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On Overdetermined Eigenvalue Problems for the Polyharmonic Operator
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS ARTICLE NO.
221, 384]404 Ž1998.
AY975906
On Overdetermined Eigenvalue Problems for the Polyharmonic Operator R. Dalmasso Laboratoire LMC-IMAG, Equipe EDP, Tour IRMA, BP 53, F-38041 Grenoble Cedex 9, France Submitted by Charles W. Groetsch Received March 17, 1997
We consider two eigenvalue problems for the polyharmonic operator, with overdetermined boundary conditions. We give an equivalent integral formulation. In some cases we obtain characterizations of open balls by means of integral identities. Q 1998 Academic Press Key Words: Polyharmonic operator; overdetermined problems
1. INTRODUCTION Let V ; R n, n G 2, be a bounded domain with sufficiently smooth boundary V. We consider the following overdetermined boundary value problem for the polyharmonic operator: m
Ž y1. Dm u s l u q m us
u n
s ??? s
mu n m
my 1 u n my 1
s c Ž const..
in V ,
s0
on V ,
on V ,
Ž 1.1. Ž 1.2. Ž 1.3.
where l ) 0, m g R, rn is the outward normal derivative and m is a positive integer. When m s 1, l s 0, and m / 0, Serrin w18x showed that if u g C 2 Ž V . satisfies Ž1.1. ] Ž1.3., then V must be a ball. The method of moving planes used there was also shown to apply to more general elliptic equations and somewhat more general boundary conditions. Serrin’s basic result was proved in an alternative manner by Weinberger w20x. 384 0022-247Xr98 $25.00 Copyright Q 1998 by Academic Press All rights of reproduction in any form reserved.
OVERDETERMINED EIGENVALUE PROBLEMS
385
When m s 2, l s 0, and m / 0, Bennett w4x modified Weinberger’s method to establish that V is a ball if u g C 4 Ž V . satisfies Ž1.1. ] Ž1.3.. Unfortunately, Bennett’s argument does not extend to more general equations. Using the method of moving planes and assuming in addition that u G 0 in V, we were able to treat more general biharmonic equations and systems Žsee w8x and w9x.. In the present work we shall first reformulate problem Ž1.1. ] Ž1.3. in an equivalent integral form. THEOREM 1.1. Let l ) 0, m / 0. Assume that V g C 3 m, a for some a g Ž0, 1x. Then the following statements are equi¨ alent: Ži. There exists u g C 2 m, a Ž V . satisfying Ž1.1. ] Ž1.3.. Žii. There exists a constant d such that
HV w dx s dH V
my 1 w n my 1
ds
Ž 1.4.
for all w g C 2 m, a Ž V . satisfying m
Ž y1. Dm w s lw ws
w n
s ??? s
my 2 w n my 2
s0
in V , on V if m G 2.
Ž 1.5. Ž 1.6.
THEOREM 1.2. Let l ) 0, m s 0. Assume that V g C 3 m, a for some a g Ž0, 1x. Then the following statements are equi¨ alent: Ži. There exists u k 0 in C 2 m, a Ž V . satisfying Ž1.1. ] Ž1.3.. Žii.
H V
my 1 w n my 1
ds s 0,
Ž 1.7.
for all w g C 2 m, a Ž V . satisfying Ž1.5., Ž1.6.. Remark 1. Let V be of class C 2 mqk, a for some k G 0, a g Ž0, 1x. If u g C 2 m Ž V . l C my1 Ž V . satisfies Ž1.1., Ž1.2., then u g C 2 mqk, a Ž V . Žsee w14x.. In the case where m s 1, Theorem 1.1 was proved by Chamberland, Gladwell, and Willms w6x. Various duality theorems were previously established by Payne and Schaefer w15, 16x. The integral duals were then used to deduce the spherical symmetry of V.
386
R. DALMASSO
We shall use Theorem 1.1 to establish the following result. THEOREM 1.3. Let l ) 0, m / 0. Assume that V g C 5m, a for some a g Ž0, 1x. Then the following statements are equi¨ alent: Ži. There exists u g C 2 m, a Ž V . satisfying Ž1.1. ] Ž1.3.. Žii. There exists u g C 4 m, a Ž V . satisfying Ž1.2. and D2 m u s l2 u q lm
u
in V ,
Ž 1.8.
on V ,
Ž 1.9.
m
n m
s c9 Ž const..
m
Ž y1. Dm u s aŽ const.. / ym D u
on V ,
Ž 1.10.
m m
Ž y1. Dm u
m
n m
on V ,
s b Ž const..
Ž 1.11.
my 1Dm u
s ??? s s0 on V if m G 2. Ž 1.12. n n my 1 A domain V with a C 2 connected boundary is said to have the Schiffer property if, for any l, the only solution to the overdetermined boundary value problem Ž1.1. ] Ž1.3. with c s 0 is the trivial solution u s 0 Žcorresponding to m s 0.. The Schiffer conjecture asserts that, if for some l ) 0, m / 0, problem Ž1.1. ] Ž1.3. with c s 0 has a solution u g C 2 Ž V . when V is a domain with a C 2 connected boundary, then V is a ball. A result due to Williams w21x asserts that any Lipschitz domain that does not possess the Schiffer property has a real analytic boundary. Therefore, using Theorem 1.1 with m s 1 and Remarks 1 and 6, the Schiffer conjecture can be restated in the following way Žsee also w6x.. If, for some fixed a g Ž0, 1x, every w g C 2, a Ž V . satisfying Ž1.5. Žwith m s 1. has vanishing mean over some fixed bounded domain V ; R n with a C 2, a connected boundary, must V be a ball? Using Theorem 1.3 with c s 0 Žsee Corollary 5.1., Remark 1, and the above result of Williams, The Schiffer conjecture can also be stated as follows. If for some l ) 0, m / 0, there exists u g C 4 Ž V . such that D2 u s l2 u q lm us0, D usa Ž const.. / m , must V be a ball?
u n D u n
in V ,
Ž 1.13.
s c9 Ž const.. , s l c9
on V ,
Ž 1.14.
387
OVERDETERMINED EIGENVALUE PROBLEMS
THEOREM 1.4. Let l ) 0, m / 0. Assume that V g C 5, a for some a g Ž0, 1x. Then the following statements are equi¨ alent: Ži. There exists u g C 4, a Ž V . satisfying Ž1.13. ] Ž1.14.. Žii. There are constants d / y1rl and d9 such that
HV w dx s dH V
w n
H V Ž Dw q lw . ds,
ds q d9
Ž 1.15.
for all w g C 4, a Ž V . satisfying D2 w s l2 w
in V .
Ž 1.16.
In some cases already mentioned Žw4x, w9x, w15x, w16x., duality theorems have been used to give characterizations of open balls in R n by means of integral identities. Since higher order elliptic equations do not have a maximum principle in general, we cannot obtain the generalization to the polyharmonic case. Nevertheless, we shall give some partial results. We first recall a theorem obtained by Grunau and Sweers w11x. We begin with the definition of closeness of domains and operators. DEFINITION 1.1 w11x. Let « ) 0, a g Ž0, 1x. V is called « close in C k, a sense to V*, if there exists a C k, a mapping g: V*¬ V such that g ŽV*. s V and 5 g y Id 5 C k , a ŽV*. F « . Now consider the higher order elliptic problem Lu G 0 us
u n
s ??? s
in V , my 1
n
u
my 1
Ž 1.17. on V ,
s0
Ž 1.18.
where V ; R 2 and
ž
Ls y
Ý 1Fi , jF2
ai j
2 xi x j
m
/
q
Ý iqjF2 my1
bi j
i
ž /ž / x1
x2
j
, Ž 1.19.
with a i j g C 2 my1, a Ž V ., bi j g C 0, a Ž V .. DEFINITION 1.2 w11x. Let « ) 0 and let L be as above. The operator L is called « close in C k, a sense to Žy1. mDm on V, if, additionally,
388
R. DALMASSO
a i j g C k, a Ž V . and 5 ai j y di j 5 C k , a Ž V . F « ,
5 bi j 5 CŽ V . F « .
THEOREM 1.5 w11x. Let n s 2 and B s x g R 2 ; < x < - 14 . There exists « 0 s « 0 Ž m. ) 0 such that, for « g w0, « 0 ., we ha¨ e the following. If V g C 2 m, a , V is « close in C 2 m sense to B, and L is « close in 2 my1, a C sense to Žy1. mDm on V, then e¨ ery u g C 2 m, a Ž V ., u k 0, satisfyŽ . ing 1.17 , Ž1.18. is strictly positi¨ e in V. We denote by lŽ1m. the first eigenvalue of Žy1. mDm in V with Dirichlet boundary conditions. Using the notations of Theorem 1.5, we now state one of our symmetry results. THEOREM 1.6.
