Interior Free Boundary Problems for the Laplace Equation ANDREW ACKER

Communicated by J. C. C. NITSCHE 1. Introduction and main results

Our results pertain to the following general free boundary problem with hydrodynamic applications. Problem 1. Given are a bounded, simply-connected region G in the plane ~2 (with closure C, and boundary 0G, the latter being a simple, closed curve) and a positive, continuous, real function a(p) defined in G. We define 1[M [[ = S~a2 (P) dx dy M

for any measurable set M e G . Given the value A, 0 < A < IIGIh we seek a simple closed curve F c G such that, if U(p) denotes the harmonic measure of •G in the doubly-connected region f2 bounded by F w 3G, then (1)

I1#11 = A

and

(2)

IVU(p)l=c.a(p)

on s for some c>O

(i.e., for each pes IVU(q)l-*c.a(p) as q--*p, qef2).

.r2: ~72U(p)=O

j,---- a6: u = 1

Fig. l

Archive for Rational Mechanics and Analysis, Volume 75, 9 by Springer-Verlag 1981

158

A. ACKER

Our existence result is Theorem 1. Assume that the set {(p,z)eG x N: 0 2 . a ( p ) + ( 1 - 2 ) . a ( q )

whenever p, qeG and 0 < 2 < 1 .

Then there exists at least one closed, convex curve F c G which satisfies (1) and (2). A number of authors have treated free boundary problems for the Laplace equation, corresponding to such physical phenomena as waves, wakes, jets, and cavitation. (See, for example, BEURLING [6], DANILJUK [8] and [9], FRIEDRICHS [10], GARABEDIAN, LEWY, & SCHIFFER [11], LAVRENT'EV [12], SERRIN [14], TEPPER [16] and WEINSTEIN [17].) However, our main purpose is to exhibit a novel method for the existence proof, which may even prove to be computationally useful (see Remark2). Moreover, the present results do not appear to follow from the literature (see Remark 3). The proof of Theorem 1 is based on the following free boundary optimization problem involving capacity. Problem 2. Let X be the set of all simple, closed curves F c G, and let X c denote the subset of X containing only convex curves. To a given F ~ X , we assign the capacity K--S[VU(p)] [dpl, 7

where U(p) is the harmonic measure of ~?G in the doubly-connected region f2 bounded by F uOG, and ~c(2 is an equip otential curve of U. (The notation (J(p),O,K A, etc. refers to other curves F , F , FA, etc. in X.) For any value 0 < A < HG II, we seek a curve Fa which minimizes capacity in Xc(A)= {FeX~: 1If2II =A}. Theorem 1 is a corollary of the following theorem. Theorem 2. (a) If G is convex, then Problem 2 has at least one solution for each value 0 < A < ]IGL (b) If the set {(p,z)eG x N: O
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159

Remark 2. The author knows of no uniqueness result applicable to Problem 1 or 2 in the general case where {(p,z)EG x R : O
Remark 3. In a more frequently studied variation of Problem 1, one seeks, for any given c > 0 , a curve FeX such that

IVU(p)l=c.a(p)

(4)

on F.

Apparently, the most general existence results for this problem in the literature are due to BEURLING [-6] and DANILJUK [-9]. DANILJUK's existence result is incorrect, as the author has recently shown in [-18]. It is easy to compare Theorem 1 with BEURLING'S results in the case where G={p~R2: [pl
2. Proof of Theorem 2 (a) For At(0, IIGll) fixed, let k = i n f { K : FsXc(A)}>O, and let (F.)cXc(A) be a sequence such that lim K , = k. It sflffices to find a subsequence ( ~ ) c (F,) and a n~oo

curve

PeX~(A) such

that

lim F~= F

(i.e., any neighborhood of/~ contains ~ for all sufficiently large n~N), since then we have /( = lim K , = k. n~oc

To define (~) and F, notice that there exists a value p > 0 so small that for each n~N, G\O, contains a disc of radius 2p. Passing to a subsequence, we can assume (for fixed po~G) that

G\f2,~Bo(Po) = {P dR2

:lp-p0l
for all n~N. By applying the theorem of Ascoli-Arzel/l to the polar coordinate representations of these curves relative to Po, we see that a further subsequence

160

A. ACKER

(~) converges to a closed convex curve /~ ~ (~, whose interior complement has area (]IGH-A). Although we omit the details, one can show that i f / ~ r then /s as n--*o% in violation of our choice of (F,). Therefore f'eX~(A), and our proof is complete.

