350
Remark 2.8. Under the assumptions
a) 8.100 ~ 0, 1001.=0 + 8~u~ ~ 0;
b) 8~/00 ~ 0, 8.1001.=0 + 8~/ooh.=0 + 8!u~ ~ 0, with 100 E H2(0, S; L2(n)),
u~ E H4(a, b);
c) I~ $; ->. < 0,100 $; ->. < 0 a.e.;
d) II/~ - looIlLl(O) $; Cv;
we have
1.
IIxII - XooIlL'(O) $; C v' , I $ ; p < +00 • D
Ideas of the proof:
i) The assumptions a) and b) garantee that 8.uoo ~ 0 and 8~uoo ~ 0;
ii) U oo solves the nonlinear elliptic problem
{
8.uoo - 8~uoo - v8~uoo = (too - v8~uoo) X{u .. >O} ,
uoo(a) = uoo(b) = 0, uool.=o = u~, 88 uoo I = 8.uoo ,
n .=5
or the equivalent variational inequality
10 8.uoo( v - uoo ) + 10 8",uoo 8",( v - uoo ) + v 10 8.uoo 8.( v - uoo ) ~
~ 10 loo(v - uoo ) - v 10 8~uoo(v - uoo ) + v l 8.uoo(S) (v - uoo(S)), 'Vv E Koo .
Then, applying a known result (see Thm. 5:4.7 of [R2]), which uses the assumption c),
we conclude the estimate from d) and
Remark 2.9. This is an interesting mathematical estimate which, however is of little
physical relevance in this model., due to the unrealistic properties 8.uoo ~ 0 and 8~uoo ~ 0
in the case of a glacier. D
2.2. The evolutionary problem
The variational inequality for the evolutionary problem is the following one:
I
UIl(O) = u O , ull(t) E K(t) for a.e. t E ]0, T[ ,
10 8tu ll (t)( v - ull(t)) + 10 8.u ll (t)( v - ull(t)) + 10 8",u ll (t) 8",( v - ull(t)) +
+ v 10 8.u ll (t) 8.( v - ull(t)) ~ 10 r(t)( v - ull(t)) , ' V v E K(t), a.e. t E ]0, T[ ,
(2.14)
where
(2.15)