361
is solution of the following parabolic variational inequality
u{(a{) =0, fora.e.7E]a{,bd,
(3.19)
L b(OrU{(7)) (v - OrU{(7)) + L V'U{(7)' V'(v - OrU{(7)) +
+ Jaw aU{(7)(v - OrU{(7)) + >.1 v+ - >. L (OrU{(7))+ ~
~ 1 f(7, 7 + ()(v - OrU«7)) + Jaw ag(7, 7 + ()(v - OrU{(7)), Vv E Hl(W) .
The problem (3.19) is nothing else than the parabolic two-phase Stefan problem defined
in w x [a{, bd for given data f{(x, z, 7) = f(x, z, 7, 7 + () and g{(x, z, 7) = g(x, Z, 7, 7 + ()
defined in w x [a{, bd and in ow x [a{, bd, respectively.
Due to the assumptions on f, 9 and a, using Theorem 2.1, we know that there exists
a unique solution u{ of problem (3.19) with the properties
u{ E W1,OO(a(, b(; Hl(w)) n H2(a{, b{; L2(W)) n LOO(a{, b{; H?a.,(w)) .
Integrating the variational inequality (3.19) with respect to 7 between a{ and b{ first
and then with respect to ( between -S and T, we obtain
Hence changing the variables (7,~) into (s, t) we have that u( x, z, s, t) = u« x, Z, 7, 7 + ()
solves the following problem
Ult=o = 0, uls=o = 0
inL b(Au(s,t))(v - Au(s,t)) + inL V'u(s,t)· V'(v - Au(s,t)) +
+' [ au(s,t)(v-Au(s,t))+>. 'lv+->. 'l(Au(s,tW~
lRlaw
lR w
lR w
(3.20)
~ [I f (s,t)(v-Au(s,t))+ , [ ag(s,t)(v-Au(s,t)), VVEL2(R;Hl(w)),
lR w
lRlaw
which can be easily seen to be equivalent to problem (3.14).
Using the regularity results obtained for u{ in w x [a{, bd, we have also
Suppose we have two solutions Ul and U2 of problem (3.14) with the properties (3.18).
Simple calculations show that, by subtraction and using (3.2), if w = Ul - U2 we have
Wlt=o = 0, wls=o = 0, and
b. [t rl1Awl 2 + ~ lIV' w(s, tW + ao , Iw(s, tW ::; 0,
10 10 w
2 w
2 law
which implies w = 0 a.e. and, therefore, the uniqueness of solutions. I