3. Area-preserving boundary perturbations which (to first order) diminish capacity We assume throughout this section that {(p, z)eG x IR: 0 <_z <_a(p)} is convex in IR3. In order to define the class of boundary perturbation transformations T~: X c ~ X r 0=
[?[=Sa(p)[dp[. .y

For any p, qeC,, d(p,q) denotes the infimum of [7] in the class of all rectifiable curves 7 c G which join p to q. For any FeXr and values 0 < a < l , v>0, we define L(r) = {pe~: u(p)=~} d,(F) = {peG\f2: d(p, F) = ~}, where ~ = closure(Q) and d(p, F) = m i n {d(p, q): qeF}. Finally, we define

T~(F)=J,(I~(F)) for all F e X c and 0 _ a-< 1. Here the value ~ =~(F, a)__>0 is uniquely determined by the requirement that T~ be area-preserving, i.e., that [[~[I = I]~[[, where we have put ~ = T~(F).

Theorem3. (a) I~: X ~ X c for each 0 < a < l . Also J~(F)eX~ if FeX~ and O<=z < z(F)=max {fl>_O: J~(F):~O}. Therefore T~: X ~ Xr for each 0-
Fa=Io{F):U:~

%(t-):d(pjG)=r Fig. 2. The transformation T~(F).

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161

where ~(r, o) =(1/o) i J" (IVU(p)l-).. a(p))ZIVU(p)l-a Idpl d~>O, or~

,~=K/IF[, and F~=I,(F) for 0 < ~ < 1. Remark4. The transformations T~ were already introduced by the author in [-4] (for the case a(p)= 1), where an analogue of Theorem 3 (only for a(p)= 1, and in the context of a different class of curves) was also proven. Proof of Theorem3(a). It suffices to show, given a curve F~X c and values 0 < cr < 1, z > 0 that the sets

2 = { p e G : O(p)
S={p~G\f2:d(p,F)>z}

are both convex. To prove that s is convex, let F(z) denote an analytic, periodic mapping of the strip ~ x (0, 1) onto f2, whose continuous extension to R x [0, 1] maps IR x {0} and IR x {1} onto F and 0G respectively. (Here, z=x +iy=(x, y) =p, i = ] f ~ . ) The function

c~(x,y) = (O/Ox)arg(V'(x + iT)) is harmonic in ~ x (0, 1). If F and 0G are analytic curves, so that qS(x, y) can be continuously extended to lRx [0,1], then (p(x,y) 0 . There exist two disjoint (possibly degenerate) line segments 11, 12c OH, each of which joins ON~(pOto #N~(p2). I f L 1 and L 2 are infinite straight lines such that l~ cL~ and L~c~H=O, /=1,2, then it is easy to see that d(p*,F)>d(p*,L~wL2). Therefore, p*~S provided that d(p*,L~)>z, i=1,2. We will now show that d(p*,LO>z in the case where L 1 c~L=l=r where L is the line containing Pl and P2- Given 6 > 0, there is a curve 7 joining p* to L 1 such that I?] < d(p*, L 1)+ 6. If we assume without loss of generality that L n L 1={0} and Ipxl<[p2l, then Pl = e p * and p2 = ( 2 - c 0 p * for some value 0 0 , then it is easy to show, using the property (3) of the function a(p), that