Let V g C 3 m, a for some a g Ž0, 1x. Assume that either
Ži. m s 1 and 0 - l - lŽ1. 1 , or Žii. n s m s 2, V is « close in C 4 sense to B and 0 - l F « - « 0 . If there exists a constant d such that
HV w dx s dH V
my 1 w n my 1
ds
for all w g C 2 m Ž V . l C my1 Ž V . satisfying m
Ž y1. Dm w s lw ws0
in V ,
on V if m s 2,
then V is a ball. In Section 2 we give some preliminary results. In particular, we recall a theorem obtained in w8x. Then, using some results of w11x and w12x, we study the eigenspace corresponding to the first eigenvalue of higher order elliptic operators, whose principal part is the mth power of a second-order elliptic operator. Theorems 1.1, 1.2, and 1.3 are proved in Sections 3, 4, and 5, respectively. Finally, in Section 6 we first show that the overdetermined boundary value problem Ž1.13., Ž1.14. has a solution when V ; R 2 is the unit ball. Then we prove Theorems 1.4 and 1.6 and another result concerning the symmetry of the domain. We also make some remarks.
OVERDETERMINED EIGENVALUE PROBLEMS
389
2. PRELIMINARIES We shall need the following lemma. LEMMA 2.1.
Let u g C 2 m Ž V . be a solution of problem Ž1.1. ] Ž1.3.. Then ml
HV u
2
dx q
n q 2m 2
m
HV u dx s
nc 2 2
< V <.
Proof. The proof follows readily from a generalization of Pohozaev’s identity Žsee w17x, pp. 701]702.. The next lemma is easily proved by an induction argument. It will be used repeatedly without referring to it. LEMMA 2.2.
Let V g C k for some k G 1. If u g C k Ž V . is such that on V ,
u s const.
u n
s ??? s
ky 1
n
u
ky 1
s0
on V if k G 2,
then
ku x j1 ??? x jk
s
ku n k
n j1 ??? n jk
on V
for j1 , . . . , jk g 1, . . . , n4 . We first recall the following theorem obtained in w8x Žtheoreme ´ ` 3.1; see also w9x.. THEOREM 2.1. Let V ; R n Ž n G 2. be a bounded domain with C 2 boundary V. Let f : R 2 ª Ž0, `. satisfy the following condition. For each ¨ g R, u ª f Ž u, ¨ . is nondecreasing, and for each u g R, ¨ ª f Ž u, ¨ . is nonincreasing. Let u g C 4 Ž V . l C 3 Ž V . be a solution of the o¨ erdetermined boundary ¨ alue problem D2 u s f Ž u, D u . us
u n
s0
D u s d Ž const..
in V , on V , on V .
If u G 0 in V, then V is a ball. If V s x g R n ; < x y x 0 < - R4 for some x 0 g R n, then uŽ x . s ¨ Ž< x y x 0 <., ¨ 9 - 0 in Ž0, R ., and Ž D¨ .9 ) 0 in Ž0, R x.
390
R. DALMASSO
Remark 2. Notice that u g C 4 Ž V . in w8x, but it is enough to assume that u g C 4 Ž V . l C 3 Ž V .. Remark 3. Assume that f ' 1, V g C 4, a for some a g Ž0, 1x and u g C 4 Ž V .. Then the assumption u G 0 in V can be removed. Indeed, this is just Bennett’s result w4x. Remark 4. Assume that f : R 2 ª w0, `. and that f Ž u, ¨ . ) 0 for u ) 0, ¨ g R. If u G 0, u k 0 in V, then the conclusion of Theorem 2.1 still holds. This readily follows from the proof. Under the assumptions of Theorem 1.5, the Green’s function of L Žgiven by Ž1.19.. for the Dirichlet problem in V is positive. Using the Krein]Rutman theorem, we have the following. COROLLARY 2.1. Under the assumptions of Theorem 1.5, there is a positi¨ e eigenfunction for the first Dirichlet eigen¨ alue of L in V. Now we consider the case n s m s 2. The Green’s function GŽ x, y . of D2 for the Dirichlet problem in B is known explicitly Žsee w5x, w10x.:
G Ž x, y . s
1 8p
½
< x y y < 2 ln < x y y < y ln < x < y y
ž
q
1 2
ž
< x< y y
x
2
< x<
x < x<
y < x y y<2
/
/5
for x, y g B, x / y Žif x s 0, replace < < x < y y xr< x < < by 1.. LEMMA 2.3.
D x GŽ x, y . ) 0 for < x < s 1, < y < - 1.
Proof. We easily get
D x G Ž x, y . s
1 4p
½
2 log < x y y < q 2 y 2 log < x < y y y2 < x < y y y1 q < y < 2
5
x < x<
y2
x < x<
Ž < x < 2 < y < 2 q < y < 2 y Ž1 q < y < 2 . ² x, y : .
391
OVERDETERMINED EIGENVALUE PROBLEMS
for x, y g B, x / y. We deduce that
D x G Ž x, y . s
1 4p
2
< x y y
for < x < s 1 and < y < - 1. Let L be given by Ž1.19. with m s 2, V s B, and a i j s d i j . Assume that L is « close in C 3, a sense to D2 on B for some a g Ž0, 1x. From w12x we know that the Green’s function GLŽ x, y . of L for the Dirichlet problem in B behaves like G if « g w0, « 0 . for « 0 small enough: 1 C
G Ž x, y . F GL Ž x, y . F CG Ž x, y . ,
x, y g B,
x / y.
With the help of Theorem 1.5 and Lemma 2.3, we immediately deduce the following lemma. LEMMA 2.4. Let n s m s 2, V s B, and let L be gi¨ en by Ž1.19. with a i j s d i j . There exists « 0 ) 0 such that, for « g w0, « 0 . we ha¨ e the following. If L is « close in C 3, a sense to D2 on B, then e¨ ery u g C 4, a Ž B ., u k 0, satisfying Ž1.17., Ž1.18. is strictly positi¨ e in B and, moreo¨ er, 2 urn 2 ) 0 on B. Now using Remark 1 of w12, Sect. 5x and Proposition 2.4 of w11, p. 92x, we obtain the following. THEOREM 2.2. Let n s m s 2 and let L be gi¨ en by Ž1.19. with a i j s d i j . There exists « 0 ) 0 such that, for « g w0, « 0 . we ha¨ e the following. If V g C 4, a , V is « close in C 4 sense to B and L is « close in C 3, a sense to D2 on V, then e¨ ery u g C 4, a Ž V ., u k 0, satisfying Ž1.17., Ž1.18. is strictly positi¨ e in V and, moreo¨ er, 2 urn 2 ) 0 on V. Using the same arguments as in Amann w1]3x, we derive the following corollary. COROLLARY 2.2. Under the assumptions of Theorem 2.2, the first eigen¨ alue of L in V with Dirichlet boundary conditions is simple.
392
R. DALMASSO
3. PROOF OF THEOREM 1.1 Let u g C 2 m, a Ž V . satisfy Ž1.1. ] Ž1.3. and let w g C 2 m, a Ž V . satisfy Ž1.5., Ž1.6.. Using repeated integrations by parts, we obtain m
l
HV uw dx q mHV w dx s Ž y1. HV wD s Ž y1 .
m
m
u dx
m
HV uD
my1
q
Ý js0
H V
ž
D jw
w dx
Dmy jy1 u n
sl
HV uw dx y H V
sl
HV
y
my 1 w m u
uw dx y c
H V
n my1 n m my 1 w n my 1
D j w n
Dmyjy1 u ds
/
ds
ds,
from which we deduce Ž1.4., with d s ycrm. Conversely, suppose that for some constant d, Ž1.4. holds for all w g C 2 m, a Ž V . satisfying Ž1.5., Ž1.6.. The Fredholm alternative w13x, w14x gives the existence of a solution u g C 2 m, a Ž V . to the boundary value problem Ž1.1., Ž1.2.. Now let w g C 2 m, a Ž V . satisfy Ž1.5., Ž1.6.. Using, as before, repeated integrations by parts, we obtain
m
HV w dx q H V
my 1 w m u n my1 n m
ds s 0.
Then condition Ž1.4. implies that
H V
my 1 w n my1
ž
md q
mu n m
/
ds s 0.
Ž 3.1.
We claim that we can find w g C 2 m, a Ž V . satisfying Ž1.5., Ž1.6. and
my 1 w n my1
s md q
mu n m
on V .
Then from Ž3.1. we deduce that Ž1.3. holds with c s ym d.
Ž 3.2.
393
OVERDETERMINED EIGENVALUE PROBLEMS
Now we prove the claim. By Remark 1, u g C 3m, a Ž V .. Therefore there exists w g C 2 m, a Ž V . such that Dmw s 0
ws
w n
s ??? s
my 1w
in V ,
w
my 2
n
mu
s md q
n my1
on V if m G 2,
s0
my 2
on V .
n m
Consider the problem m
Ž y1. Dmc s lc q lw cs
c n
s ??? s
c
in V ,
Ž 3.3.
my 1
n
on V .
s0
my 1
Ž 3.4.
If l is not an eigenvalue of the polyharmonic operator Žy1. mDm in V with Dirichlet boundary conditions, then Ž3.3., Ž3.4. has a unique solution c g C 2 m, a Ž V . by the Fredholm alternative. Therefore w s w q c g C 2 m, a Ž V . satisfies Ž1.5., Ž1.6. and Ž3.2.. The claim is proved in this case. Now if l is an eigenvalue of Žy1. mDm in V with Dirichlet boundary conditions, we denote by q its multiplicity and by Ž w j .1 F jF q the corresponding orthonormalized set of eigenfunctions. Problem Ž3.3., Ž3.4. has a solution if and only if
HV ww
dx s 0,
j
j s 1, . . . , q.
