Free boundary problems for the laplace equation - Wiley Online Library

Math. Meth. in the Appl. Sci. 3 (1981)38-69 A M S subject classification: 35 J99,35J05

Free Boundary Problems for the Laplace Equation: A Priori Estimates, Existence Theorems, Asymptotic Behaviours A. Dervieux, Le Chesnay, and H. Kawarada, Tokyo

Communicated by H. Fujita This work is devoted to Free Boundary Problems with Laplace equation Au = f in the domain and the two conditions on the Free Boundary u = 1 and lgrad u I = I = const. In the model problems which we study, three cases arise: 1)f = 0, A is given, 2)f = 0,I is unknown and the length of the Free Boundary is given, 3)f # 0,I = 0 (Obstacle Problem).

Introduction

This work contains two parts. The first part concerns the two following Free Boundary Problems (F.B.P.): P r o b l e m 1: Find a function u and a curve y such that u = 0

on a (given) boundary r

.

lgrad u I = k

on the curve y (k is given)

Au = 0 in the domain between r a n d y P r o b l e m 2: Find function u and a curve ysuch that u = 0

on r(given)

u = l

Igrad u I is constant

on the curve y au I---do = C(given) av (v is the outward normal vector) Au = 0 in the domain between r a n d y. 0170-4214/81/01 0038-32 $ 06.40/0

0 1981 B. G. Teubner Stuttgart

Free Boundary Problems for the Laplace Equation

39

Problem 1 has a hydrodynamical origin and is a generalization of a wave or jet problem. Problem 2, which can be interpreted as an inverse probiem with respect to the first one, is connected with the Stefan problem and occurs in cryogenics experiments; that kind of problem is also introduced to modelize plasma configurations in tokamak machines when the plasma is subject to skin effect (see for example A. S. Demidov [l]). It is interesting to note that these two problems have not been transformed into Variational Inequalities (nor Quasi-Variational Inequalities; see C. Baiocchi [1]) so that methods developped for that family of problems are not, as far as we know, available for the above two problems. Our main point of view is to regard the Free Boundary as the most important unknown variable. A very simple a priori estimate is exhibited for the Free Boundary of Problem 1 and consequences are derived for some asymptotic behavior of this problem. From continuity and asymptotic properties of Problem 1, existence and uniqueness of a solution to Problem 2 is showed, as well as an iterative process is constructed for a convergent approximation of the solution of Problem 2 by solutions of Problem 1. Many authors studied Problem 1: for instance A. Beurling, D. E. Tepper, G. Wildenberg studied it when the Free Boundary is outside of the fixed one. The “inner case” has been considered by I. I. Daniljuk. Conformal mappings have been used by these authors whose results we use; but, on the other hand, our methods do not use that kind of arguments, which allows us to hope that they could be widely generalized. In the second part we study a particular obstacle problem

Au

=finQ

u = 1 on the curve A B t3U = OonAD

ax

u =

v BC

au = 0 on the free boundary D C

av

The important point of this study is to obtain an a priori estimate for the slope of the Free Boundary;

40

A. Dervieux and H. Kawarada

Indeed this can be considered only if the geometrical data have a distinguished direction: the slope of CD is said to be zero if C D a straight segment perpendicular to A D and B C. It is interesting to note that this estimate may imply the regularity (and even smoothness) of the free boundary. Naturally this estimate has some classical-type applications, two examples of which we give, concerning optimal control and asymptotic behaviour. It is important to note with D. G. Schaeffer [2] that the regularity of the data does not generally imply the regularity of the free boundary; as in D. Kinderlehrer, G. Stampacchia [l] (see also C . Baiocchi et al. [l]) we start from assumption concerning only the data.

