x.
It is clear that f (0) = lpq,c (0), f (x) ~ lpq,c (x) in a small neighborhood of
0, and
\J\J<Pq,c (q) = \J\J f (q) +cg (q).Thus if \J\Jf (q) ~ kqg (q), then Hesssupp (f) (q) ~ kq.
(iii) Again assuming \J\J f (q) = D^2 f (0) exists, for any c > 0, define
'Pq,- 6 () x =f () 0 +Df(O)·x+2x lT ·\J\Jf(q)·x-2x_.x. cT
We have f (0) = rpq, 6 (0) and f (x) 2: rpq, 6 (x) in a small neighborhood of 0.
If Hess supp (f) (q) ~ kq < oo, then for any 0 > 0 there exists a C^2 local
upper barrier function cp : U-+ JR such that \J\Jcp (q) ~ (kq + o) g (q). Since
rpq, 6 ~ f ~ tp in a neighborhood of 0 and rpq, 6 (0) = tp (0), we have
\J\J f (q) - cg (q) = \J\Jrpq,c ~ \J\Jtp (q) ~ (kq + o) g (q).
Taking o -+ 0 and c -+ O, we conclude
\J\Jf (q) ~ kqg (q)'
and tracing, we have t:i.f ( q) ~ nkq.
9.4. The equivalence of notions of supersolution for nonsmooth
functions. Note that (Mn, fj) is a complete Riemannian manifold. Recall
the following definition.
DEFINITION 7.124 (Weak-type notions of supersolution). Let f: M -+
JR be a continuous function.
(i) Suppose f satisfies Hesssupp (f) ~ C in the support sense. Then f is
· said to satisfy t:i.f ~ k in the support sense for some continuous
function k if for every p E M there are a neighborhood U of p
and a constant C 1 which have the following property. For anyc > 0 and any q E U there are a constant r > 0 and C^2 function
<.p : B (q, r) -+ JR such that tp (q) = f (q), f (x) ~ tp (x) for all
x EB (q, r), l"Vcpl (q) ~ C1, and
t:i.1.p(q) ~ k+c.(ii) A continuous function f : M -+JR is said to satisfy t:i.f ~ kin the
weak sense for some function k E Lfoc (M) if for any nonnegative
C^2 function cp with compact support, we have(7.144) JM f f:i.cpdμ9 ~ JM cpkdμ9 ..