- HEAT BALLS AND THE SPACE-TIME MEAN VALUE PROPERTY 373
Hence, from the absolute continuity of the function I(r), we haveu(xo, to) = r~ j j u(y, t) JV' log HJ^2 (x 0 , y, T) dμ dt
Er(26.115)
The theorem follows from this equality. DA consequence of the mean value property for harmonic functions is the
strong maximum principle (seep. 27 in Evans [58]). Similarly, a consequence
of the mean value property for solutions of the heat equation is the strong
maximum principle for the heat equation. In particular, tracing through
the proof of the theorem, one sees that if u is a subsolution of the heat
equation, then formula (26.106) holds with'=' replaced by ':S:'. Moreover, if
u(x, t) is a subsolution to the heat equation and if u achieves its maximum
on M x (0, T] at some point (xo, to), then one may infer from (26.106) that
u(x, t) = u(xo, to) for all (x, t) with t <to.
3.3. Mean value property using heat spheres.
In this subsection we present the mean value theorem for the heat equa-
tion on Riemannian manifolds in terms of integrals on heat spheres by adapt-
ing an argument of Fabes and Garofalo [60].
Without loss of generality, assume that to= 0. Let M ~ M x (-oo, O]
be equipped with the Riemannian product metric
n
g(x, t) = L 9ij(x)dxi 0 dxj + dt^2 ,
i,j=lwhere t is the global (time) coordinate on (-oo, O]. Again let H(xo, y, T)
denote the heat kernel centered at (xo, 0), where T ~ -t.
Applying the divergence theorem to the space-time vector field given by
8
uH
8
t + u\7H -H\7u
on a bounded (space-time) domain Din M, we obtain
(26.116)L ( ~~ -~u) H dμdt = L ( ( ~~ -Liu) H + u (
8
8 ~ + ~H)) dμdt= L div g ( u Hgt + u \7 H - H\7 u) dμ dt