346
where u!o ~ 0, u!o(a) = u!.,(b) = 0, is a given function.
As it was seen in the last section, the steady-state problem can be formulated as
the following elliptic variational inequality (recall (1.23»
u~ E Koo: av( u~, v - u~) ~ 10 1:"( v - u~), \I v E Koo ,
(2.3)
where, for II > 0,
av(u,v) = 10 O.UV+II 10 o.uo.v+ 10 o"'uo"'v, \lu,v E Hl(O) ,
and I:" = I:"(x,s) is a prescribed function.
Proposition 2.1. Suppose that I:" E L2(0), u!o E HJ(a, b), u!o ~ O. Then the
variational inequality (2.3) has a unique solution u~, which satisfies
a.e. in 0,
(2.4)
where XA denotes the characteristic function of the set A.
Proof: Since av is a continuous and coercive bilinear form defined in V, the existence
and uniqueness of solution of the variational inequality (2.3) is a direct consequence of
Stampacchia Theorem (see [R2J, for instance).
By standard regularity results for elliptic variational inequalities, we know that u~ E
H~(O). So, in the coincidence set {u~ = O} we have o.u~ - IIO~U~ - o;u~ = 0 a.e.
and, since in the non-coincidence set {u~ > O} we have o.u~ - IIO~U~ - o;u~ = I:" a.e.,
(2.4) is satisfied. I
Consider now the limit parabolic problem, obtained by letting II = 0. It is formulated as follows
{
uoo(O) = u!o , uoo ( s) E K, for a.e. s E ]0, S[ ,
t O.Uoo(S; (v - uoo(s)) + t o",uoo(s)o",(v - uoo(s)) ~
~lloo(s)(v-uoo(s)), \lVEK, a.e.sE]O,S[,
where K = {v E HJ(a,b): v ~ O}.
(2.5)
Proposition 2.2. Let u!., E K and suppose that 100 E L2(0). Then the variational
inequality (2.5) has a unique solution U oo E Hl(O) n L2(0, S; H2(a, b)) which satisfies
O.uoo - o~uoo = 100 X{uoo>o} a.e. in 0,
(2.6)
Proof: Existence and uniqueness are obtained directly from known results for parabolic variational inequalities (see [L2,3], for instance).
Let
,8 E C 2 (R) , ,8' ~ 0, ,8(u) = ° ifu ~ 1, ,8(0) = -1,
(2.7)