360
Lemma 3.2. Let
(3.15)
Then
hi Av(x,z,s,t)dxdzdsdt= Li aT(V(X,Z,7,7+0)dxdzd~d7, V'vEW, (3.16)
where A = {(7,~) E R2: 0::; 7::; S, -7::; ~::; T - 7}.
Proof: Consider the change of variables
{
7 = s,
~=t-s.
(3.17)
The set R in the (s, t )-coordinate plan corresponds to the set A in the (7, ~)-coordinate
plan (see figure below),
t
T
o
and we have
R
S
T-S
s
v(x, z, s, t) = v(x, Z, 7, 7 +~) ,
Av(s, t) = (at + as) v(s, t) = aT[V(7, 7 + ~)J .•
Theorem 3.3. Suppose that
are, such that, 0°1.=0 = 01It=0.
S
7
Then there exists a unique solution u of the variational inequality (3.14) which satisfies
Proof: Consider the change of variables introduced in (3.17). Let a~ = max{O,-O
and b~ = min{S, T - O.
We write the variational inequality (3.14) along the characteristics of A. We have then,
for each fixed ~ E [-S, TJ that, if u solves (3.14), then
u((x, z, 7) = u(x, Z, 7, 7 +~), (x, Z, 7) E wx Ja(, bd