- EQUATIONS AND INEQUALITIES SATISFIED BYLAND£ 325
PROOF. (i) From (7.85) we have:r (L-2nr) +L1(L-2nr) ~O.
If L ( ·, ·) were C^2 , then the lemma would follow from the maximum principle.
Since we only know that L (·,·)is locally Lipschitz, we proceed with the max-
imum principle argument with some care. Define the function h: (0, T) --+IR
by
h(r) ~min (L(x,r)-2nr).
xEMCLAIM. For any r > 0 there exists q 7 EM such that
h(r) = L (qn r) - 2nr,and h ( r) is a continuous function.
If M is closed, then the claim follows from L(x, r) being a continuous
function when r > 0. When Mis not closed, the claim follows from Lemma
7.13(i), which says that limx---+oo L (x, r) = +oo. This and the local Lipschitz
property of L (x, r) imply the claim.
Now we estimate the right limsup derivative of hat r E (0, T):
d+h(_) _,_ 1. h(r + s) - h(f)
dr -=- imsup
T s---+0+ S1. L(qr+sJ + s) - L(q7, f)
= imsup - 2 n
s---+O+ S
<^1
. L(q7, f + s) - L(q7, f)
_ imsup - 2 n,
s---+O+ S
where the last inequality follows from the definition of qT+s. If L is C^2 at
(q 7 , f), then by using (7.84), we have
1. L(q7,f+s)-L(q7,f)8L( )
imsup - -
8
q7, r
s---+0+ S T
~ 2n - L'.lL(q 7 , f) ~ 2n.The reason why L'.lL(q 7 , f) ~ 0 is that q 7 is a minimum and smooth point of
L(·, f). We have proved that d;Th(f) ~ 0 when Lis C^2 at (q7, f).
When L is not C^2 at (q 7 , f), let L be a smooth barrier function of L
at (q 7 , f) as in (7.56). Setting L(x, r) ~ 2y'rLJ,x, r), we se~ that q7 is a
minimum point of the locally defined function L(·, f) since L is a barrier
function of L from above at (q 7 , f) and q7 is a minimum point of L(·, f).