"he plan is following: Part I: Jet-like Problems 1. Inner Problem 1.1 Formulation of the Free Boundary Problem 1.2 The circular case 1.3 An a priori estimate for the length of the Free Boundary 1.4 Consequences for asymptotic behaviour 2. Outer problems 2.1 Position of the problems 2.2 Some known properties 2.3 Main results 2.4 Proofs

Part II: An Obstacle Problem 1. A regularity theorem via an a priori estimate 1.1 Preliminaries 1.2 Main results: An a priori estimate for the Free Boundary 1.3 Proof 2. Application to an optimal control problem 2.1 Position of the general problem 2.2 Definition of our problem 2.3 Existence of a n optimal control 2.4 Proof 3. Application to an asymptotic study 3.1 Formulation of the problem and main result 3.2 Proof

Part I: Jet-like Problems 1

Inner Problem

1.1 Formulation of the Free Boundary Problem Tis a smooth curve without self-intersection points which lies in the 2-dimensional space; Q(x, y) is a real-valued, non negative function defined in the region 52 bounded by the curve T(52is bounded). The F.B.P. is (Po) Find an other simple closed sufficient& smooth curve y c 52 which, together with rforms the boundary of a region 52, for which there exists a function u such that


41

Au = 0 in l.2, (11

u = Oonr u = l

]gradu I = Q

jony.

Fig. 2

Theorem 1 (Daniljuk [l], [2], [3]) Wesuppose that Q isafunction of C”(d) such

that VX,Y

(2)

~ f i , Q ~ Y 2) Qo > 0

Inf j (IgradzI2 + Q2}dxdy < j Q’dxdy Y.2

R,

R .

where the infimum is taken over every smooth curve y and every smooth function that z = 0 on r a n d t = 1 on y. Then problem Poadmits at least one solution such that y is a CODcurve.

.Z such

The solution of Problem Podoes not always exist, and is generally not unique; to illustrate this, we shall study a particular case which gives a good intuitive idea of what happens in the general case as well for the number of solutions as for their asymptotic behaviour. 1.2 The Circular Case

The supplementary assumptions are

(3)

Q = k = const. Tis a circle with radius R ; We put kmin= exp(1 - 1nR)

Proposition 1 (trivial) Under the assumptions (3), (i) Problem Po admits

no symmetric solution for k < kmin one symmetric solution for k = kmin two symmetric solutionsfor k > kdn

42


(kh, corresponds to a turning point for the branch of symmetric solutions); (ii) If k tends to infinity, afirst symmetric free boundary converges to the center of r while the second converges toward K 1.3 An a Priori Estimate for the Length of the Free Boundary 1.3.1 Estimate for the Distance between the Boundaries Proposition 2 For every pair ofpositive constants (c0,Qo), for any rand Q such

that Q(x) < Qo

VXS.t. d(x, r ) < EO

there exists a positive constant E~ = el (e0, Qo,r ) such that the corresponding Free Boundary y satisfies d(Y>r ) > E l .

(4)

P r o o f . Let us first consider the case where Tis a circle with the origin 0 as center and R > 0 as radius, and let us introduce the domain

{x;d(xJ r )

=

< %}

and the function

where K is chosen sufficiently large so that

--6P > Qo

(5)

6r

withr = 1x1,

and

(6)

v,(x)

> 1 forax#= R - %;

we shall show that (7)

v, 2 u inQo n 0,.

For this purpose, we consider the two following cases ; (7) follows trivially from (6) and the Maximum Principle. (i) d(y, r ) 2 E ~ then (ii) d(y, r ) < En; we use a contradiction argument: if (7) does not hold, then moreover, since v, depends only Max - v,) is positive and occurs on y n 0,;

(u

X€lb

on 1Ix Itand since u is constant on y , this maximum occurs in point(s) xosuch that d(x,, r ) = d(y, r ) and y is tangent at xoto a circle with the origin as center, so

but, using ( 5 ) and the assumption of Proposition 2 we get


43

On the other hand, by the strong Maximum Principle, we have

which is in contradiction with (8). Thus (7) is proved. From the definition of E ~ R, [ such that