Ž 3.5.
On integrating by parts, we obtain
l
HV ww
j
dx s Ž y1 .
m
s Ž y1 .
m
HV wD w m
j
dx
HV w D w dx m
j
my1
q
Ý ks0
sy
H V
my 1w mw j
H V n
my1
n m
ž
Dw k
Dmy ky1w j n
ds s y
H V
ž
y
md q
Dkw n
Dmyky1w j ds
/
m u mw j n m
/
n m
ds
394
R. DALMASSO
for j s 1, . . . , q. Then Ž3.5. is equivalent to
H V
ž
md q
m u mw j n m
/
n m
ds s 0,
j s 1, . . . , q.
Ž 3.6.
Now we write q
us
Ý ck w k q u0 ,
Ž 3.7.
ks1
where c k , k s 1, . . . , q are arbitrary constants, and u 0 g C 2 m, a Ž V . is the particular solution of Ž1.1., Ž1.2. orthogonal to w j for j s 1, . . . , q, i.e., satisfying
HV u w 0
j
dx s 0,
j s 1, . . . , q.
Let M s Ž Mi j .1 F i, jF q with Mi j s
H V
mw i mw j n m n m
ds;
let e s Ž e j .1 F jF q with ej s y
ž
H V
md q
m u 0 mw j n m
/
n m
ds,
and x s Ž c j .1 F jF q . Using Ž3.7., we can write Ž3.6. in the following form: Mx s e.
Ž 3.8.
Since
mw j
q
x Mx s T
½
H V Ý c js1
j
n m
2
5
ds,
we deduce that M is positive semidefinite. Assume that M is not invertible. Then there exist c j , j s 1,???, q with c j / 0 for some j and such that the Dirichlet eigenfunction q
¨s
Ý c j wj js1
satisfies
m¨ n m
s0
on V .
395
OVERDETERMINED EIGENVALUE PROBLEMS
Therefore ¨ satisfies Ž1.1. ] Ž1.3. with m s c s 0. From Lemma 2.1 we deduce that ¨ s 0 in V. Since the w j ’s are linearly independent, we obtain that c j s 0 for j s 1,???, q, a contradiction. Therefore M is invertible. We deduce that Ž3.8. has a solution. Then problem Ž3.3., Ž3.4. has a solution c . Finally, w s w q c g C 2 m, a Ž V . satisfies Ž1.5., Ž1.6., and Ž3.2.. The proof is complete. Remark 5. Let V be of class C 3 mqk, a for some k G 0, a g Ž0, 1x. Then we can replace w g C 2 m, a Ž V . in Theorem 1.1 Žii. by w g C 2 mqk, a Ž V .. This readily follows from the proof, using the elliptic regularity theory and Remark 1. Remark 6. The proof of Theorem 1.1 shows that c s 0 if and only if d s 0.
4. PROOF OF THEOREM 1.2 Let u g C 2 m, a Ž V ., u k 0, satisfy Ž1.1. ] Ž1.3. with m s 0 and let w g C V . satisfy Ž1.5., Ž1.6.. Using repeated integrations by parts, we obtain 2 m, a Ž
m
l
HV uw dx s Ž y1. HV wD s Ž y1 .
m
m
u dx
m
HV uD
w dx
my1
q
Ý js0
H V
sl
HV uw dx y H V
sl
HV uw dx y cH V
ž
D jw
Dmy jy1 u n
my 1 w m u n my1 n m my 1 w n my 1
y
D j w n
Dmyjy1 u ds
/
ds
ds.
Since by Lemma 2.1 c / 0, we deduce Ž1.7.. Conversely, suppose that Ž1.7. holds for all w g C 2 m, a Ž V . satisfying Ž1.5., Ž1.6.. Assume that l is not an eigenvalue of Žy1. mDm in V with
396
R. DALMASSO
Dirichlet boundary conditions. Let r g C 2 m, a Ž V . be the solution of Dmr s 0
rs
r n
s ??? s
r
in V ,
my 2
n
my 2
my 1r n my 1
on V if m G 2,
s0
on V .
s1
The Fredholm alternative gives the existence of a solution ¨ g C 2 m, a Ž V . to the boundary value problem m
Ž y1. Dm ¨ s l¨ q lr ¨s
¨ n
my 1 ¨
s ??? s
n my 1
in V , on V .
s0
Then w s ¨ q r g C 2 m, a Ž V . satisfies Ž1.5., Ž1.6. and
my 1 w n my 1
on V .
s1
Since Ž1.7. does not hold, we reach a contradiction. Therefore l is an eigenvalue of Žy1. mDm in V with Dirichlet boundary conditions. Let u g C 2 m, a Ž V . be a solution of the boundary value problem Ž1.1., Ž1.2. with m s 0, and let w g C 2 m, a Ž V . satisfy Ž1.5., Ž1.6.. Using, as before, repeated integrations by parts, we obtain
H V
my 1 w m u n my1 n m
ds s 0.
Then condition Ž1.7. implies that
H V
my 1 w n my1
ž
cy
mu n m
/
ds s 0
Ž 4.1.
for any c g R. We claim that we can find w g C 2 m, a Ž V . satisfying Ž1.5., Ž1.6. and
my 1 w n my1
scy
mu n m
on V
for any c g R. Then from Ž4.1. we deduce that Ž1.3. holds. If we choose c / 0, then Lemma 2.1 implies that u k 0. To prove the claim, we use the same arguments as in the proof of Theorem 1.1 with u 0 s 0 in Ž3.7..
397
OVERDETERMINED EIGENVALUE PROBLEMS
5. PROOF OF THEOREM 1.3 If u g C 2 m, a Ž V . satisfies Ž1.1. ] Ž1.3., then u g C 4 m, a Ž V ., since V is assumed to be sufficiently smooth Žsee Remark 1.. Clearly u satisfies Ž1.8. ] Ž1.12. with c9 s c, a s m , and b s l c. Conversely, let u g C 4 m, a Ž V . satisfy Ž1.8. ] Ž1.12.. Let w g C 4 m, a Ž V . be a solution of Ž1.5., Ž1.6.. On integrating by parts, we get
l2
HV uw dx q lmHV w dx s
HV wD
s
HV uD
2m
u dx
2m
w dx
2 my1
q
H V
Ý js0
s l2
ž
D jw
D2 myjy1 u
y
n
D j w n
D2 myjy1 u ds
/
HV uw dx q Q,
where Qs
H V
ž
my 1 w mDm u n m
n my1
q
my 1Dm w m u n my1
n m
y
Dmy 1 w n
Dm u ds
/
if m s 2 p q 1, and Qsy
H V
ž
my 1 w mDm u n my1
n m
q
my 1Dm w m u n m
n my1
q
Dmy 1 w
if m s 2 p. Then
lm
HV w dx s Q.
From Ž1.5. we get
my 1Dm w n my 1
m
s Ž y1 . l
my 1 w n my1
,
and using the divergence theorem, we can write
H V
Dmy 1 w n
m
ds s Ž y1 . l
HV w dx.
n
Dm u ds
/
398
R. DALMASSO
Then we deduce that
lŽ m q a.
HV w dx s y Ž b q lc9. H V
my 1 w n my 1
ds,
and Ž1.4. holds, with d s yŽ b q l c9.rlŽ m q a.. Then Theorem 1.1 and Remark 5 imply that there exists u g C 2 m, a Ž V . satisfying Ž1.1. ] Ž1.3.. The following corollary is an immediate consequence of the proof of Theorem 1.3 and Remarks 1, 5, and 6. COROLLARY 5.1. Let l ) 0, m / 0. Assume that V g C 5m, a . Then the following statements are equi¨ alent: Ži. There exists u g C 2 m, a Ž V . satisfying Ž1.1. ] Ž1.3. with c s 0. Žii. There exists u g C 4 m, a Ž V . satisfying Ž1.2. and Ž1.8. ] Ž1.12. with b s yl c9.
6. PROOFS OF THEOREMS 1.4, 1.6 AND ANOTHER SYMMETRY RESULT We first show that the overdetermined boundary value problem Ž1.13., Ž1.14. has a solution when V ; R 2 is the unit ball. We recall some basic properties of Bessel functions Žsee w19x.. Let J 0 and J1 denote the Bessel functions of the first kind of order 0 and 1, respectively. Let I0 and I1 denote the modified Bessel functions corresponding to J 0 and J1: Ik Ž z . s eyi kp r2 Jk Ž iz ., k s 0, 1. The functions J 0 and I0 satisfy the equations d2 dz
2
J0 q
1 d z dz
J0 q J0 s 0
and d2 dz
2
I0 q
1 d z dz
I0 y I0 s 0,
respectively. We have the following relation between the functions of order 0 and 1: d dz
J 0 s yJ1
and
d dz
I0 s I1 .
J 0 , J1 , I0 , and I1 are all real valued on Ž0, `.. Moreover, I0 and I1 are positive on Ž0, `.. I0 is strictly increasing on Ž0, `. and lim x ª` I0 Ž x . s `. J 0 and J1 have only real zeros z 0, n and z1, n , respectively, and lim z 0, n s lim z1, n s `. Furthermore, the zeros of J 0 and J1 separate each other and J 0 is bounded on Ž0, `..
