The parameterized obstacle problem

Nonlinrar Analysts, Theory, Methods & Applicotmns. Vol. 4, No. 1, PP. 73-91 Pergamon Press Ltd 1980. Printed in Great Britain

THE PARAMETERIZED OBSTACLE PROBLEM

SHUI-NEE CHOW*

Michigan State University, East Lansing, MI 48823, U.S.A

and

JOHN MALLET-PARETt

Brown University, Providence, RI 02912, U.S.A.

(Received 9 August 1978)

Key words: Obstacle problem, obstacle with two peaks, asymptotic forms, upper and lower solutions, varying obstacles, bifurcation, conformal mapping, scaling.

1. INTRODUCTION

LET R s R2 BE a smooth bounded region and $ : n + R a smooth (say PO) function satisfying

$(x) < 0, all x E 8R.

We regard the graph of $ in R3 as an obstacle and consider the problem of stretching a membrane over the obstacle, fixing the membrane at zero on 80. This is to be done so as to minimize some quantity such as energy or surface area of the membrane.

Mathematically this is the variational problem of minimizing a functional, say the energy

over the convex set J(u) = n Iv u’2 s

It is known [l], [2], [3] that this problem has a unique solution U* which in fact is C’ and has a Lipschitz continuous lirst derivative. Of particular interest is the nature of the set of contact between the membrane and obstacle

I = {x E R( u*(x) = $(x)},

and the boundary I- = 1?1 of this set. Some results on this problem are given in [4], [S], [6]. In particular, Lewy and Stampacchia [5] show that if R is convex and tj is strictly concave and analytic then r is an analytic Jordan curve, and Q - I is homeomorphic to an annulus. Kinder- lehrer [4] proves similar results for the problem of minimizing surface area.

Recently, Schaeffer [7] has proved a stability theorem for the free boundary I-, namely, if I- is a C” curve and

Au=AII/<Q inI (1.1)

*This research was supported in part by the National Science Foundation under MCS 76-06739AOl. tThis research was supported by the United States Army under AROD AAG 29-76-6-0052, and in part by the National

Science Foundation under MPS 71-02923.

73

74 S.-N. CHOW AND J. MALLET-PARE?

then for sufficiently small C” perturbations of $, the boundary I’ will still be a smooth curve, and near I in the C” topology. We note that (1.1) is a non-degeneracy condition in the sense that we always have Au = A$ < 0 in I.

In this paper we shall consider the case where t,Q represents two strictly concave obstacles, of heights s1 and E?. Moreover, we treat s1 and s2 as parameters, allowing them to vary independently but restrict them to lie near zero. There is no assumption on the shape of Q. Figure 1 schematically illustrates a typical solution when s1 and s2 are of comparable magnitude; here each peak of the obstacle makes contact with the membrane and I consists of two Jordan curves. Ife, is relatively large compared with Ed then only the first peak touches the membrane as in Fig. 2. Here I is a single Jordan curve. We shall prove these assertions below, and quantitatively determine the asymptotic relation between &I and s2 describing the transition from Fig. 1 to Fig. 2, as well as describing I.

Fig. I.

Fig. 2

The techniques of proof are roughly as follows. One considers the problem near each peak, and judiciously guesses the local forms of the solution-for example, near a peak with circular symmetry about the origin a reasonable choice might be some multiple of log r. For peaks with an elliptical cross section, conformal mapping reduces this to the case of circular symmetry. These guesses serve as approximations to the true solution u* in R. By means of the maximum principle rigorous estimates are obtained. Now by letting s1 and s2 tend to zero, and scaling appropriately near each peak, one obtains a perturbation of some problem whose solution is known exactly. Schaeffer’s result can then be used to describe the perturbed situation.

2. THE VARIATIONAL PROBLEM AND FORM OF THE OBSTACLE

Using the fact that u* minimizes J and is of class C’, one obtains an equivalent characterization of u* as the unique solution of the following problem:

u:a -+ R is C’

u 3 $ in R, u = 0 on IX

Au d Oinn

Au = 0 in (x E Q/ u(x) > $(x)}.

The parameterized obstacle problem 15

This formulation of the problem, together with the maximum principle, allows us to obtain bounds for u*. In particular, note that a* 2 0 in R.

LEMMA 2.1. Let u+ satisfy

Then u* d u+ in Q.

