- HEAT BALLS AND THE SPACE-TIME MEAN VALUE PROPERTY 367
Note that the condition t;:::: ~; is equivalent toEr(x, t) C ffi.n X [O, oo ).Theorem 26.39 follows from (26.95) and the following (compare this with
(26.87) in the elliptic case).LEMMA 26.40. For any (x, t) E ffi.n x (0, oo) and r < .J41ft,
[)(26.97) n-<Pxt(r) = 0.
ur '
The proof of Lemma 26.40 uses integration by parts. Since we shall
discuss its generalization to the heat equation on Riemannian manifolds in
the next subsection, we omit its proof.
Note that since
(
a ) 2 Ix -yl
2-n-- ~Y logHm; = IV'ylogHm;I = 2 ,
us 4(t-s)
we may rewrite the mean value property (26.96), in terms of integrating
against the heat kernel, as
(26.98)u (x, t) = r~ Jr r u(y, s) IV' y log Hm;l^2 (x, t; y, s) dμm; (y) ds
} Er(x,t)
(26.99)= r~ f lr(x,t) u(y, s) ( (-:s - ~Y) log Hm;) (x, t; y, s) dμm; (y) ds.
This observation is useful in formulating its generalization to Riemannian
manifolds, where one does not have an explicit formula for the heat kernel.
Note also that in (26.99) we may replace log Hm; by 'I/Jr since they differ by
a constant (by (26.90)), i.e.,
(26.100)
u (x, t) = _!__Jr r u(y, s) (-~ - ~y) 7/Jr(x, t; y, s) dμm; (y) ds.
rn j Er(x,t) us
We now derive the mean value property for heat spheres. Since 'I/Jr = 0
on 8Er(x, t), replacing log Hm; by 'I/Jr and integrating by parts on (26.100) in
the time and space directions separately, we have
u (x, t) = ~Jr r (~u -~yu) (y, s)'l/Jr(x, t; y, s) dμm; (y) ds
r j Er(x,t) us_ _!__ft { u(y,s)(V'y'l/Jr,v)(x,t;y,s)d(J's(Y)ds,
rn lt-~! j 8Er,s(x,t)where Er,s(x, t) is the time slice of Er(x, t) defined by (26.92), v is the
outward unit normal to its boundary 8Er,s(x, t), and d(J' 8 is the induced