399
OVERDETERMINED EIGENVALUE PROBLEMS
Then there exists l0 ) 0 such that for infinitely many values of l G l 0 we have J1Ž'l . s 0 and I0 Ž'l . q J 0 Ž'l . ) 0. Now let l G l0 be such that J1Ž'l . s 0. Define ¨ Ž r . s I0 Ž 'l r . q J 0 Ž 'l r . y I0 Ž 'l . y J 0 Ž 'l . .
Then uŽ x . s ¨ Ž< x <., x g V, is a radial solution of Ž1.13., Ž1.14. with
m s l Ž I0 Ž 'l . q J 0 Ž 'l . . / 0, c9 s 'l I1 Ž 'l . / 0, and a s l Ž I0 Ž 'l . y J 0 Ž 'l . . / m . Now we consider the case c9 s 0. For any l ) 0 such that J1Ž'l . s 0, let ¨ Ž r . s J 0 Ž 'l r . y J 0 Ž 'l . .
Then uŽ x . s ¨ Ž< x <., x g V, is a radial solution of Ž1.1. ] Ž1.3. with m s 1, m s l J 0 Ž'l . / 0, and c s 0. Clearly u satisfies Ž1.13., Ž1.14. with m s l J 0 Ž'l ., c9 s 0, and a s yl J 0 Ž'l . s ym. Proof of Theorem 1.4. Let u g C 4, a Ž V . satisfy Ž1.13., Ž1.14. and let w g C 4, a Ž V . satisfy Ž1.16.. On integrating by parts, we get
l2
HV uw dx q lmHV w dx s HV wD u dx 2
s
HV uD w dx q H V 2
q
H V
s l2
ž
Dw
u n
y
ž
w
D u n
Dw n
y
w n
D u ds
/
/
u ds
HV uw dx q lc9H V w ds y aH V
w n
ds
q c9
H V Dw ds,
from which we deduce Ž1.15., with d s yarlm / y1rl and d9 s c9rlm. Conversely, suppose that for some constants d / y1rl and d9, Ž1.15. holds for all w g C 4, a Ž V . satisfying Ž1.16.. Let w g C 4, a Ž V . satisfy Ž1.5. with m s 1. Since Ž1.16. holds for w, Ž1.15. also holds with d / y1rl and Dw q lw s 0 on V. From Ž1.5. with m s 1, using the divergence theo-
400
R. DALMASSO
rem we can write
w
1
HV w dx s y l H V
n
ds,
and we deduce that
HV w dx s 0. By Theorem 1.1 and Remarks 1, 5, and 6, there exists u g C 4, a Ž V . satisfying Ž1.1. ] Ž1.3. with m s 1 and c s 0. Clearly, u also satisfies Ž1.13., Ž1.14. with c9 s 0 and a s ym. Remark 7. Clearly, Theorem 1.4 also holds with Žii. replaced by: Žii.9 There exists a constant d / y1rl such that
HV w dx s dH V
w n
ds,
for all w g C 4, a Ž V . satisfying Ž1.16. and Dw q lw s 0
on V .
Proof of Theorem 1.6. Ži. By Theorem 1.1 there exists u g C 2, a Ž V . such that D u q lu q 1 s 0 in V , us0 on V , u s c Ž const.. on V . n Ž Since u k 0 and 0 - l - lŽ1. 1 , we conclude that u ) 0 in V see Amann w3x.. Then we can apply the result of Serrin w18x to conclude that V is a ball. Žii. By Theorem 1.1 there exists u g C 4, a Ž V . such that D2 u s l u q 1 u us s0 n
2u n 2
s c Ž const..
in V , on V , on V .
OVERDETERMINED EIGENVALUE PROBLEMS
401
Since u k 0 and 0 - l F « - « 0 , Theorem 1.5 implies that u ) 0 in V. Then we can apply Theorem 2.1 to get the conclusion Žsee Remark 4.. Remark 8. In Theorem 1.6, necessarily d ) 0 in case Ži. and d - 0 in case Žii.. Indeed, the proof of Theorem 1.1 shows that c s yd. Since u ) 0 in V in both cases, Lemma 2.1 implies that c / 0. Then the result follows easily from Taylor’s formula. COROLLARY 6.1. Assume that m s 2. Let E« s Ž x 1 , x 2 .; x 12 q Ž1 q . « x 22 - 14 denote an ellipse close to the unit ball and let 0 - l F « - « 0 . Then, for any d g R, there exists w g C 4 Ž E« . l C 1 Ž Ee . such that D2 w s lw
in E« , on E« ,
ws0 and
HE w dx / dH E «
PROPOSITION 6.1. Assume that either
w
«
n
ds.
Let l, m ) 0 and V g C 2 m, a for some a g Ž0, 1x.
Ži. m s 1, or Žii. n s 2, m G 2, and the assumptions of Theorem 1.5 hold with L s Žy1. mDm . Let u g C 2 m, a Ž V . be a solution of problem Ž1.1. ] Ž1.3. with the sign condition l u q m G 0 in V. Then 0 - l - lŽ1m., and the constant d in Ž1.4. is positi¨ e when m is odd and negati¨ e when m is e¨ en. Proof. Let w be an eigenvalue corresponding to lŽ1m.. In case Ži. we can assume that w ) 0 in V by the Courant nodal line theorem Žw7x, p. 452.. In case Žii. there exists w ) 0 in V by Corollary 2.1. The maximum principle implies that u ) 0 in V. By integration by parts, it follows immediately that
lŽ1m.
m
HV uw dx s Ž y1. HV uD s Ž y1 .
m
m
w dx
m
u dx
HV wD
sl
HV uw dx q mHV w dx,
i.e.,
Ž lŽ1m. y l . H
V
uw dx s m
HV w dx,
402
R. DALMASSO
from which we deduce that lŽ1m. ) l. Since u ) 0 in V and l, m ) 0, Lemma 2.1 implies that c s yd m / 0. Then, using Taylor’s formula, we obtain the last part of the proposition. PROPOSITION 6.2. Let l ) 0, m / 0, and V g C 2 m, a for some a g Ž0, 1x. Assume that either Ži. m s 1, or Žii. n s 2, m G 2 and the assumptions of Theorem 1.5 hold with L s Žy1. mDm . If problem Ž1.1. ] Ž1.3. has a solution u g C 2 m, a Ž V ., then l / lŽ1m.. If, moreo¨ er, c s 0, then l g lŽj m. ; j G 24 . Proof. Assume that l s lŽ1m. and let w be a corresponding eigenfunction. As before, in both cases we can assume that w ) 0 in V. Now the Fredholm alternative implies that
HV w dx s 0, and we obtain a contradiction. When c s 0, ur x j is a Dirichlet eigenfunction for Žy1. mDm on V, j s 1, . . . , n, corresponding to the eigenvalue l. The first part of the proof implies that l g lŽj m. ; j G 24 . Now we consider problem Ž1.1. ] Ž1.3. with m s 0 and V a bounded connected domain with smooth boundary. Clearly, the overdetermined eigenvalue problem has radial solutions on balls for infinitely many values of l. The problem is to prove the conjecture that the overspecification of data on the boundary forces the domain to be a ball. When m s 1 and c / 0, u is an eigenfunction corresponding to lŽ1. 1 . We can assume that u ) 0 in V. Then by the classical result of Serrin w18x, V is a ball. By Theorem 1.2 we can state: If
H V w dx s 0 for all w g C 2 Ž V . l C Ž V . satisfying Dw q lŽ1. 1 w s 0 in V, then V is a ball. The situation is quite different when m G 2, since the eigenfunctions corresponding to lŽ1m. are not necessarily of one sign. We refer to w11x for a detailed discussion of this problem. However, when m s 2, we have a partial result.
OVERDETERMINED EIGENVALUE PROBLEMS
403
THEOREM 6.1. Assume that n s m s 2 and that V g C 6, a for some a g Ž0, 1x. Let the assumptions of Theorem 2.2 be satisfied with L s D2 . If
H V
w n
ds s 0
for all w g C 4 Ž V . l C 1 Ž V . satisfying D2 w s lŽ2. 1 w ws0 then V is a ball.
in V , on V ,
Proof. By Theorem 1.2 and Corollaries 2.1 and 2.2 there exists u g C 4, a Ž V . such that D2 u s lŽ2. 1 u
in V ,
u)0 in V , u us s0 on V , n
2u n 2
s c Ž const..
on V .
Then Theorem 2.1 implies that V is a ball Žsee Remark 4..
ACKNOWLEDGMENT The author is grateful to the referee for his comments and suggestions on the first draft of this paper.
REFERENCES 1. H. Amann, On the number of solutions of nonlinear equations in ordered Banach spaces, J. Funct. Anal. 11 Ž1972., 346]384. 2. H. Amann, Existence of multiple solutions for nonlinear elliptic boundary value problems, Indiana Uni¨ . Math. J. 21 Ž1972., 925]935. 3. H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Re¨ . 18 Ž1976., 620]709. 4. A. Bennett, Symmetry in an overdetermined fourth order elliptic boundary value problem, SIAM J. Math. Anal. 17 Ž1986., 1354]1358. 5. T. Boggio, Sulle funzioni di Green di ordine m, Rend. Mat. Palermo 20 Ž1905., 97]135. 6. M. Chamberland, G. M. Gladwell, and N. B. Willms, A duality theorem for an overdetermined eigenvalue problem, Z. Angew Math. Phys. 47 Ž1995., 623]629. 7. R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol. I, Interscience, New York, 1953.