Au+ d 0 in Sz

u+ 3 *inn

u+ Z Oon32

Proqf: Define the open set 0 = {xc s1( u*(x) > u’(x)}.

Then u* - u+ = 0 on 80, and A(u* - u+) > 0 in 0. Thus, u* - U+ < 0 in 0, a contradiction if

0 #0.

LEMMA 2.2. Let Q* c Q be a region such that aR c aQ*, and let u- satisfy

Au- > OinQ*

Then u- < u* in a*.

U- G I) on an* - an

~-<00naQ

ProoJ Let

0 = {x E Q*[ Us > max($(x), u*(x))).

The fact that u* > 1,9, and the boundary conditions for u- imply u* - u- = 0 on a0. Also, A(u* - u-) d 0 on Co, hence u* > u- in 0, a contradiction if 0 # 4.

We refer to u+ and u- as upper and lower solutions, respectively. For simplicity, we place one peak of the obstacle near the origin, and the other near the point

(LO). Assume then $(x1, x2, c 1, EJ satisfies the following conditions:

$:n x .1”^-t R is C” 1

$(x1, x2,0,0) < 0 on a - {(O,O), (LO)}

$(X1,XZ,$,&J = -+f - 23,x,x, - c,x; + E1 + o(Ix13 + 1q + (I( 1x1) near (xi, x2) = (0,O) ’ (2.1)

9(x,, X2,“1>&2) = -A,@, - 1Y - 24(x, - 1) x2 - c,x: + E2 + 0(1x - (1, 0)13 + 1812

0 < Aj, 0 < Cj, B; < A,C, + I&l Ix - (l,O)l) near (x1, x2) = (LO)

i

where N G R” is a neighbourhood of the origin and A, Bj and Cj are constants. Also, assume that varying E~ and e2 affects only the first and second peaks respectively, that is:

In a neighborhood of (O,O), $ is independent of Ed, and in a neighborhood of (1,0) is independent of sl.

76 S.-N. CHOW AND J. MALLET-PARET

We may solve the equations 8$/8x, = a$/ax, = 0 for x in terms of E, near x = (0,O) and (1,O). Thus, there are functions x = x*(&r) and x = x**(F& of the form

x*(&J = (070) + C&l)

x**(EJ = (LO) + O&-al)

at which 8$/8x, and @/ax, both vanish. These are thus the locations of the two local maxima of $ near (0,O) and (l,O). The values of Ic/ at these maxima are

m*(EJ = $(x*(&J 8) = El + 0(&f)

m**(&J = ti(x**(&& E) = E2 + O(E$

By considering m* and m** as parameters instead of E, and c2 we may, in fact, assume

m*(E1) = El’ m**(EJ = EZ,

Because of the arbitrariness in labelling the peaks, we shall usually assume c1 3 Ed 3 0.

3. UPPER AND LOWER SOLUTIONS FOR AN OBSTACLE WITH ONE PEAK

Consider first the situation where Ed > 0 but Em = 0. Only the first peak has a positive maximum and makes contact with the membrane; the second lies below the membrane. We shall exhibit below upper and lower solutions u+ and u- respectively, for this problem. Throughout this and the next section simply write E = Ed and suppress s2.

We see that near (0,O) the obstacle has the form

$(x1, x2, E) = c - A,@, - x:(s))’ - 2&(x, - x:(E)) (x, - X;(E)) - C,(x, - X;(E))2

+ o(lEl IX - X*(E))’ + IX - X*(E)13). I (3.1)

Use complex notation z = x1 + ix,, Z*(E) = x:(e) + OX: and introduce the rotated coordinates

z” = 1, + i.??, = (z - z*(~))e~’ (3.2) so that

$ = E - A$ - c%; + o(lEl [21’ + (z13)

where 0 < A d C. Observe that

2A = A, + C, - [(C, - A,)* + 4B:]“2

2c = A, + c, + [(C, - A,)2 + 4B:]‘!2

C, - A, + 2B,i e2ia = [(cl _ A,)2 + 4~;]1/2’ I

(3.3)

We let w be a complex variable and construct the functions uf first as functions of w. Then via a conformal mapping between the w and z planes they can be considered as functions of z. Of course the property of being an upper or lower solution is invariant under conformal mappings.

First let ((E) be the continuous function defined by

-<(&)log C(E) = e 0 < E < e-l

t(o) = 0.

