366 26. BOUNDS FOR THE HEAT KERNEL FOR EVOLVING METRICSAnother useful way to describe the heat ball is
Er(x, t) = {(y, s) E JR.n x (-oo, t): 'I/Jr (x, t; y, s) > O}.
Hence
(26.91) 8Er(x, t) = {(y, s) : 'I/Jr (x, t; y, s) = O}.
Each time slice of Er(x, t) is a round ball. In particular, givens E (-oo, t),
(26.92) Er,s(X, t) ~ {y E JR.n: (y, s) E Er(X, t)}= {y E lR.n: Ix - YI < t-s (r)},
where
<I>t-s ( r) ~ v 4 ( t - s) ( n log r - ~ log ( 4n ( t - s)))is the radius of the times slice of Er(x, t). Fixing r, note that the supremum
of <I>t-s ( r) occurs when t - s = ;;e. Hence the maximal radius of a time
slice of Er(x, t) issup <I>t-s (r) =
sE[O,t)ne r2 ( nlogr - n (r2))
2 1og ~.
Now we are ready to discuss the space-time mean value property.u: JR.n x [O, t) ~JR. be a C^2 solution to the heat equation. Define
(26.93) </>x,t (r) :::;=. 1 - fl lx-yl2
4u(y, s) ( ) 2 dμE (y) ds,
rn Er(x,t) t - s
where dμE denotes the Euclidean measure on JR.n.EXERCISE 26.38. Show that(26.94)^1 ft lx-yl
2
- 4 ( ) 2 dμE (y) ds = 1.
rn Er(x,t) t - s
HINT: See p. 409 of [185].
LetNote that (26.94) implies that <f>x,t (r) is a weighted average of u on the
heat ball of radius r based at (x, t).
Since limr-+O Er(x, t) = {(x, t)} in the Gromov-Hausdor:ff sense and since
u is continuous, by (26.94) we obtain
(26.95) u (x, t) = lim <f>x t (r).
r-+0 '
With these preliminaries, we may state the following mean value prop-
erty for heat balls.THEOREM 26.39 (Space-time mean value property for the heat equation- Euclidean space). If u : JR.n x [O, t] ~ JR. is a classical solution to the heat
equation and if t 2 ~: > 0, then
(26.96) u (x, t) =^1 ft Jx-yl2
(^4) r n Er(x,t) u(y, s) ( t - s )^2 dμE (y) ds.