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R. DALMASSO
8. R. Dalmasso, Un probleme pour une ´ equation biharmonique, Ann. Fac. Sci. ` de symetrie ´ Toulouse 11 Ž1990., 45]53. 9. R. Dalmasso, Symmetry problems for elliptic systems, Hokkaido Math. J. 25 Ž1996., 107]117. 10. H-C. Grunau, The Dirichlet problem for some semilinear elliptic differential equations of arbitrary order, Analysis 11 Ž1991., 83]90. 11. H-C. Grunau and G. Sweers, Positivity for perturbations of polyharmonic operators with Dirichlet boundary conditions in two dimension, Math. Nachr. 179 Ž1996., 89]102. 12. H-C. Grunau and G. Sweers, Positively for equations involving polyharmonic operators with Dirichlet boundary conditions, Math. Ann. 307 Ž1997., 589]626. 13. O. A. Ladyzhenskaya, The boundary value problems of mathematical physics, Appl. Math. Sci. 49, Ž1977., 183]187. 14. C. Miranda, ‘‘Partial Differential Equations of Elliptic Type,’’ Springer-Verlag, BerlinrHeidelbergrNew York, 1970. 15. L. E. Payne and P. W. Schaefer, Duality theorems in some overdetermined boundary value problems for the biharmonic operator, Math. Methods Appl. Sci. 11 Ž1989., 805]819. 16. L. E. Payne and P. W. Schaefer, On overdetermined boundary value problems for the biharmonic operator, J. Math. Anal. Appl. 187 Ž1994., 598]616. 17. P. Pucci and J. Serrin, A general variational identity, Indiana Uni¨ . Math. J. 35 Ž1986., 681]703. 18. J. Serrin, A symmetry problem in potential theory, Arch. Rational Mech. Anal. 43 Ž1971., 304]318. 19. G. N. Watson, ‘‘A Treatise on the Theory of Bessel functions,’’ 2nd Ed., Cambridge Univ. Press, Cambridge, England, 1962. 20. H. F. Weinberger, Remark on the preceding paper of Serrin, Arch. Rational Mech. Anal. 43 Ž1971., 319]320. 21. S. A. Williams, Analycity of the boundary for Lipschitz domains without the Pompeiu property, Indiana Uni¨ . Math. J. 30 Ž1981., 357]369.
221, 384]404 Ž1998.
AY975906
On Overdetermined Eigenvalue Problems for the Polyharmonic Operator R. Dalmasso Laboratoire LMC-IMAG, Equipe EDP, Tour IRMA, BP 53, F-38041 Grenoble Cedex 9, France Submitted by Charles W. Groetsch Received March 17, 1997
We consider two eigenvalue problems for the polyharmonic operator, with overdetermined boundary conditions. We give an equivalent integral formulation. In some cases we obtain characterizations of open balls by means of integral identities. Q 1998 Academic Press Key Words: Polyharmonic operator; overdetermined problems
1. INTRODUCTION Let V ; R n, n G 2, be a bounded domain with sufficiently smooth boundary V. We consider the following overdetermined boundary value problem for the polyharmonic operator: m
Ž y1. Dm u s l u q m us
u n
s ??? s
mu n m
my 1 u n my 1
s c Ž const..
in V ,
s0
on V ,
on V ,
Ž 1.1. Ž 1.2. Ž 1.3.
where l ) 0, m g R, rn is the outward normal derivative and m is a positive integer. When m s 1, l s 0, and m / 0, Serrin w18x showed that if u g C 2 Ž V . satisfies Ž1.1. ] Ž1.3., then V must be a ball. The method of moving planes used there was also shown to apply to more general elliptic equations and somewhat more general boundary conditions. Serrin’s basic result was proved in an alternative manner by Weinberger w20x. 384 0022-247Xr98 $25.00 Copyright Q 1998 by Academic Press All rights of reproduction in any form reserved.
OVERDETERMINED EIGENVALUE PROBLEMS
385
When m s 2, l s 0, and m / 0, Bennett w4x modified Weinberger’s method to establish that V is a ball if u g C 4 Ž V . satisfies Ž1.1. ] Ž1.3.. Unfortunately, Bennett’s argument does not extend to more general equations. Using the method of moving planes and assuming in addition that u G 0 in V, we were able to treat more general biharmonic equations and systems Žsee w8x and w9x.. In the present work we shall first reformulate problem Ž1.1. ] Ž1.3. in an equivalent integral form. THEOREM 1.1. Let l ) 0, m / 0. Assume that V g C 3 m, a for some a g Ž0, 1x. Then the following statements are equi¨ alent: Ži. There exists u g C 2 m, a Ž V . satisfying Ž1.1. ] Ž1.3.. Žii. There exists a constant d such that
HV w dx s dH V
my 1 w n my 1
ds
Ž 1.4.
for all w g C 2 m, a Ž V . satisfying m
Ž y1. Dm w s lw ws
w n
s ??? s
my 2 w n my 2
s0
in V , on V if m G 2.
Ž 1.5. Ž 1.6.
THEOREM 1.2. Let l ) 0, m s 0. Assume that V g C 3 m, a for some a g Ž0, 1x. Then the following statements are equi¨ alent: Ži. There exists u k 0 in C 2 m, a Ž V . satisfying Ž1.1. ] Ž1.3.. Žii.
H V
my 1 w n my 1
ds s 0,
Ž 1.7.
for all w g C 2 m, a Ž V . satisfying Ž1.5., Ž1.6.. Remark 1. Let V be of class C 2 mqk, a for some k G 0, a g Ž0, 1x. If u g C 2 m Ž V . l C my1 Ž V . satisfies Ž1.1., Ž1.2., then u g C 2 mqk, a Ž V . Žsee w14x.. In the case where m s 1, Theorem 1.1 was proved by Chamberland, Gladwell, and Willms w6x. Various duality theorems were previously established by Payne and Schaefer w15, 16x. The integral duals were then used to deduce the spherical symmetry of V.
386
R. DALMASSO
We shall use Theorem 1.1 to establish the following result. THEOREM 1.3. Let l ) 0, m / 0. Assume that V g C 5m, a for some a g Ž0, 1x. Then the following statements are equi¨ alent: Ži. There exists u g C 2 m, a Ž V . satisfying Ž1.1. ] Ž1.3.. Žii. There exists u g C 4 m, a Ž V . satisfying Ž1.2. and D2 m u s l2 u q lm
u
in V ,
Ž 1.8.
on V ,
Ž 1.9.
m
n m
s c9 Ž const..
m
Ž y1. Dm u s aŽ const.. / ym D u
on V ,
Ž 1.10.
m m
Ž y1. Dm u
m
n m
on V ,
s b Ž const..
Ž 1.11.
my 1Dm u
s ??? s s0 on V if m G 2. Ž 1.12. n n my 1 A domain V with a C 2 connected boundary is said to have the Schiffer property if, for any l, the only solution to the overdetermined boundary value problem Ž1.1. ] Ž1.3. with c s 0 is the trivial solution u s 0 Žcorresponding to m s 0.. The Schiffer conjecture asserts that, if for some l ) 0, m / 0, problem Ž1.1. ] Ž1.3. with c s 0 has a solution u g C 2 Ž V . when V is a domain with a C 2 connected boundary, then V is a ball. A result due to Williams w21x asserts that any Lipschitz domain that does not possess the Schiffer property has a real analytic boundary. Therefore, using Theorem 1.1 with m s 1 and Remarks 1 and 6, the Schiffer conjecture can be restated in the following way Žsee also w6x.. If, for some fixed a g Ž0, 1x, every w g C 2, a Ž V . satisfying Ž1.5. Žwith m s 1. has vanishing mean over some fixed bounded domain V ; R n with a C 2, a connected boundary, must V be a ball? Using Theorem 1.3 with c s 0 Žsee Corollary 5.1., Remark 1, and the above result of Williams, The Schiffer conjecture can also be stated as follows. If for some l ) 0, m / 0, there exists u g C 4 Ž V . such that D2 u s l2 u q lm us0, D usa Ž const.. / m , must V be a ball?
u n D u n
in V ,
Ž 1.13.
s c9 Ž const.. , s l c9
on V ,
Ž 1.14.
387
OVERDETERMINED EIGENVALUE PROBLEMS
THEOREM 1.4. Let l ) 0, m / 0. Assume that V g C 5, a for some a g Ž0, 1x. Then the following statements are equi¨ alent: Ži. There exists u g C 4, a Ž V . satisfying Ž1.13. ] Ž1.14.. Žii. There are constants d / y1rl and d9 such that
HV w dx s dH V
w n
H V Ž Dw q lw . ds,
ds q d9
Ž 1.15.
for all w g C 4, a Ž V . satisfying D2 w s l2 w
in V .
Ž 1.16.