The parameterized obstacle problem

Note that t(s) N 2 as E + 0. Set log E

p _ C - A _ [(C, - A,)’ + 4Bf]“’ 2AC 2(A,C, - B:)

R=A+C AI + C, ~ = 2(A,C, - B:) 2AC

D = AC = A,C, - B;

and consider the mapping

iil-+z=z*(s)+(il.+~)e-ia. (3.4)

This maps the set IwI > P1’25(s)1’2 conformally and one-to-one onto the z-plane with the line

segment 1 = [Z*(E) - 2P”25(~)“2 eCia, Z*(E) + 2P1’25(~)1’2 eCia]

deleted. The concentric circles 1~1 = constant are mapped onto the family of confocal ellipses with foci at the endpoints of 1. Thus, there is defined in the exterior of 1 the inverse mapping

w = w(z, F).

Now let

U,(x, E) = -log)o(z, &)I2 - P<(E) Re(w(z, E)-~)

be defined for x E n - I, and observe it is harmonic there since as a function in the w-plane it is harmonic. Let U,(x, E) satisfy

AU,(x, E) = 0 in R

and define

U,(x, E) + UZ(x, F) = 0 on an

U(x, E) = U,(x, E) + U,(x, E).

We observe here that near (0, 0),

U,(x, 4 = U,(O, 0) + O(lxl + I&I,

which follows, in part, from t(s) = O(E). Also set

Q = log R - U,(O, 0) - 1

p*(E) = 1 + 50 E Q++

where the constants S* are chosen to satisfy

s- < Q(Q + 1) - IQW

s+ > Q(Q + 1) + IQW. We now define, in R - 1,

U’(X, E) = 5(e) U*(E) U(x, E).

(3.5)

(3.6)

7x S.-N. CHOW AND J. MALLET-PARET

The functions U’ and $ are as follows in a region of the form

(constant) &)“’ d IwI d (constant) or”.

LEMMA 3.1. Let /WI = KR’12 5(~)“~. Then given positive K, and K,, we have uniformly for K, d K 6 K,(e”2/&)1’2)

-logK2-1-&zRe< I4 1

S’-Q(logK’+Q+l)-;; 7Re$] + Q[$Iog&j

-R2K2+?P2-$+ PRK2-ZPR+g2 > 1 Re$ D + O(C~‘~).

The proof is straightforward when we note that, in the calculation of II/,

4(x, - X$9)2 + =5(x, - xT(s))(x2 - X;(F)) + C,(x, - x:(6))2

Observe that Lemma 3.1 implies

!!?&+ogK”+(R2K2-P2-R2+$)D-PRD(K-;)2Re&] ]

+%I i.

S’-QQ(~O~K~+Q+~)- QP PRez] + Q(y log&)] 1 (3’7) RK’ l\v12

The proof that 1.4~ are upper and lower solutions involves examining the sign of U* - $. Con- sequently, consider the leading term

cp(K,i)= -logK2+ R2K2-P2-R2+c K’) D-PRD K-1 2’

for -1 d /I d 1.

( K) ’ } (3’8)

LEMMA 3.2. For any - 1 < A 6 1, we have

The parameterized obstacle problem

where 4’ means d4/aK.

Proof: Verifying 4 and 4’ vanish at K = 1 is easy. We also have

@‘(1,/2) = 2 + 2R2D i- 6P2D - 8PRDR 3 2 -I 2R2D -I 6P2D - 8PRD = 4 ( >

E > 0.

Now 4 has the form, for fixed ,I,

4(K, A) = y1K2 + y2 + “r’3K-2 + y4 log K2.

One easily sees that 4 can have at most one positive zero other than K = 1. This and the fact that

y1 = R2D - PRDR > R2D - PRD > 0

implies 4 is positive on (1, co). To show 4 is positive on [P1/2/R1’2, 1)) we need only observe

= -log;-(R-P)2(1+,?)D> -log;-2(R-P)‘D

which holds because the function

log(1 + x) - log(1 - x) - 2x

is positive on (0, 1).

LEMMA 3.3. Given K, > 1, then for small E, (1.4’ - $)I<(&) is positive for K E [K,, KI(txl “/{(.z)‘~“)].

Proof Given 6 > 0, then for small E we have from (3.7) the lower bound

Uf - *

C(E) 2 qb(K, 1) - 6K2 (3.9)

in the range of K considered. Arguments similar to those in the proof of Lemma 3.2 show that for small d the right-hand side of (3.9) is positive for all K 2 K,.