(p

]R (9)

and from (6), there exists a number ro belonging to

~ ( x=) 1 if 1x1 = ro

therefore, if we chose

el = Min{Eo,R

- ro},

the conclusion follows from (7) and (9). Let us now deal with the general case (fis no more a circle). The region 52 is a regular simply connected domain; then there exists a (fixed) regular conformal mapping T which maps D into the disk { e c 1). Let us denote by ii the transported solution:

ii = u

o T-I;

by invariance of harmonicity by conformal mapping we have AC = 0 inT(S2,) on the other hand we have

ii = 0 o n ( @= 1) = T ( T ) ii = 1 on = T(y); thus if we introduce

Q - = - -aii ( av

;

an L"-estimate for Q implies an L" estimate for 8 because the conformal mapping T belongs to W'," and its inverse too. Thus C fulfills the conditions of the above particular case: we deduce an estimate for the distance d(7, {e = 1)) and therefore an estimate for d(y, 0. 1.3.2 Estimate for the Length of the Free Boundary Proposition 3 Under the assumptions of Proposition 1, we have, f o r every solution y of Problem Po

(10)

jQda Y

< K(eO,Qo,r);

Kindep. of Q .

44

A. Dervieux and H.Kawarada

P r o o f . We have (11)

d(y, r ) > c1,

cl fixed.

Let u be defined by u E H’(52)

(12)

Ei v(x) = 0 for d(x, r ) < 2

u(x) = 1 for d(x, I) > cl. Let us estimate the difference z = u the datum Q; we have (13)

zI~=

AZ = - A u ,

thus

2

lz Iln;(nr) =

-I

4

0,

- u, where u is the solution of the F.B.P. z(,= 0

A u dx G Az UL2(ny)I A v IL2(Qy)*

Then (by Friedrichs-PoincarC inequality)

(14) IlZtlHl,(*) Q K(Q)I A+(*) where we use the same notation for the zero extension of z . Let us also denote by u the 1-extension of u in 52; the statement (14) implies (15)

twHl,(n) Q (w) + 1)

U U U ~ ~ ( ~ ) .

By Green’s formula

au ju-da

av

=

I [ U A U+ (gradu, gradu)]dxdy 4

- jv-da au

r av

and by (15 )

G (m-3+ 1) RvllSrz(n) so that we have proved Q d o Q const. Y

1.4 Consequences for Asymptotic Behavior

The following result is a direct consequence of Proposition 3. Proposition 4 We suppose that

(9 (ii)

d(x, r ) < GI * Q n b ) < QO V X E S C QQ,, ( x ) > k , , + a a s n - ,

+a

for


45

for a fixed domain $ then MaxI(ynS)-+O Y

where the maximum is taken over the solution(s) of Problem Po P r o o f . Indeed we have for every solution y of (1): kn * I(Y n 9)
no = d(x0, y ) > E .

E

> 0 and no E N such that

P r o o f of the Lemma 1. We have

but

d(x0, y n ) 2 d(x0, x * )

- Max d o , x*) YEYn

Max d(y; x*) < I(y,,)

+ d(y,,

x*)

+

0

YEYn

which implies the conclusion. End of the proof of Proposition 5 : Using the above Lemma we deduce

v n > no,

W O ,E ) = {Y; 1x0 - Y

I c EI C a;,

46

A. Derview and H. Kawarada

for any function q E 9 (B(xo,E ) ) we have, from the weak convergence (cf. (16))

5 (grad u,, grad q ) dx

(grad u*, grad q ) dx as n + m.

--+

R

R

Now the first member is zero, thus

Au* = 0 inB(xo, E ) . The point xo was arbitrary chosen, then (17)

n u * = 0 p.p.inSZ,

so that u* is the unique, vanishing (from (16) and (17)) limit. 2

Outer Problems 2.1 Position of the Problems

Let 9 be a doubly connected region limited by the point at infinity and a convex boundary component r with a continuous tangent and which does not reduce to a point; if k is a given positive number then it is known (A. Beurling [l]) that there exists an unique annulus SZ C 9 having r as one boundary component and an other boundary component y, the Free Boundary, such that there exists a function u defined in SZ satisfying (cf. Fig. '3) (18)

(PI

Au

=0

onQ

(19) u = 0 o n f (20) u = 1 on y (21) lgraduI= k o n y .