In some cases already mentioned Žw4x, w9x, w15x, w16x., duality theorems have been used to give characterizations of open balls in R n by means of integral identities. Since higher order elliptic equations do not have a maximum principle in general, we cannot obtain the generalization to the polyharmonic case. Nevertheless, we shall give some partial results. We first recall a theorem obtained by Grunau and Sweers w11x. We begin with the definition of closeness of domains and operators. DEFINITION 1.1 w11x. Let « ) 0, a g Ž0, 1x. V is called « close in C k, a sense to V*, if there exists a C k, a mapping g: V*¬ V such that g ŽV*. s V and 5 g y Id 5 C k , a ŽV*. F « . Now consider the higher order elliptic problem Lu G 0 us
u n
s ??? s
in V , my 1
n
u
my 1
Ž 1.17. on V ,
s0
Ž 1.18.
where V ; R 2 and
ž
Ls y
Ý 1Fi , jF2
ai j
2 xi x j
m
/
q
Ý iqjF2 my1
bi j
i
ž /ž / x1
x2
j
, Ž 1.19.
with a i j g C 2 my1, a Ž V ., bi j g C 0, a Ž V .. DEFINITION 1.2 w11x. Let « ) 0 and let L be as above. The operator L is called « close in C k, a sense to Žy1. mDm on V, if, additionally,
388
R. DALMASSO
a i j g C k, a Ž V . and 5 ai j y di j 5 C k , a Ž V . F « ,
5 bi j 5 CŽ V . F « .
THEOREM 1.5 w11x. Let n s 2 and B s x g R 2 ; < x < - 14 . There exists « 0 s « 0 Ž m. ) 0 such that, for « g w0, « 0 ., we ha¨ e the following. If V g C 2 m, a , V is « close in C 2 m sense to B, and L is « close in 2 my1, a C sense to Žy1. mDm on V, then e¨ ery u g C 2 m, a Ž V ., u k 0, satisfyŽ . ing 1.17 , Ž1.18. is strictly positi¨ e in V. We denote by lŽ1m. the first eigenvalue of Žy1. mDm in V with Dirichlet boundary conditions. Using the notations of Theorem 1.5, we now state one of our symmetry results. THEOREM 1.6.
Let V g C 3 m, a for some a g Ž0, 1x. Assume that either
Ži. m s 1 and 0 - l - lŽ1. 1 , or Žii. n s m s 2, V is « close in C 4 sense to B and 0 - l F « - « 0 . If there exists a constant d such that
HV w dx s dH V
my 1 w n my 1
ds
for all w g C 2 m Ž V . l C my1 Ž V . satisfying m
Ž y1. Dm w s lw ws0
in V ,
on V if m s 2,
then V is a ball. In Section 2 we give some preliminary results. In particular, we recall a theorem obtained in w8x. Then, using some results of w11x and w12x, we study the eigenspace corresponding to the first eigenvalue of higher order elliptic operators, whose principal part is the mth power of a second-order elliptic operator. Theorems 1.1, 1.2, and 1.3 are proved in Sections 3, 4, and 5, respectively. Finally, in Section 6 we first show that the overdetermined boundary value problem Ž1.13., Ž1.14. has a solution when V ; R 2 is the unit ball. Then we prove Theorems 1.4 and 1.6 and another result concerning the symmetry of the domain. We also make some remarks.
OVERDETERMINED EIGENVALUE PROBLEMS
389
2. PRELIMINARIES We shall need the following lemma. LEMMA 2.1.
Let u g C 2 m Ž V . be a solution of problem Ž1.1. ] Ž1.3.. Then ml
HV u
2
dx q
n q 2m 2
m
HV u dx s
nc 2 2
< V <.
Proof. The proof follows readily from a generalization of Pohozaev’s identity Žsee w17x, pp. 701]702.. The next lemma is easily proved by an induction argument. It will be used repeatedly without referring to it. LEMMA 2.2.
Let V g C k for some k G 1. If u g C k Ž V . is such that on V ,
u s const.
u n
s ??? s
ky 1
n
u
ky 1
s0
on V if k G 2,
then
ku x j1 ??? x jk
s
ku n k
n j1 ??? n jk
on V
for j1 , . . . , jk g 1, . . . , n4 . We first recall the following theorem obtained in w8x Žtheoreme ´ ` 3.1; see also w9x.. THEOREM 2.1. Let V ; R n Ž n G 2. be a bounded domain with C 2 boundary V. Let f : R 2 ª Ž0, `. satisfy the following condition. For each ¨ g R, u ª f Ž u, ¨ . is nondecreasing, and for each u g R, ¨ ª f Ž u, ¨ . is nonincreasing. Let u g C 4 Ž V . l C 3 Ž V . be a solution of the o¨ erdetermined boundary ¨ alue problem D2 u s f Ž u, D u . us
u n
s0
D u s d Ž const..
in V , on V , on V .
If u G 0 in V, then V is a ball. If V s x g R n ; < x y x 0 < - R4 for some x 0 g R n, then uŽ x . s ¨ Ž< x y x 0 <., ¨ 9 - 0 in Ž0, R ., and Ž D¨ .9 ) 0 in Ž0, R x.
390
R. DALMASSO
Remark 2. Notice that u g C 4 Ž V . in w8x, but it is enough to assume that u g C 4 Ž V . l C 3 Ž V .. Remark 3. Assume that f ' 1, V g C 4, a for some a g Ž0, 1x and u g C 4 Ž V .. Then the assumption u G 0 in V can be removed. Indeed, this is just Bennett’s result w4x. Remark 4. Assume that f : R 2 ª w0, `. and that f Ž u, ¨ . ) 0 for u ) 0, ¨ g R. If u G 0, u k 0 in V, then the conclusion of Theorem 2.1 still holds. This readily follows from the proof. Under the assumptions of Theorem 1.5, the Green’s function of L Žgiven by Ž1.19.. for the Dirichlet problem in V is positive. Using the Krein]Rutman theorem, we have the following. COROLLARY 2.1. Under the assumptions of Theorem 1.5, there is a positi¨ e eigenfunction for the first Dirichlet eigen¨ alue of L in V. Now we consider the case n s m s 2. The Green’s function GŽ x, y . of D2 for the Dirichlet problem in B is known explicitly Žsee w5x, w10x.:
G Ž x, y . s
1 8p
½
< x y y < 2 ln < x y y < y ln < x < y y
ž
q
1 2
ž
< x< y y
x
2
< x<
x < x<
y < x y y<2
/
/5
for x, y g B, x / y Žif x s 0, replace < < x < y y xr< x < < by 1.. LEMMA 2.3.
D x GŽ x, y . ) 0 for < x < s 1, < y < - 1.
Proof. We easily get
D x G Ž x, y . s
1 4p
½
2 log < x y y < q 2 y 2 log < x < y y y2 < x < y y y1 q < y < 2
5
x < x<
y2
x < x<
Ž < x < 2 < y < 2 q < y < 2 y Ž1 q < y < 2 . ² x, y : .
391
OVERDETERMINED EIGENVALUE PROBLEMS
for x, y g B, x / y. We deduce that
D x G Ž x, y . s
1 4p
2
< x y y
for < x < s 1 and < y < - 1. Let L be given by Ž1.19. with m s 2, V s B, and a i j s d i j . Assume that L is « close in C 3, a sense to D2 on B for some a g Ž0, 1x. From w12x we know that the Green’s function GLŽ x, y . of L for the Dirichlet problem in B behaves like G if « g w0, « 0 . for « 0 small enough: 1 C
G Ž x, y . F GL Ž x, y . F CG Ž x, y . ,
x, y g B,
x / y.
With the help of Theorem 1.5 and Lemma 2.3, we immediately deduce the following lemma. LEMMA 2.4. Let n s m s 2, V s B, and let L be gi¨ en by Ž1.19. with a i j s d i j . There exists « 0 ) 0 such that, for « g w0, « 0 . we ha¨ e the following. If L is « close in C 3, a sense to D2 on B, then e¨ ery u g C 4, a Ž B ., u k 0, satisfying Ž1.17., Ž1.18. is strictly positi¨ e in B and, moreo¨ er, 2 urn 2 ) 0 on B. Now using Remark 1 of w12, Sect. 5x and Proposition 2.4 of w11, p. 92x, we obtain the following. THEOREM 2.2. Let n s m s 2 and let L be gi¨ en by Ž1.19. with a i j s d i j . There exists « 0 ) 0 such that, for « g w0, « 0 . we ha¨ e the following. If V g C 4, a , V is « close in C 4 sense to B and L is « close in C 3, a sense to D2 on V, then e¨ ery u g C 4, a Ž V ., u k 0, satisfying Ž1.17., Ž1.18. is strictly positi¨ e in V and, moreo¨ er, 2 urn 2 ) 0 on V. Using the same arguments as in Amann w1]3x, we derive the following corollary. COROLLARY 2.2. Under the assumptions of Theorem 2.2, the first eigen¨ alue of L in V with Dirichlet boundary conditions is simple.