LEMMA 3.4. Consider in x-space the set

Z = (x E R2) lx - x*(&)1 < Mu”}

for any M. Then for small enough E,

u+ > $onZ.

Proofi Under the conformal map (3.4), Z is contained in the image of the annulus

{wIp”25(s) 1’2 d [WI d K,R”2 &)“‘}

80 S.-N. CHOW AND J. MALLET-PARI..~

for sufficiently large K,. We see from (3.7) (3.8) that if we let

4+(K)=S+-Q(logK’+Q+l)- $$ I I

then for any 6 > 0 we have for small c

mm% B min b(K, 1) +$)(4+(K) - 6) 1 taken over pi2 , R”2 < K < K,. (3.10)

The choice (3.6) of S+ implies we may assume 4+(l) - 6 > 0. This and the properties of 4 given in Lemma 3.2. imply the right hand side of (3.10) is positive, as required.

LEMMA 3.5. In 2 coordinates define the ellipse

(3.11)

Then for small enough E,

u- < $onE.

Proof: The ellipse E is the image under (3.4) of the circle [WI = R”‘@c)‘~~. By (3.7) we have at such points

- -Q(Q+l)-$Ref$]+O(T).

This and the choice (3.5) of S- proves the lemma.

Remark. We shall show that for the solution u*, the free boundary F is a smooth curve within ~(&s)i’~) of E. Asymptotically then I depends only on the shape of the obstacle, and not on the shape or size of the domain n. Observe also that the ratio of the major and minor axes of E is C/A, which is roughly the square of the corresponding ratio for a cross section $ = constant of the obstacle.

THEOREM 3.6. The functions u+ and z.- satisfy

U+ 3 u* in0

U- < u* in Q*

where n* is Q with the closed region bounded by the ellipse E (3.11) removed.

Proof. We show u+ and U- satisfy the hypotheses of Lemmas 2.1 and 2.2 respectively. First observe that u+ (and u-) are positive in Sz. This follows from the maximum principle since they vanish on an, are positive on I, and are harmonic in fi - 1. In particular, we need only verify U+ 2 $ in the region where $ is positive, which is contained in a ball IwI d (constant)E’!‘. But this follows from Lemmas 3.3 and 3.4 with appropriate choices of K,, K, and M. Lemma 3.5 says u- c $ on E = Xl* - 82.


Remark. We note that the asymptotic form of the solution U* can be given on any compact sub- set of a obtained by removing a neighborhood of the origin. This is given by the asymptotic form of u*, and, in fact

u*(x) - 5(s) G(x, 0)

where G(x, y) is the Green’s function for 0 normalized so that G(x, y) + log)x - y12 is continuous.

4.THE FREE BOUNDARY FOR AN OBSTACLE WITH ONE PEAK

We wish to determine I, the boundary of the set I where U* = $. Lemma 3.3 implies that I does not lie in the region

l&R”’ &) 1’2 < 1~1 < K1R”2 ElJ2,

for any K, > 1, K, > 0. Also, for large enough K, we have U* > u- > 0 > $ for 1~1 3 K,R”’ x E”‘. Thus, for any K, > 1, I lies in the region (for small E)

P”2t(&) “’ < (~1 f K,R’12 &)“l,

or equivalently, inside the ellipse

E(K,) = I F R2( A2T2

[$K, + l/K,) + (AW)(K, - 1/K,,)12

C2Z2

+ [$Wo + UK,) + I&W,- l/K,)]” We shall, therefore, restrict our attention to such a region, say the one bounded by E(2).

By scaling the I coordinate by the transformation X + Z<(E)~‘~ we obtain a fixed region

A25T2

(2 + 3A;4C)2 +

c2z2 2AC

(2 + 3C74A)’ < ___ A+C ’

The functions uf and u* may be considered as being defined on nz,. In particular, by Lemma 3.1 we have for the values on the boundary an,

u*,u* = ( Pcos 28 E - 5(F) log 4 + 4R - 1)+0(Y)

at w = 2R”’ ei8, that is, where

Zl = 2R’12 + & >

cos e

g?, = 2R’:2 - & >

sin 8.

In Q,, the obstacle has the form

$ = E - 5(s) (Al; + CZ;) + O(E<(E)).