2 ~(2-c01~1_->(2-~) 17 ~/I + ~ 1(2- ~) 71. Since 1 ~ 7 1 > d ( p l , L 0 = r and 1(2-oO?[>d(p2,LO=r, we conclude that I~1 >(z/~(2-7))=>z. Thus d(p*,L1)>'c-6 for any 6 > 0 , implying d(p*,LO>z. The proof that d(p*,LO>r in the case where Lc~L~ = r is similar. Proof of Theorem 3(h). We begin by proving

(6)

K--K~>~(K2/IF~I)d~ 0

for all F~X~

and

z~[O,z(F)),

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A. ACKER

where F~=J~(F). It suffices to show, for each (7)

lim sup +-o+

K+-a-K~>= -K2 6

FeX+,

that

for all ze(0,~(F)),

IFJ

since K s and IF~l depend continuously on ~ in [0,z(F)). For FeX~ and -cE(0, v(F)) fixed, and for any 6s[0,~], let V~(p) denote a continuous function in closure (~2+_+) which is harmonic in ~2~_+ and satisfies 0 < V6(p)< U+(p)/b on F~_6 and V+(p)=0 on ~?G. Clearly 0 < V6(p)=

(8)

OG

~ Ivga(P)lldpl,

0<6<~.

~G

Let aa(p) = ([ VU+(p)]/V~(p))>0 in (2+_auF~ a for each 3e(0,v]. Also define F~,~=Ip(F~)=I~(J+(F))for all 0 < f i < l . For a fixed value of fie(O, 1), one sees by using (8) and applying Green's second identity to the functions (U+(p)-fi)/(1- fi) and ~(p) in f2~,~ that

K, +-K+> ~ U~(p)-fi 1

(9)

-1-// __

f

]VVa(p)]]dp]

IVU+(p)[Vo(p)ldpl

l Ff o IVS~(p)[ 2 1 - fi +, a+(p)

[dp[,

provided that 6e(0, ~] is sufficiently small (so that (2+,~ c Q+_ ~). By setting

q=K+/ S ao(p)ldpl F~,

in the inequality [(I VU+(p)[-rlaa(p))2/aa(P)]]dp[> O, F ,:, !1

we obtain (10)

~ a+(p)ldp[. ~ ]VU~(p)l__]dpl>=Kf.2 r+,~ r+.e aa(P)

By combining (9) and (10), we obtain an inequality whose form strongly suggests (7), namely (ll)

(1-/3) ~ F,,/~

ao(p)ldpl.K'-~TK+>=K~

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163

for 3 > 0 sufficiently small. The inequality (11) continues to hold when ~?G does not have bounded curvature, as is seen by a direct limiting procedure. We can uniquely specify the related functions V6(p) and aa(p ), 0 < 3 < z, in the above calculations by setting Va(p)=lVU+(p)l/f6(p) on ~-a, where the functions f6: G-+ R+, (to be defined later) are continuous and uniformly convergent to the function a(p) as 3 ~ 0 . In order to see that this step does not violate the condition that O
b'pl
(12)

whenever p~F~_~, 3~(0,z].

Also, [VU+(q)I> [VU+(p)I for qev;, peO+. Both inequalities follow from the convexity of the curves F~,~, 0 < f l < 1, which are orthogonal to the curves 7p. We define f~(p) = max {a(q): qeCJ, d(p, q) <__M 3} in G, 0<3__
whence

1

ga(p) = IVgAp)lfa(p)=
U+(p)

7,~ IVgdq)l Idql- 6

on F~_a for each 0 < 6 < ~. Now to prove that (11) implies (7), we observe that for any given e > 0, there exists a function V(p), continuous in closure (f2+) and harmonic in ~2+, such that r=0

on

and

c~6, O
[(I vg+(p)l-a(p) g(p))/g(p)] [dpl
where ]VU~(p)[ at pEF~ denotes the limit of IVU~(q)[ as q ~ p along a curve of steepest descent of U+. Using fa(p)~a(p) (for 3 ~ 0 ) and the monotonicity of IVg+l, one sees that there is a function t/(3): [0, r)--,IR with t/(3)-+ 0 as 3-+0, such that Va(p)> V(p)-rl(3) on F~_a and hence (by the maximum principle) throughout f2,_a, 0 < 3 < ~ . Therefore, for the function a*(p)=IVU+(p)I/V(p), we have

max{aa(p),a*(p)}~a*(p)

as 3 ~ 0 ,

uniformly in each compact subset of f2+. By substituting a*(p) for aa(p) in (11) and passing to the limit, first as 3 ~ 0, then as fl ~ 0, we obtain the inequality (IF~[+ e) lim sup a+o