392
R. DALMASSO
3. PROOF OF THEOREM 1.1 Let u g C 2 m, a Ž V . satisfy Ž1.1. ] Ž1.3. and let w g C 2 m, a Ž V . satisfy Ž1.5., Ž1.6.. Using repeated integrations by parts, we obtain m
l
HV uw dx q mHV w dx s Ž y1. HV wD s Ž y1 .
m
m
u dx
m
HV uD
my1
q
Ý js0
H V
ž
D jw
w dx
Dmy jy1 u n
sl
HV uw dx y H V
sl
HV
y
my 1 w m u
uw dx y c
H V
n my1 n m my 1 w n my 1
D j w n
Dmyjy1 u ds
/
ds
ds,
from which we deduce Ž1.4., with d s ycrm. Conversely, suppose that for some constant d, Ž1.4. holds for all w g C 2 m, a Ž V . satisfying Ž1.5., Ž1.6.. The Fredholm alternative w13x, w14x gives the existence of a solution u g C 2 m, a Ž V . to the boundary value problem Ž1.1., Ž1.2.. Now let w g C 2 m, a Ž V . satisfy Ž1.5., Ž1.6.. Using, as before, repeated integrations by parts, we obtain
m
HV w dx q H V
my 1 w m u n my1 n m
ds s 0.
Then condition Ž1.4. implies that
H V
my 1 w n my1
ž
md q
mu n m
/
ds s 0.
Ž 3.1.
We claim that we can find w g C 2 m, a Ž V . satisfying Ž1.5., Ž1.6. and
my 1 w n my1
s md q
mu n m
on V .
Then from Ž3.1. we deduce that Ž1.3. holds with c s ym d.
Ž 3.2.
393
OVERDETERMINED EIGENVALUE PROBLEMS
Now we prove the claim. By Remark 1, u g C 3m, a Ž V .. Therefore there exists w g C 2 m, a Ž V . such that Dmw s 0
ws
w n
s ??? s
my 1w
in V ,
w
my 2
n
mu
s md q
n my1
on V if m G 2,
s0
my 2
on V .
n m
Consider the problem m
Ž y1. Dmc s lc q lw cs
c n
s ??? s
c
in V ,
Ž 3.3.
my 1
n
on V .
s0
my 1
Ž 3.4.
If l is not an eigenvalue of the polyharmonic operator Žy1. mDm in V with Dirichlet boundary conditions, then Ž3.3., Ž3.4. has a unique solution c g C 2 m, a Ž V . by the Fredholm alternative. Therefore w s w q c g C 2 m, a Ž V . satisfies Ž1.5., Ž1.6. and Ž3.2.. The claim is proved in this case. Now if l is an eigenvalue of Žy1. mDm in V with Dirichlet boundary conditions, we denote by q its multiplicity and by Ž w j .1 F jF q the corresponding orthonormalized set of eigenfunctions. Problem Ž3.3., Ž3.4. has a solution if and only if
HV ww
dx s 0,
j
j s 1, . . . , q.
Ž 3.5.
On integrating by parts, we obtain
l
HV ww
j
dx s Ž y1 .
m
s Ž y1 .
m
HV wD w m
j
dx
HV w D w dx m
j
my1
q
Ý ks0
sy
H V
my 1w mw j
H V n
my1
n m
ž
Dw k
Dmy ky1w j n
ds s y
H V
ž
y
md q
Dkw n
Dmyky1w j ds
/
m u mw j n m
/
n m
ds
394
R. DALMASSO
for j s 1, . . . , q. Then Ž3.5. is equivalent to
H V
ž
md q
m u mw j n m
/
n m
ds s 0,
j s 1, . . . , q.
Ž 3.6.
Now we write q
us
Ý ck w k q u0 ,
Ž 3.7.
ks1
where c k , k s 1, . . . , q are arbitrary constants, and u 0 g C 2 m, a Ž V . is the particular solution of Ž1.1., Ž1.2. orthogonal to w j for j s 1, . . . , q, i.e., satisfying
HV u w 0
j
dx s 0,
j s 1, . . . , q.
Let M s Ž Mi j .1 F i, jF q with Mi j s
H V
mw i mw j n m n m
ds;
let e s Ž e j .1 F jF q with ej s y
ž
H V
md q
m u 0 mw j n m
/
n m
ds,
and x s Ž c j .1 F jF q . Using Ž3.7., we can write Ž3.6. in the following form: Mx s e.
Ž 3.8.
Since
mw j
q
x Mx s T
½
H V Ý c js1
j
n m
2
5
ds,
we deduce that M is positive semidefinite. Assume that M is not invertible. Then there exist c j , j s 1,???, q with c j / 0 for some j and such that the Dirichlet eigenfunction q
¨s
Ý c j wj js1
satisfies
m¨ n m
s0
on V .
395
OVERDETERMINED EIGENVALUE PROBLEMS
Therefore ¨ satisfies Ž1.1. ] Ž1.3. with m s c s 0. From Lemma 2.1 we deduce that ¨ s 0 in V. Since the w j ’s are linearly independent, we obtain that c j s 0 for j s 1,???, q, a contradiction. Therefore M is invertible. We deduce that Ž3.8. has a solution. Then problem Ž3.3., Ž3.4. has a solution c . Finally, w s w q c g C 2 m, a Ž V . satisfies Ž1.5., Ž1.6., and Ž3.2.. The proof is complete. Remark 5. Let V be of class C 3 mqk, a for some k G 0, a g Ž0, 1x. Then we can replace w g C 2 m, a Ž V . in Theorem 1.1 Žii. by w g C 2 mqk, a Ž V .. This readily follows from the proof, using the elliptic regularity theory and Remark 1. Remark 6. The proof of Theorem 1.1 shows that c s 0 if and only if d s 0.
4. PROOF OF THEOREM 1.2 Let u g C 2 m, a Ž V ., u k 0, satisfy Ž1.1. ] Ž1.3. with m s 0 and let w g C V . satisfy Ž1.5., Ž1.6.. Using repeated integrations by parts, we obtain 2 m, a Ž
m
l
HV uw dx s Ž y1. HV wD s Ž y1 .
m
m
u dx
m
HV uD
w dx
my1
q
Ý js0
H V
sl
HV uw dx y H V
sl
HV uw dx y cH V
ž
D jw
Dmy jy1 u n
my 1 w m u n my1 n m my 1 w n my 1
y
D j w n
Dmyjy1 u ds
/
ds
ds.
Since by Lemma 2.1 c / 0, we deduce Ž1.7.. Conversely, suppose that Ž1.7. holds for all w g C 2 m, a Ž V . satisfying Ž1.5., Ž1.6.. Assume that l is not an eigenvalue of Žy1. mDm in V with
396
R. DALMASSO
Dirichlet boundary conditions. Let r g C 2 m, a Ž V . be the solution of Dmr s 0
rs
r n
s ??? s
r
in V ,
my 2
n
my 2
my 1r n my 1
on V if m G 2,
s0
on V .
s1
The Fredholm alternative gives the existence of a solution ¨ g C 2 m, a Ž V . to the boundary value problem m
Ž y1. Dm ¨ s l¨ q lr ¨s
¨ n
my 1 ¨
s ??? s
n my 1
in V , on V .
s0
Then w s ¨ q r g C 2 m, a Ž V . satisfies Ž1.5., Ž1.6. and
my 1 w n my 1
on V .
s1
Since Ž1.7. does not hold, we reach a contradiction. Therefore l is an eigenvalue of Žy1. mDm in V with Dirichlet boundary conditions. Let u g C 2 m, a Ž V . be a solution of the boundary value problem Ž1.1., Ž1.2. with m s 0, and let w g C 2 m, a Ž V . satisfy Ž1.5., Ž1.6.. Using, as before, repeated integrations by parts, we obtain
H V
my 1 w m u n my1 n m
ds s 0.
Then condition Ž1.7. implies that
H V
my 1 w n my1
ž
cy
mu n m
/
ds s 0
Ž 4.1.
for any c g R. We claim that we can find w g C 2 m, a Ž V . satisfying Ž1.5., Ž1.6. and
my 1 w n my1
scy
mu n m
on V
for any c g R. Then from Ž4.1. we deduce that Ž1.3. holds. If we choose c / 0, then Lemma 2.1 implies that u k 0. To prove the claim, we use the same arguments as in the proof of Theorem 1.1 with u 0 s 0 in Ž3.7..
397
OVERDETERMINED EIGENVALUE PROBLEMS
5. PROOF OF THEOREM 1.3 If u g C 2 m, a Ž V . satisfies Ž1.1. ] Ž1.3., then u g C 4 m, a Ž V ., since V is assumed to be sufficiently smooth Žsee Remark 1.. Clearly u satisfies Ž1.8. ] Ž1.12. with c9 s c, a s m , and b s l c. Conversely, let u g C 4 m, a Ž V . satisfy Ž1.8. ] Ž1.12.. Let w g C 4 m, a Ž V . be a solution of Ž1.5., Ž1.6.. On integrating by parts, we get
l2
HV uw dx q lmHV w dx s
HV wD
s
HV uD
2m
u dx
2m
w dx
2 my1
q
H V
Ý js0
s l2
ž
D jw
D2 myjy1 u
y
n
D j w n
D2 myjy1 u ds
/
HV uw dx q Q,
where Qs
H V
ž
my 1 w mDm u n m
n my1
q
my 1Dm w m u n my1
n m
y
Dmy 1 w n
Dm u ds
/
if m s 2 p q 1, and Qsy
H V
ž
my 1 w mDm u n my1
n m
q
my 1Dm w m u n m
n my1
q
Dmy 1 w
if m s 2 p. Then
lm
HV w dx s Q.