Fig. 3

That problem will be called Problem P; we shall also consider a second problem, Problem (P'), in which (21) is replaced by (21')

To find (u, y ) such that (18) (19) (20) (21') hold, with (PI)

c

(21') lgrad u I = -on y 1(Y) where C is a given positive number and I ( y ) is the length of y .


47

Since I gradu I is constant on y , the number C denotes the circulation of this quantity along y ; from (18) (19) (20) and from Green's formula, it represents too the total energy of the system C = Jlgradul'dxdy. R

2.2 Some Known Properties After A. Beurling, D. E Tepper [l], [2] (cf. also more recently Tepper, Wildenberg [l]) studied the qualitative properties of the Free Boundary of Problem (P); we give here some results of those authors which will be useful in the sequel.

Proposition 6 (Regularity property) The solution y of Problem P is an analytic closed con vex curve. Proposition 7 (Monotonicity property) If kl > k2 > 0, then the corresponding domains verify fikl

c 0,.

Proposition 8 (Surjection property) For any point zo of 9 , there exists some k such that 20 E Y k j

where Y k denotes the Free Boundary corresponding to k, that is to say (22)

9 =

u

yk.

k >O

Proposition 9 (Asymptotic behaviour) Let (23)

z

0

=f ( w ) = cw

+ a. + C

anw-"

n=1

be a conformal mapping which transforms 9 into the exterior circle; the annulus f-' (Ok) will be denoted by Of; then for k > 0 there exists two positive numbers ak, PkSUCh that and ( p k -

ak) tends

to zero when k tends to zero.

2.3 Main Results

Theorem 2 (Existence and uniqueness) Under the assumptions of Paragraph 2.1, and for every given positive number C, there exists a unique solution (ii, 7) of Problem P'. We need, for the statement of the next result, to introduce the following notions:

Definition The polar parametrization of the regular boundary y of a starlike set is defined as a functions still denoted y (in the sequel this simplification of notation will be used systematically) such that

48 (24)


y E C”(0,2 R ) , periodic withperiod2~

the boundary y = {(rcos 8, rsin 8), B E [0,2x],r = y(O)}.

This parametrization is a fortiori possible for regular boundaries of convex sets: the origin of the axis is chosen inside r. Now we shall show how the solution of Problem P’ can be approximated by solutions of Problem P; let us consider the following iterative process: Step I : Choose kosuch that co = ko . I( yo) > C fi

L

Step 2: Define k, + = yn+l

[(Y”) = ~(ki+l)

Cn+l

=

kn+l *I(Yn+l)

where y(k) denotes the F.B. solution of Problem P corresponding to the datum k. Theorem 3 (Approximating process) Under the assumptions of Theorem 2, the above algorithm converges monotonically to the unique solution of P‘for the datum C: y in C O (0,2 R ) (i.e. uniform&)

y,/

l(Ynb-QY)

c

\ ,c En\

ii in C o ( 9 )a n d H ’ ( 9 ) .

Remark 1 In a numerical implementation, Problem P, whose solution is used in Step 2, can be solved for example by “trial methods” (see the complete survey of C. W. Cryer [l]). 8 2.4 Proofs 2.4.1 Monotonicity and Convergence Results for Problem P Let us first recall the following classical result (see for example M. Berger [I]). Lemma 2 Let sZi (i = 1, 2) be two closed convex regions of R2 with smooth boundaries yi = aQi such that sZ2

then

c

s17

sZ2

#

sZ1

~ ( Y I> ) ~(Yz).

Lemma 3 Let (k,)( n = 1 , 2 , 3 , (25)

...) be a sequence of positive numbers such that

kl > k2 > ... > k, > k,+’

ko €10,

a[

when n

-+

+ a.

Let us denote by y, (resp. l,, u,) the corresponding Free Boundary (resp. its length, and the distributed solution) f o r Problem P ( n = 0, 1, 2, ...). Then, when n tends to infinity we have


(i) (ii)

< Yn+ 1

yn

I,

+

49

yo in ~ ~ ( 0 , 2 1 1 (see ) (24))

-,fo.

Moreover if ii, denotes the I-extension of u, in the open set iR whose boundaries are f and yo (n = 0 , 1 , 2, ...), then (iii) (iv)

ii, ii,

+ uo in

~'(0)

+ uo in

N' (a).

