1. WEAK MAXIMUM PRINCIPLES FOR SCALARS AND 2-TENSORS 3
associated to g (t). Given a C°° function u : M x [0, T) M, we define the
semilinear second-order parabolic operator:
Ov
(io.i)
Lu^Yt~ As(i)U
~
x ( t )
'
V u )
~
F (n'*}'
where X (t) is a time-dependent vector field and the function F : Ex [0, T)
K. is locally Lipschitz in the M factor and continuous in the [0, T) factor.
The essence of the scalar maximum principle is contained in the following
comparison result.
THEOREM
10.1 (Comparison principle). Suppose that u and v satisfy
Lu Lv on M x [0, T) and u (x, 0) v (x, 0) for all x G M. Then
u (x, t) v (or, t) onMx [0, T).
This theorem can be proved in the same manner as for Theorem 4.4
on p. 96 of Volume One using the second derivative test. As a special case
we may take v = v (t) to be a function depending only on time and may
consider the ODE Lv ^ F (v,i) = 0, which is no longer a PDE since
the Laplacian term and the gradient term vanish. Hence, as a consequence,
we have the following two scalar weak maximum principles. We have the
following result for subsolutions.
THEOREM
10.2 (Scalar maximum principle (SMP)). Suppose that u sat-
isfies Lu 0; where L is defined by (10.1), and suppose u(x,0) c for all
x G M and some constant c G R. Let (f)(t) be the solution to the correspond-
ing ordinary differential equation:
(10.2) ^ = F ( 0 , t ) ,
(10.3) 0(0) = c.
Then
(10.4) u{x,t) cj)(t)
for all x G Ai and t G [0,T); as long as the solution to the ODE exists.
We have the analogous result for supersolutions where we replace by
in the statement of the above theorem. For a solution we obtain both
upper and lower bounds.
THEOREM
10.3 (SMP—upper and lower bounds). Suppose that u satis-
fies Lu = 0 for L as in (10.1) and suppose c\ u (x, 0) C 2 for all x G M,
where c\,C2 G M. Let (fi(t), i 1,2; be the solutions to the associated ODE;
K0) = ci.
Then
Mt)u(x,t)42(t)
for all x G M and t G [0, T), as long as the solutions to the ODE exist.
Previous Page Next Page