- HEAT BALLS AND THE SPACE-TIME MEAN VALUE PROPERTY 371
Noticing that the set Er,t is empty for t sufficiently close to -T, by the
co-area formula (see Evans and Gariepy [59] for example), we have
J(r) = 1-~lr,t u(y,t)[V''l/Jr[
2
dμdt= 1° (1
00
( r u(y, t) \V''lf;[ da) de) dt,
-oo -nlogr J'lj;(y,r)=cwhere
'lj;(y, r) ~ logH(xo, y, r) = 'l/Jr(Y, r) - nlogr
and dais the induced volume element on 'lj;(y, r) = c.
Hence, for almost every r, using the fact that the outward unit normal
v of 8E~ is given by - 1 ~~~ 1 , we have
dd J = 1° '!!:. ( r u(y, t)\V''l/Jr\ da) dt
r -oo r J 8Er,t= _'!!:.1° ( r u(y, t)(V''l/Jr, v) da) dt
r -oo j EJEr,t= _'!!:.1° ( r (('\Ju, V''l/Jr) + ub.'l/Jr) dμ) dt
r -oo }Er,t(26.112) = '!!:.1° ( r ( 'l/Jrb.u - ub.'l/Jr) dμ) dt,
r -oo }Er,twhere we have used the divergence theorem and where we have also used
the fact that 'l/Jr = 0 on 8Er,t·^8 Now we may apply (26.109) to (26.112) and(^8) Note that
div(u'V'¢r) ={'Vu, 'V'¢r) +u6.'¢r,
div ('¢r \Ju)= ('V'¢r, 'Vu)+ '¢r6.u.