It is thus natural to change variables by introducing the functions

*_ U* - E UO -5(E)

*o = *<(;) E

v >

(4.1)

both defined in no. We let g(0, E) denote the value of ug on aa,, so

g@, E) = - log 4 + P cos 28 -----+ l)+,(Y). 4R

LEMMA 4.1. The solution of the obstacle problem of minimizing jQ,(Vu12 over the class

x0 = {U 8 H’(R,)( u > $, in a0 and u = g on aR,J

is the function z$.

This lemma, whose proof is easy, implies the free boundary r for U* is, after scaling, the free boundary To for ug. The problem of studying To for E > 0 is essentially a question of perturbing the situation at E = 0, where To and uz are known. The following theorem implies that I-, is stable under such perturbations.

THEOREM 4.2. (Schaeffer [7]). Let u’ be the solution of minimizing jo,lVul over

X’ = (24 8 H’(R’)j u 3 9’ in Q’ and u = g’ on 80’)

where II/‘: Ti’ -+ R and g’: KY + R are C” with I+Y < g’ on XY, and Q’ is a smooth bounded region in R2. Let

I’ = {x E 0’1 u’(x) = l/Y(x)}

r’ = art

and assume r’ is a C” curve. Assume also that

A$’ < 0 on I’.

Then if $“ and g” are sufficiently near II/’ and g’ in the C” topology, the free boundary I-” of the corresponding problem is a C” curve near I? in the C” topology (in normal coordinates about r’).

We see that the smoothness conditions on 9, and g hold, as does the non-degeneracy condition

A$o = -2A - 2C < 0.

All that is left is to determine To when E = 0.

LEMMA 4.3. Let oO(z) be the inverse of the conformal map MJ --f M) + P/w from the annulus


defined by P1~2 < Iw( < 2R”’ onto R, - [-2P 1’2, 2P1j2J. Let E, be the ellipse defined by 1~1 = R”2, namely

2AC E, = R E R2jA2$ + C21; = ___

A-CC

Then at s = 0,

P Re(o,(Z)-2) - 1 in that part of 0, exterior to E,

*= UO -Ax: - Cxi inside E,

To = E,.

Proof. Lemma 3.1 and Theorem 3.6, and the change of variables defining uz yield the formula for ug in the area exterior to E,; the function given there is harmonic and equals g on dQ,. It must be shown that the difference a: - $, is C’ across E,, and we verify this working in the w coordinates. In the region exterior to E,, following previous calculations we have from (3.8) that

Lemma 3.2 implies that this function and its first derivative vanish on (~11 = R’j2, which defines E,. Since U: = II/, inside E,, the lemma is proved.

We may now state the main result for an obstacle with one peak.

THEOREM 4.4. Let $ be the obstacle (3.1), and I the corresponding free boundary of the set where u* = $. Then, in rotated coordinates (3.2), (3.3) I is a C” Jordan curve within ~(((a)~‘~) of the ellipse

A’$ + C2f; = gg &). (4.2)

5. TWO PEAKS OF COMPARABLE HEIGHT

Here we let both &I and s2 be positive and near zero but restrict our values to a segment

El a, < - Q a2 &2

(5.1)

where the aj are fixed but arbitrary positive constants. The set of such (.sl, a2) is denoted d c R2. In this case the construction of U* and the scaling near each peak is similar to that of the previous sections.


For the peak of height ~~ we obtain quantities Pj, R, Dj and clj defined as before. The conformal maps, however, are slightly different. Consider the map

and, its inverse

wj = Wj(Z, E).

The difference is the appearance of a new quantity dj = 8,(&r, c2) defined implicitly by

6, + G@,) = c1

6, + G&Y,) = Ed. (5.2)

Here G > 0 is the value of the Green’s function G(x, y) (normalized as before) at x = (O,O),

Y = (LO).

LEMMA 5.1. For (sr, EJE& there is a unique solution dj = dj(e,, EJ to (5.2) near zero.

Proof Fix sl, Ed and in the (6,, b,)-plane graph the functions

6j = ij(S,_j) = ~j - G5(6,_j)

on some fixed interval 0 < 6, _ j < constant. The slopes of the graphs are nearly vertical and horizontal, and consequently, there is a unique intersection, respresenting the solution of (5.2).