K~-a-K~> 6 K~z'

from which (7) follows, since e is arbitrarily small. For the remainder of the proof of (5), we redefine F~=I~(F) and F~,~ =J~(I~(C)) for FeXc, 0_-0. In this new notation, (6) states that (13)

K~--K~,+>~(K~,~/IF~,~I)dfl 0

for

rex~,

0_
and

0_-
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A. ACKER

We again fix FeX~. One sees using a simple argument based on I-2, Lemma4] and the convexity of the curves F~, 0__<~ < 1, that

IVU(p)l>=(1-

U(p))/d>O

in f2,

where d = m a x { I p - q l : p , qsG}. It follows that max{d(p,F):peF~}=O(~r) and hence IIf2\f~l[ =O(~r) as ~--,0. Since the curves F~, 0 < a < l , are convex, it also follows from the first estimate that I~l=l/'l+O(a) as a--,0. If we now relate a and ~ by the condition z = r ( F , a ) , so that ~=F~,~=T,(F) for all 0_
(14)

i'll, el d/~= II~?~\s%ll = I1~\~11,

0 < a < 1,

0

from which it easily follows that z = O(a). Therefore, IF~,pl= IFI + O(a) and K~,p =K+O(a), assuming 0__
K,, - IC,~ >= ( K 2/IFI) r + ,~ O(cr)

and (16)

= ( I I Q ~ II/Irl) + ~ O(a).

By expanding the integral defining ~b(F, a) and using the identity IIQ~O~II-

O~a~l,

~ a2(p)IVU(p)I - Idpldo:, 0 F~

we obtain (17)

tb(r,a)=K

2K ~

K2 I I ~ l l

b

a Irl 2

alFI .[IF~Id~-t

'

0
which implies that ~(F,a)=O(1). By solving (17) for IIO~O~11 and substituting the resulting expression into (16), we find that

(18)

z=~-~ (1 +' 4~(F'a)] ~ ] o-+~ o(o).

By combining (15) and (18), one obtains (19)

K~ - / ~ __>(K + ~(r, a)) a + a O(a).

Now (5) follows directly from (19), in view of the trivial identity

K ~ - K = (a/(1 --a)) K = a K + ,:rO(a).

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165

4. Proof of Theorem 2(b) Let _PeX~(A) solve Problem2 for some fixed value As(0, llGll). Under the assumption that {(p,z)E(~ x R: O<_z<_a(p)} is convex in R 3, we will show that

]gO(p)l=~.a(p)

(20)

on /~,

where )~=/r (i.e., for any psF, IVO(q)l-~i.a(p) as q-.p, qe(2). Since /~ minimizes capacity in X~(A), we conclude using (5) that (21)

lim(1/a) S ~ (IVO(p)1- 2. a(p)) 2 IVO(p)l- * ldpl d~ =0, a~O

0 f"

where/~ = I~(f), 0_< c~< 1. Therefore there is a null sequence (a,) of values in the interval (0, 1/3) such that (22)

lim ~ (IVU(p)I-,~. a(p)) 2 IVO(p)l-~ldpl =0, n~

co

tF n

where we define F , = Q = I ~ . ( f ) , n e N . Since x(log(x))Z/(x-1) 2 is uniformly bounded over (0, oo), we conclude that lim ~ (V(p)-0(p)) 2 Idpl =0,

(23)

n~

co

irn

where V(p)=log(IVO(p)l)in t} and O(p)=log(2.a(p))in (~. To prove (20), we will show that V(p)= W(p) throughout co = {pet~: 0(p)<89 where W(p) solves the boundary value problem V2 W = 0 in co, W = 0 on F, W = V on 7 = {Pe~" O(p)=89 For each heN, let co. = {pe~: ~. < O(p) <89 ~co

