34
Proof.- We apply the proposition 2 for the following choices:
U = ]RN, W = C(A x [0, T]),
C={wEW:w(x,t)::;u, VxEA, VtE[O,T]}
G(b) = CbIAx[O,Tj.
Then there exist .A ~ 0 and a measure I' E C(A x [0, T])' such that
.A+ I I I' II> 0
(1', w - c;;) ::; 0, Vw E C(A x [0, T]), w(x, t) ::; u,
.A(b - a, b - b)+ (1', V",c(b, .).(b - b)) ~ 0, Vb E Uad.
(3.96)
(3.97)
(3.98)
(3.99)
(3.100)
(3.101 )
Let us consider the problem of finding p E 1'([0, TJ, w1,q(O)) such that, for every
zEYonC1(Q)
h ~;pdxdt+ h({JVz.Vp+u.Vzp)dxdt+ hCl.zpdxdt= hzdp.(x,t) +
~ zdp.( x, t) + in z( x, T)dp.( x). (3.102)
where
Yo = {z E C(A x [0, T]) n 12(0, T, Hl(O)): z(x,O) = 0, Vx EO}.
(3.103)
In Casas[1984] existence and uniqueness of solution for this problem is proved, Vs, q E [1,2)
with 2/s + N/q > N + 1.
Now we are going to show the equality
(1', V",c(b, .), h) = loT V",p(b, t).h, Vh E ]RN.
(3.104)
For this let us take a regularizing sequence {8n } for the Dirac measure 5 at point zero.
Suppose the support of 8n is included in the ball B(O, ~). Let Znh be the solution of the
problem
8znh
-
at + U.VZnh - {Jb.znh + Cl.Znh = -m(t)V8n(x - b).h
Then we have, for x E A, t E [0, T]
8Znh = 0
8n
Znh(X, 0) =0
in Q,
on ~,
on O.
(3.105)
(3.106)
(3.107)
(3.108)