Now let

urj(x, s) = -log Iwfi, s)[’ - Pt(‘j) Re(ojz, E)-~)

AUzj(x, E) = 0 in Iz

li,j(s, i,) + U2j(x, E) = 0 on aR

where Qj and S,? are constants satisfying

Q, = log R, - U,,((O, Q, 0) - 1

Q2 = log R, - U,,((LO),o) - 1

where zj is the finite quantity

(5.3)

(5.4)

(5.5)

The pacameterized obstacle problem 85

The upper and lower solutions U+ and u- are then

u’(x, 6) = r($)P:(E)[&,(& s) + U,,(.% s)]

+ M,)P;(c)I:q& i-:) + U,,(% &)I.

To see this, we consider a neighborhood of x = (0,O) of radius O(F.~‘~) = O{S:/“). Letting IWI 1 = KR;‘25(6,)“2 and K,d K d KI(S~‘2/&5,)“2) we have (compare with Lemma 3.1)

u* = s, -I- &S,) - logK2 - 1 - 1 + %!I%!! 8,

SF - QI(logK2 + Q + 1)

‘(’

QIP1 Re w’ -m m 1 5@J2

+ @,)G + 6 GQ2 -t_ 0 Alog- ) 61

2 6: i(J,)

4 = El + t(d,) - RfK* + 2P; - ;;

+ P,R,K2 - 2P,R, +

+ O(ty).

Now using (5.2) and (5.3), we have

By noting (5.4) and (5.S), and arguing as before, we may show that ui and u- are upper and lower solutions. By scaling as before, we obtain the following:

THEOREM 5.2. Let E~ and a2 be positive, satisfy (5.1) for some positive a, and a2, and be near zero. Then r consists of two C” Jordan curves JYI and r2 about the two local maxima of 3. Near the first peak, IYl is within o(<(E#‘~) of the ellipse (in rotated coordinates as before}

A2x2 I- C2x2 = 1 2

and the analogous result holds for r2.

CHOW AND J. MALLET-PARET

6. TWO PEAKS OF INDEPENDENT HEIGHT

Here we allow a1 and s2 to be positive, and vary independently near zero. Above, we have considered these parameters in certain restricted regions, so here we fill in the gaps to let them lie in a full neighborhood of the origin. In particular, we shall describe the transition of I from one to two Jordan curves.

LEMMA 6.1. Let u*(si, sZ) denote the solution of the variational problem for $ = I&, Em). There is a monotone increasing function

s2 = E;(E~) - GQsi) as s1 + 0

such that

0 d s2 d e:(si) implies ~*(a~, EZ) = u*(s,, 0),

that is, for such s2 the second peak lies below the membrane. There is a unique local maximum of $(si, Ed) - a*(&,, 0) in a neighborhood of x = (LO), located at a point x = x” (I:~, ez), and with the value

[if% s2) - u*(c,> O)llx=*“,,I,E,) = Yh %I where

7(&i, Q) = Ez - E&) + O(EZ - E2*(Q2.

The functions x R and y are continuous.

Proof: First note that u*(x, &i, 0) is continuous in (x, EJ; moreover it is monotone increasing in s1 since u*(E,, 0) is a lower solution for u*(E,, 0) when s1 < Ei. By (3.12) its value, and first and second derivatives are of order O(<(E ,)) in a neighborhood of x = (l,O). The form (2.1) of $ there implies the existence of ,$(si) and x$(&r, E&, as well as the form of 7(&r, s2). As s1 --f 0, xZ(cl, a;(~~)) is within 0(5(&i)) of (0, l), hence by (3.12) the asymptotic form of E;(E~) follows.

At the parameter value .s2 = Q&i) the second peak touches the membrane in one point, which we denote

We have then near there

x*(i$ = xfl(i:,, &$(E1)).

$(Ei, EZ*($)) - u*($, 0) = -4sr)(x1 - x&))”

- 2%)(x, - x#:,))(x, - x:&i)) - C&)(x2 - ~:(EI))~

+ G(Jx - x%:,)j3),

where A, B(si) and C(si) are continuous functions which equal A,, B, and C, at s1 = 0. The values P(si), R(EJ, D(EJ and CI(E~) are defined as before, and we may consider the map

w--f z = Z”(I:1,E2) + ( w + P(E~HY(E1~ ‘2)) ,-ia

W >

and its inverse

w = dz, El, 4


where zn is the complex notation for xg. As before, set

U,b, 4 = - log(&, 41’ - f’(~,)t(y(r)) Re(o(z, E)-~).