(i.e., co, is the doubly-connected region bounded by 7 uF,), let G,(p, q) denote the Green's function for the Laplacian in co,, and let W,(p) solve the boundary value problem V2 W,=0 in co,, W , = 0 on F,, W , = V on 7. Clearly lim W,(p)= W(p)

in co

n~oo

by the maximum principle, since lira max {IW (q)-r

qeF.} =0.

n~cc

Therefore, to prove (20), it suffices to show that (24)

lim W,(p) = V(p),

peco.

n~co

For peco fixed, and neN sufficiently large (so that pea),), we have

V(p)-- W~(p)= ~(V(q)- 0(q))IVq G,(p, q)l Idql, rn

166

A. ACKER

whence, by the Schwarz inequality, (25)

IV(p)-W,(p)12<= ~ (V(q)-@(q))Zldql ~ IVqG,(p,q)J2ldql. F~

F~

One can show by an adaptation of the argument given in [4, w that the second integral in (25) is uniformly bounded over all sufficiently large n~N. Therefore (24) follows from (23) and (25), completing the proof of (20).

5. How to construct curves in

Xc(A ) which

almost satisfy (2)

Again, let {(p, z)~ G x IR: 0 < z < a (p)} be convex. The proof of Theorem 3 (b) can easily be extended to show that the estimate (5) actually holds in the following uniform sense. For any values A~(0, IIG][) and B, there exists a bound M(A,B) such that (26)

Ir < K - q~(F, 8) 3 + M(A, B) 8 2,

uniformly over all F~Xc(A ) satisfying K
Given 0 < 6 < 1 and a curve F~Xc(A), one can construct, by successive applications of the transformation T~ to F, a curve lYEXc(A ) such that

q~(.f, 6)< (M(A, K) + 1) 3.

(27)

To construct F, define the sequence (~)=Xc(A) such that F l = F and ~+1 =T~(~), n = 1 , 2 ..... If F satisfies (27), we simply define F = F . Otherwise, (26) implies / ( 2 < / < 1 - 8 2 . Let S be the largest set of consecutive natural numbers starting with 1 such that / ( , + 1 < / < , - 6 2 for each n6S. The set S is bounded, since for n6S the inequalities K =/(1 > / ( 2 +82 >/<3 +2 62 > ... > / ( , + ( n -

1) 82

imply

n<(K/82)

and

/<,
If m denotes the largest element in S, then the curve/~=/~,,+1 satisfies (27), as immediately follows from the inequalities

/~m+ 1- (~2~_~/~rn+2 ~_~/~m+1-~P(I~m+I,O)8+M(A,K)8 2. Clearly, the expression

o ro

Ivu[o)l

(where/~ T I , ( f ) and i=g/lPI) is a measure of the deviation of IV0(p)l from the function 2-a(p) in a neighborhood of f (relative to ~ t j f ) . By further refinements based on the above result, one can construct, for any 0 < 6 < 1, an analytic curve FEXc(A ) such that [IV0(p)[-)~-a(p)[<6 on F.

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167

6. Appendix. A related flee-boundary existence result (added in proof) The methods given above can be easily adapted to prove the following result closely related to Theorem 1 (see Remark 3). Theorem4. Assume that the set {(p,z)~G x R : 0-
IVU(p)l =a(p)

on f .

Proof sketch. By a variation of the argument in w one sees that there exists a curve/~eX~ such that (29)

/( + 11811=min {K + Ilf2[[ : F~ X~}.