P r o o f . Using (24), then sequence of boundaries yl, ..., yn corresponds to a sequence of functions yl, ..., y,; from Proposition 7, this sequence is strictly increasing and converges to a function ym: y n r y m everywhere in [0,211] Ym

G

YO*

Let us show that (26)

yo.

70;

we use a contradiction argument: if (26) does not hold, then, if we denote by SZ, the bounded domain with ym as boundary, there exists a point P such that (27)

P E 52,\fim ;

but from Proposition 8 there exists a positive number k* such that P belongs to the Free Boundary y(k*) of Problem P for k = k*; if ko 2 k*, then, by Proposition 7, we have y(k*) 2 y(ko),which contradicts (27); thus k* > k,; therefore there exists some k, in the sequence (k,) such that k* > k, and yk. < yk,, which contradicts (27); thus (26) is proved. The limit of yn is then smooth; we can use Dini's theorem so that (i) is proved. (ii) From (i) we deduce (28)

for every E > 0 there exists no such that for n > no, (1

- E ) yo g

ykn

< yo;

thus, from Lemma 2

- &)YO] < I(Yk,,)

(29)

= In

0 then euery solution (u, y ) of (1) (2) (3) such that u >oinam,

y ( x ) Max a(x) xot0,11 the inequality (4) has an unique solution. (ii) Moreover, i f a < f < p, 0 belong to Co and r > then the solution of

G,

(4) satisfies

( Q + ( u ) isdefinedin(6)).

P r o o f of (ii). Let us introduce the following functions tl =

L/3 (y + a 2

ii = L 2

a (y - b -

Ey

if y

0) aY au wo < 0 on o from (13) and -= 0, where rdenotes an unit tangential vector a7

thus the maximum of u on 52, occurs on the boundary aQ+; by a symmetry argument, we see that it does not occur on a vertical side so that from (14) and from wo = 0 on ,Rg,,\,R+ we derive (15)

wo < 0 on 52, ;

For an arbitrary

(16)

E,

we have

w, = wo < 0 on vertical sides (using (15)) w, = 0 on 1;2,,\52+

on a we have

;

58

A. Dervieux and

H.Kawarada

do the second term of the sum is identically zero; from the bound of -(see the dx definition of C) we derive ((E,l),V> =

(( & , I ) , (1 + [3 - 1 ’ 2 ( - 3 ) ) 4 0

1) so that, using (13), weget

(becausel&$l
= suP{YEIO,rl, (X>Y)EQ+).

For every point (xo, y(x0)) of aQ+ we have u(x0, y(xo)) = 0; since u is a non negative function and is non increasing in the direction ( E , l), u vanishes also on the half lines {(XOf A E , y(xo) + A ) , 1 2 0} with (xo, y(xo)) as origin; choosing every point of those half-lines as a new origin, we derive that u vanishes in the set {(xo + p ~ y(x0) , + A ) , A 2 0, p Q A } so that (17) can be interpreted as follows 1 (x, y ) , y 4 -(x &

- XO)+ y(xo)

or Y Q

1

- -(x &

- xo) +

and similarly

1

Y 2 --(x

E

- xo) + Y(X0)i .

Now, those statements imply that y is a Lipschitz continuous function satisfying

and also, using (15), that

aQ+

= {(X>v(x)),xe ro,

111 u (0) x

[o(O), Y(011 u (1) x

[di)r(1)l , u 0;

thus the conclusion of Proposition 3 follows. Remark 1 Similarly the estimate (18) also holds for every curve u = const. i.e. if we denote by yk the following set (1 > k > 0)

Free Boundary Problems for the Laplace Equation Yk

59

= { ( & Y ) E n + , u(x,Y) = k }

then there exists a parametrization, still denoted yk, such that

and (20) the Set Yk = {(x, yk(x)),x E [1,0]}; moreover it is easy to see that yk converges uniformly to y when k tends to infinity. Remark 2 The conclusions of Proposition 3 and Remark 1 still hold if the set 5 is replaced by