Remark. If as E, approaches zero there are points where A - C(EJ = B(E~) = 0, then it may not be possible to make a continuous choice of CI(E~). However, by explicitly writing the formula for U, in terms of z, we observe that a(~~) occurs only as a term P(.si) e’2ia(E’). By (3.3), this quantity is well defined and continuous, hence so is U,.

Before defining U,, we recall that by Theorem 4.4, the free boundary Qi) of u*(si,O) is a Jordan curve with a certain asymptotic form. We let 0*(&i) denote that part of R exterior to Q1). Now let 0, satisfy

AU,(x,e) = 0 in R*(r.,)

U, (X,E) + U,(x, E) = 0 on ?Q*(r,,).

LEMMA 6.2. On lY(.si), the first derivative of U, + U, is of order O({(E~)~~~/E~).

Proof Let U,(x, E) satisfy

AU&x, F) = 0 in Q

U,(x, E) + U,(x, E) = 0 on dQ.

Consider the function U,(x, E), harmonic in R*(F~), defined by

On an, U, = O(1) as E + 0. On I(&,), we note from the asymptotic formula (4.2) that

u, = (u, - u,) log t(E1) + [u,(o, 8) - u,(o, E)] log t($) f o(1)

= o(t(&,)“2 log 5(&J) + O(1) = O(1).

Hence U, has a uniform bound in 0*(&i) as E + 0. Now restrict U, to a disc about the origin of radius Mr(si) li2, for M large enough that r(~r)

lies in the disc, and scale the disc up to one of radius M. In the scaled region corresponding to the disc intersect R*(E,) observe that the first derivative DU, of Ui, is of order O(l), hence in unscaled coordinates is of order O(~(E,)-“~). Returning to (6.1) shows that D(U, - U,) and hence D(U, + U,) is of order

as required.

0 i &,)“2 :,, &i)) = O(i)

As in Section 3, upper and lower solutions exist.

88

LEMMA 6.3. Let

S.-N. CHOW AI\D J. MALLET-PARE-I

S-h) < Q-(MQ-(EJ + 1) - IQ-;;l~~'l

P-(E) = 1 + F Q-(ar) + q S-(EJ

and set

u- = u*(a,, 0) + 5(Y(E))K(&) [L&(x, s) + LJ,(x, &)I.

T’mlrieze; ,< ~*(a,, EJ in the region sZ*(s,) with the interior of the ellipse Iw(z, E)( = R(E~)“‘&(E))“”

The proof of this lemma follows the ideas of Section 3, so will not be given. Construction of an upper solution u+ however involves some new techniques. We must have U+ defined over all of a, including the interior of I(al), and must be careful how this is done.

To do this, recall G(x, 0) is the Green’s function for a with a singularity of the form -log/x(‘. Let p(x, E) be defined in the region a - R*(E,) bounded by I(si), satisfying

P(x, a 1) = Gk 0) on rh)

4%~ ~1) = 0 infi - a*(&,).

Observe that G(x, 0) - j?(x, al) is positive in Sz - a*(.sJ, hence its outward normal derivative on I is negative. In fact, via a scaling argument, as in the proof of Lemma 6.2, we have

g (G(x,O) - B(x, e,)) ,< (constant) &1)-“2 < 0 on I-(&i)

where a/&i is the outward normal derivative. This inequality and Lemma 6.2 imply the following.

LEMMA 6.4. There is a constant K > 0 such that on I(eJ

a -[U,(x,c) + U,(X,E) + F K(G(x, 0) - /?(x, e,)) -=c 0 an 1

where dldn is the outward normal derivative. Now let

Q+(q) = log Nd - U,(X"(~:,),E~,E~*(E~))

4(&l) - __ KG(xr(c,), 0)

El

S+(&,) > Q+M(Q+(d + 1) + 1Q+WW1

p+(c) = 1 + FQ+(i,) + ++(i,)


and define

zl+ = u*(Er,o) + S(Y(E))~+(E)[Ur(X, 6) + L:,(x,L)

+ 5(&I) __ KG@, O)l,

&I in 0*(&r) (6.2)

LEMMA 6.5. In Sz, u+ 2 u*(sl, E*).