For/~ fixed, l e t / ~ = T~(f)eX~ for each 0 < a < l . (5) implies

(30)

Since

R + [lS~ll < R + 11811-~(/~,~r)o-+~rO(o-)

Now (29) and (30) imply q~(f,~)~0 as a ~ 0 + , argument in w that (31)

[V(J(p)l=X.a(p)

as ~--,0+. and we conclude by the

on F,

where Y=g/I/~l. If we set ~ = l , ( f ) e X c , 0 < a < l , identities displayed following (16) and (19) that

s

II~ll = 11811,the estimate

then it follows from the

I1~11= R + 11811+ ( R - ( l f l 2 / R ) ) c r + ~ O ( a )

as ~--,0+.

In view of (29), we conclude that / ( > IFI, whence 2 > 1. Similarly, if we set =Jr c for ~ > 0 sufficiently small, then it follows using (6) and the identity

II<\Oll=i
that

/~+ I1~11= R + I1~11+(I/~l-(K2/IFI))r+ro(~)

as r ~ 0 + ,

from which we conclude, again using (29), that IF[ > / ( implies 2__<1. Therefore = 1 in (31), i.e., f satisfies (28).

References 1. ACKER,A., Heat flow inequalities with applications to heat flow optimization problems. SIAM J. Math. Anal. 8, 604-618 (1977). 2. ACKER,A., A free boundary optimization problem. SIAM J. Math. Anal. 9, 11791191 (1978).

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A. ACKER

3. ACKER, A., Some free boundary optimization problems and their solutions, Numerische Behandlung von Differentialgleichungen mit besonderer Beriicksichtigung freier Randwertaufgaben. J. ALBRECHT, L. COLLATZ, G.I-I~MMERLIN, eds. 9-22. Basel: Birkh~iuser Verlag 1978. 4. ACKER, A., Free boundary optimization-a constructive iterative method, Z. Angew. Math. Phys. 30, 885-900 (1979). 5. ACKER, A., An area-preserving domain perturbation which diminishes capacity. Notices Amer. Math. Soc. 25, A 592 (1978). 6. BEURLING,A., On free boundary problems for the Laplace equation. Seminars on Analytic functions, Institute for Advanced Study, Princeton, N J, 1, 248-263 (1957). 7. CARLEMAN,T., fJber ein Minimalproblem der mathematischen Physik. Math. Zeitschrift 1, 208-212 (1918). 8. DANILJUK,I.I., On a non-linear problem with a free boundary. Dokl. Akad. Nauk SSSR 162, 979-982 (1965). 9. DANILJUK,I.I., An existence theorem in a certain nonlinear problem with free boundary. Ukrain. Math. Z., 20, 25-33 (1968). 10. FRIEDRICHS,K., fJber ein Minimalproblem fiir Potentialstr6mungen mit freiem Rande. Math. Annalen 109, 60-82 (1934). 11. GARABEDIAN,P.e., U. LEWY & M. SCH1FFER, Axially symmetric cavitational flow. Annals of Math. 56, 560-602 (1952). 12. LAVRENT'Ev,M.A., Variational Methods. Groningen: Noordhoff 13. SCHIFFER,M.M., Partial differential equations of the elliptic type, Lecture series of the symposium on partial differential equations held at the University of California at Berkeley (June 20 - July 1, 1955), 97-149. 14. SERRIN,Jr., J. B., Existence theorems for some hydrodynamical free boundary problems. J. Rational Mech. Anal. 1, 1-48 (1952). 15. SZEGO,G., fJber einige Extremalaufgaben der Potentialtheorie. Math. Zeitschrift 31, 583-593 (1930). 16. TEPPER, D. E., Free boundary problem - the starlike case. SIAM J. Math. Analysis 6, 503-505 (1975). 17. WEINSTEIN,A., Jets & Wakes. Tullio Levi-Cevita, Convegno Internazionale Celebrativo del Centenario della Nascita (Roma, 17-19 dicembre 1973), Atti dei Convegni Linci 8; Accademia Nazionale dei Lincei, Roma 1975, pp. 269-296. 18. ACKER,A., Concerning Daniljuk's existence theorem for free boundary-value problems with the Bernoulli condition, Proc. Amer. Math. Soc. (to appear). Mathematisches Institut I Universit~it Karlsruhe Karlsruhe

(Received April 10, 1979)