P r o o f of Corollary 1. From H. Brezis, D. Kinderlehrer [l], we have (21)

u E c’.’([o, 11 x

[o, r ] ) ;

now, from the Lipschitz property of y, we deduce that this free boundary has no zero density point with respect to the support of u ; this jointly with (21), is sufficient to apply the theorem of L. Caffarelli [l] firstly, and then those of D. Kinderlehrer, L. Nirenberg [l], from which the smoothness of y is derived (the corners are dealt with by symmetry arguments). 2

Application to an Optimal Control Problem 2.1 Position of the General Problem

Let us consider a Free Boundary Problem in which a datum (the control variable) is supposed to vary: to an arbitrary value of this datum corresponds an unique free boundary and an unique value of a cost functional (to minimize) expressed with respect to this Free Boundary (the F.B. is “observed”). The existence of a control value minimizing the cost (an optimal control) is generally an open problem. The arguments to deal with this problem need schematically two steps: (i) A priori estimates for the Free Boundary. (ii) Continuity of the F.B. with respect to the control variable. Step (i) is strongly related with the close problem of controlability, i.e. looking for data such that the corresponding F.B. is identical (or close) to a given one; for that point see for instance D. G. Schaeffer [3].

60


The second step deals with dependence with respect to data: with arguments analogous to thoses of D. G. Schaeffer 111, some differentiability results with respect to data are proved in Dervieux [l] for C”’norms (with rn large enough). For the two steps, see 0. Pironneau, C. Saguez [l], [2]: the cost function introduced by the authors is the Hausdorff distance between the free domain and a “desired” one. Now the two topologies above (C“ norms, Hausdorff distance) make the results difficult to apply to very concrete cases (for instance numerical experiments); in fact, it seems necessary to use a reasonable intermediary topology: this paragraph is an attempt in this direction for a particular problem, with the hope of a possibility of generalization.

2.2 Definition of Our Problem The notations and assumptions are identical to the ones in Paragraph I ; we consider the following subset of F(M‘ is a strictly positive constant):

x C, then the From the above section if the pair of data V; a) belongs to Faad corresponding Free Boundaries will belong to the set

Let J be a functional defined on &d:

J: ‘‘I

-+

R+;

we assume that J is lower semi-continuous on‘‘I topology of C7(0,1) with 0 < 7 < 1. Let us consider the functional j:

Fad

x C

-+

equipped with the strong

R+

v ( f ,0) E F a d x C Y

jyl a) = J ( Y c f , a))

where V; a) is the parametrization of the Free Boundary which corresponds to the datafand a in Problem (1) (2) (3). We consider the following problem Problem PI To find a pair

(fop,

, goopt) belonging to

Fad

x C such that

Example if we want the free boundary to be as close as possible to a so-called “desired” curve the parametrization of which is y = yd(x), we may choose 1

JW

= S[YW - Yd(X)IZdX. 0


61

2.3 Existence of an Optimal Control Proposition 4 Problem PIadmits at least one solution. Remark 3 It is also possible to consider, instead of J, a functional fdepending both on the control variables (f and a) and on the states functions a), ycf, a)) solutions of (1) (2) (3) (4)). Then Proposition 8 still holds if the x x rad x mapping ( J a, y, ii) + .T((f, a, y, ii) is continuous from H' (]0,1[ x 10, r [ )equipped with the norm of Lz(]O,1[ x [C5(0,l)]' x H1(]O,l[ x 10, r [ ) into R (ucf, a) is extended by 1 and 0).