Proqf: We show a+ satisfies the conditions of Lemma 2.1. On aQ, a+ = 0. The proof that u+ > tj near (LO) (near the second peak) uses the techniques of Section 3, and will not be given. To see that U+ 2 I/I near (0,O) (near the first peak) use the fact that U, + U,, K, G and /? are all positive, and that

There remains to show that Au+ d 0; this certainly holds in Q*(E,), and also in R - R*(c,) since p is harmonic. However, note that U+ is continuous, but generally not C’ across F(E~). We interpret Au+ in the sense of distributions, and to show it is non-positive across MY we must verify that

a pu+ a --u an- + <o on U-4

where a/an+ and a/an- are both the outward normal derivatives on l-(&r), but computed for function values in Q*(E,) and R - R*(E,) respectively. To see that (6.3) holds, observe that the first derivatives of u*(E,, 0) and $(sI, Ed) agree on J?(E~), hence by (6.2)

a pu+ a 5(& 1 --?A an- + = t(Y(E))P++(d; [u, + u, + *K(G - /I)] < 0,

by Lemma 6.4. This then proves Lemma 6.5.

We now restrict s2 to satisfy

where 0 > 0 is a sufftciently small constant, to be fixed later. With these parameter values, and the ones considered previously, we have essentially analyzed the obstacle problem for s1 and .s2 varying independently near zero. What therefore remains to be done is to show that for E as in (6.4), r is the union of two Jordan curves rI and TZ about the first and second peaks. As before, this is done by scaling uf and u- near each peak, then applying Schaeffer’s Theorem.

Near the second peak, one considers the scaled solution

U&, EZ) = u*(&,, EZ) - u*tq 0) - Y(F)

5(Y(4


in the region bounded by

I+, s)( = 2~(s,)1’25(Y(s))“2,

as is done (4.1) in Section 4. Near the first peak more care must be paid to the scaling. Our object here is to show u*(c,, a2)

can be considered as a perturbation of u*(s,, 0), hence so can its free boundary. Recall the asymptotic form (4.2) of T(Q); also note that by Lemma 3.1, we may consider u*(sr, 0) on a region 1x1 d M~(E,)“~ (for fixed, large AI), scale this region up to unit size, and scale the solution u by considering (u*(c,, 0) - e,)/t(~r). What results is a solution of the obstacle problem satisfying the hypothesis of Schaeffer’s Theorem for .sl small. Hence, if we show the quantity

(6.5)

is sufficiently small on 1x1 = Map”, we may regard ~*(a,, cZ) as a perturbation of ~*(a,, 0), which implies rI is a C” Jordan curve near Qe,).

Thus on Ix/ = Mu” we have (6.5) in absolute value bounded by

(constant) #[ [p++(e) - P-(E)) - 50 log l(EI)]

< (constant) v E1 El

d (constant) 2. “1

Thus, if 0 is chosen small enough, the above argument is valid. We assemble this and previous results to give a qualitative description of r. It would be pos-

sible to glean from the results of this section a more precise quantitative description, but we refrain from doing that.

THEOREM 6.6. For &I and Ed near zero, T has the form T = Tl u T2 where Tj is either a C” Jordan curve, a single point, or is empty, and lies near thejth peak. There are continuous monotone functions FT(E~) and Ed with

ETCF3 - j) - Gl(&x-j)

for some constant G > 0 such that

1. for s2 < sT(ar), rl = curve, r2 = or 2. for a2 = F~(F~), r, = curve, r2 = point 3. for Ed > IT and c1 > IT, Tl and T2 are curves and c1 > In;;

with analogous results holding for s1 < a:(~~).


REFERENCES

1. BK~ZIS H & KINDERLEHREK D., The smoothness of solutions to non-linear variational inequalities, Indiana Math. J. 23, 831-844 (1974).

2. LIONS J. L. & STAMPACCHIA G., Variational inequalities, Communr purr appl. Math. 20, 493-519 (1967). 3 STAMPAU WA G., Formes hilinkarea coercitives SW le\ ensembles convexes, C.r. hebd. Scum. -lad. Sci., Paris 258.

44134416 (1964). 4. KINDERLEHRER D., The coincidence set of solutions of certain variational inequalities, Archs ration. Mech. Analysis

40, 231-250 (1971). 5. LEWY H. & STAMPACCHIA G., On the regularity of the solutions of a varktional inequality, Communspure app[. Math.

22, 153-188 (1969). 6. SCHAEFFER D. G., Some examples of singularities in a free boundary, Ann. Scuola Norm. Sup. Piss, Ser. IV 4, 133-144

(1977). 7. SCHAFFFER D. G., A stability theorem for the obstacle problem, Adr. Math. 17, 34-47 (1975).

Documents

The parameterized obstacle problem