(uv,

c

Remark 4 In accordance to what is said in Section 2.1 we have also the following non-controlability result: if the desired boundary yd in the above example does not verify (26) (27), then there does not exist a pair (fopt, aopt) in Fadx C such that A f o p t , aopt) = 0. Remark 5 The case where f is fixed (the control variable is only a) is studied in Dervieux [l] where are given some further informations concerning physical motivations and necessary optimality conditions, and also numerical results. Remark 6 Using Remark 1, we can get a result similar to Proposition 4 with the following functional: i k ( J 0)=

J(Yk(J

6))

where yk is the isovalue curve {(x,y),u(x,.y) = k}. 2.4 Proof We shall use the following lemma Lemma 1 Let us assume that f,,, for n belonging to N, and f belongs to L 2 ( D ) with D = 10, T[x I-a,r[, r> E + b and

f ( x ) > 0 a.e. in D,

that f,,converges to f in L z ( D )when n tends to infinity, that a,,,for n belonging to N, and a belongs to Co(O,l), satisfies

- a < a,,,a < b, that a,,converges to a in Co(O,1) when n tends to infinity, and that the Free Boundaries y,, corresponding to (f,,, a,,)as solution of (1) (2) (3) (4) exist and satisfy b < y. ,..< r (22)

< C,where C is a constant not depending on n .

Then the solutions (u,,, y,,) of (1) (2) (3) (4) converge to the solution (u, y ) corresponding to ( J a) in the following ways:

62

A. Derview and H. Kawarada

a(x)}; from the convergence of a,,to a,the righthand side tends to zero as n tends to infinity, and, using the convergencesoff, and ii, (see (W),we get

1 fnfinbdy

--*

j- f u * b d ~ *

D\Qun

Let us now consider the remaining terms of (25): using the H' (0) weak lower semicontinuity of the HA@) canonical semi-norm, we obtain


(gradu*, grad(y, - u*))dxdy 2

(26)

- 5 f(p

Qa

63

- u*)dxdy;

Ql7

making e tend to zero, and by a density argument with respect to p, the inequality (26) still holds for every nonnegative function p of H'(52,) equal to 1 on a. Trivially we have (27)

u* 2 0

onD.

On the other hand, we can easily verify that (28) U * l ~ \ f = i ~1 . From (26) (27) (28) we deduce u* = u and the whole sequence u, converges weakly to u ; the strong convergence is shown later. (ii): Convergence of y,. From (22), after having extracted a subsequence still denoted y,, there exists a function y* such that

,

y,

-+

y* strongly in C'(0,l)

y,:

-+

(y*)' weakly star in L"(0,l);

in particular, y, converges uniformly to y * ; by an analogous argument as in the proof of (i), we can deduce that the solutions u, of the Dirichlet problems in

u, =

0 on y,,

u, = 1 o n a ,

have extensions U, (by 1 for y < a(x), by 0 for y > y,(x)), which converge weakly in H ' ( D ) to the extension G** of the solution u** of the following Dirichlet problem: AU** = f

,..,

in nu,

u * * = 0 on y*,

au** --

av

- 0 f o r x = Oandx = 1

u * * = 1 on a.

Now, from Step 1, we derive u** = u ; from the Maximum Principle, we have

nu,,. = supp u, nu,,.* = supp u** then 52u,y = nu,+, and y = y* is the unique limit of the whole sequence y,,

and (ii) is proved. Using (mutatis mutandis) the Lemma 4 of Part I we deduce of (ii) the strong w convergence of 12, toward u .

End of the proof of Proposition 4. From Proposition 3 and Lemma 1, we derive that the mapping V; a) -+ y V ; a) is sequentially continuous from Fadx C, equipped with the L 2 ( ] 0 1 , [ x 10, rD x Co(O,1) norm, into CT(O,1); on the other hand, it is easy to see that the set Fd x C is compact in L 2 ( ] 0 ,1[ x 10, r n x Co(O,l); thus every minimizing sequence for Problem P converges to a minimum.

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3

Application to an Asymptotic Study

The a priori estimates established in the above section can be used for a lot of asymptotical studies; one interest of the example presented here is the special importance of these estimates for the derivation of convergence of the free boundary. There are quite few papers concerning the asymptotic behaviour of the Free Boundary for the Obstacle Problem; see as a recent work, W. Eckaus, H. J. K. Moet [I]. 3.1 Formulation of the Problem and Main Result

For f belonging to .F (see Proposition 3) we consider the following Variational Inequality indexed by E > 0.

The above methods apply (directly or with the substitution 2 = fix) so that we have the following result. Proposition 5 Under the above assumptions, for every estimates

O