From Ž1.5. we get
my 1Dm w n my 1
m
s Ž y1 . l
my 1 w n my1
,
and using the divergence theorem, we can write
H V
Dmy 1 w n
m
ds s Ž y1 . l
HV w dx.
n
Dm u ds
/
398
R. DALMASSO
Then we deduce that
lŽ m q a.
HV w dx s y Ž b q lc9. H V
my 1 w n my 1
ds,
and Ž1.4. holds, with d s yŽ b q l c9.rlŽ m q a.. Then Theorem 1.1 and Remark 5 imply that there exists u g C 2 m, a Ž V . satisfying Ž1.1. ] Ž1.3.. The following corollary is an immediate consequence of the proof of Theorem 1.3 and Remarks 1, 5, and 6. COROLLARY 5.1. Let l ) 0, m / 0. Assume that V g C 5m, a . Then the following statements are equi¨ alent: Ži. There exists u g C 2 m, a Ž V . satisfying Ž1.1. ] Ž1.3. with c s 0. Žii. There exists u g C 4 m, a Ž V . satisfying Ž1.2. and Ž1.8. ] Ž1.12. with b s yl c9.
6. PROOFS OF THEOREMS 1.4, 1.6 AND ANOTHER SYMMETRY RESULT We first show that the overdetermined boundary value problem Ž1.13., Ž1.14. has a solution when V ; R 2 is the unit ball. We recall some basic properties of Bessel functions Žsee w19x.. Let J 0 and J1 denote the Bessel functions of the first kind of order 0 and 1, respectively. Let I0 and I1 denote the modified Bessel functions corresponding to J 0 and J1: Ik Ž z . s eyi kp r2 Jk Ž iz ., k s 0, 1. The functions J 0 and I0 satisfy the equations d2 dz
2
J0 q
1 d z dz
J0 q J0 s 0
and d2 dz
2
I0 q
1 d z dz
I0 y I0 s 0,
respectively. We have the following relation between the functions of order 0 and 1: d dz
J 0 s yJ1
and
d dz
I0 s I1 .
J 0 , J1 , I0 , and I1 are all real valued on Ž0, `.. Moreover, I0 and I1 are positive on Ž0, `.. I0 is strictly increasing on Ž0, `. and lim x ª` I0 Ž x . s `. J 0 and J1 have only real zeros z 0, n and z1, n , respectively, and lim z 0, n s lim z1, n s `. Furthermore, the zeros of J 0 and J1 separate each other and J 0 is bounded on Ž0, `..
399
OVERDETERMINED EIGENVALUE PROBLEMS
Then there exists l0 ) 0 such that for infinitely many values of l G l 0 we have J1Ž'l . s 0 and I0 Ž'l . q J 0 Ž'l . ) 0. Now let l G l0 be such that J1Ž'l . s 0. Define ¨ Ž r . s I0 Ž 'l r . q J 0 Ž 'l r . y I0 Ž 'l . y J 0 Ž 'l . .
Then uŽ x . s ¨ Ž< x <., x g V, is a radial solution of Ž1.13., Ž1.14. with
m s l Ž I0 Ž 'l . q J 0 Ž 'l . . / 0, c9 s 'l I1 Ž 'l . / 0, and a s l Ž I0 Ž 'l . y J 0 Ž 'l . . / m . Now we consider the case c9 s 0. For any l ) 0 such that J1Ž'l . s 0, let ¨ Ž r . s J 0 Ž 'l r . y J 0 Ž 'l . .
Then uŽ x . s ¨ Ž< x <., x g V, is a radial solution of Ž1.1. ] Ž1.3. with m s 1, m s l J 0 Ž'l . / 0, and c s 0. Clearly u satisfies Ž1.13., Ž1.14. with m s l J 0 Ž'l ., c9 s 0, and a s yl J 0 Ž'l . s ym. Proof of Theorem 1.4. Let u g C 4, a Ž V . satisfy Ž1.13., Ž1.14. and let w g C 4, a Ž V . satisfy Ž1.16.. On integrating by parts, we get
l2
HV uw dx q lmHV w dx s HV wD u dx 2
s
HV uD w dx q H V 2
q
H V
s l2
ž
Dw
u n
y
ž
w
D u n
Dw n
y
w n
D u ds
/
/
u ds
HV uw dx q lc9H V w ds y aH V
w n
ds
q c9
H V Dw ds,
from which we deduce Ž1.15., with d s yarlm / y1rl and d9 s c9rlm. Conversely, suppose that for some constants d / y1rl and d9, Ž1.15. holds for all w g C 4, a Ž V . satisfying Ž1.16.. Let w g C 4, a Ž V . satisfy Ž1.5. with m s 1. Since Ž1.16. holds for w, Ž1.15. also holds with d / y1rl and Dw q lw s 0 on V. From Ž1.5. with m s 1, using the divergence theo-
400
R. DALMASSO
rem we can write
w
1
HV w dx s y l H V
n
ds,
and we deduce that
HV w dx s 0. By Theorem 1.1 and Remarks 1, 5, and 6, there exists u g C 4, a Ž V . satisfying Ž1.1. ] Ž1.3. with m s 1 and c s 0. Clearly, u also satisfies Ž1.13., Ž1.14. with c9 s 0 and a s ym. Remark 7. Clearly, Theorem 1.4 also holds with Žii. replaced by: Žii.9 There exists a constant d / y1rl such that
HV w dx s dH V
w n
ds,
for all w g C 4, a Ž V . satisfying Ž1.16. and Dw q lw s 0
on V .
Proof of Theorem 1.6. Ži. By Theorem 1.1 there exists u g C 2, a Ž V . such that D u q lu q 1 s 0 in V , us0 on V , u s c Ž const.. on V . n Ž Since u k 0 and 0 - l - lŽ1. 1 , we conclude that u ) 0 in V see Amann w3x.. Then we can apply the result of Serrin w18x to conclude that V is a ball. Žii. By Theorem 1.1 there exists u g C 4, a Ž V . such that D2 u s l u q 1 u us s0 n
2u n 2
s c Ž const..
in V , on V , on V .
OVERDETERMINED EIGENVALUE PROBLEMS
401
Since u k 0 and 0 - l F « - « 0 , Theorem 1.5 implies that u ) 0 in V. Then we can apply Theorem 2.1 to get the conclusion Žsee Remark 4.. Remark 8. In Theorem 1.6, necessarily d ) 0 in case Ži. and d - 0 in case Žii.. Indeed, the proof of Theorem 1.1 shows that c s yd. Since u ) 0 in V in both cases, Lemma 2.1 implies that c / 0. Then the result follows easily from Taylor’s formula. COROLLARY 6.1. Assume that m s 2. Let E« s Ž x 1 , x 2 .; x 12 q Ž1 q . « x 22 - 14 denote an ellipse close to the unit ball and let 0 - l F « - « 0 . Then, for any d g R, there exists w g C 4 Ž E« . l C 1 Ž Ee . such that D2 w s lw
in E« , on E« ,
ws0 and
HE w dx / dH E «
PROPOSITION 6.1. Assume that either
w
«
n
ds.
Let l, m ) 0 and V g C 2 m, a for some a g Ž0, 1x.
Ži. m s 1, or Žii. n s 2, m G 2, and the assumptions of Theorem 1.5 hold with L s Žy1. mDm . Let u g C 2 m, a Ž V . be a solution of problem Ž1.1. ] Ž1.3. with the sign condition l u q m G 0 in V. Then 0 - l - lŽ1m., and the constant d in Ž1.4. is positi¨ e when m is odd and negati¨ e when m is e¨ en. Proof. Let w be an eigenvalue corresponding to lŽ1m.. In case Ži. we can assume that w ) 0 in V by the Courant nodal line theorem Žw7x, p. 452.. In case Žii. there exists w ) 0 in V by Corollary 2.1. The maximum principle implies that u ) 0 in V. By integration by parts, it follows immediately that
lŽ1m.
m
HV uw dx s Ž y1. HV uD s Ž y1 .
m
m
w dx
m
u dx
HV wD
sl
HV uw dx q mHV w dx,
i.e.,
Ž lŽ1m. y l . H
V
uw dx s m
HV w dx,
402
R. DALMASSO
from which we deduce that lŽ1m. ) l. Since u ) 0 in V and l, m ) 0, Lemma 2.1 implies that c s yd m / 0. Then, using Taylor’s formula, we obtain the last part of the proposition. PROPOSITION 6.2. Let l ) 0, m / 0, and V g C 2 m, a for some a g Ž0, 1x. Assume that either Ži. m s 1, or Žii. n s 2, m G 2 and the assumptions of Theorem 1.5 hold with L s Žy1. mDm . If problem Ž1.1. ] Ž1.3. has a solution u g C 2 m, a Ž V ., then l / lŽ1m.. If, moreo¨ er, c s 0, then l g lŽj m. ; j G 24 . Proof. Assume that l s lŽ1m. and let w be a corresponding eigenfunction. As before, in both cases we can assume that w ) 0 in V. Now the Fredholm alternative implies that
HV w dx s 0, and we obtain a contradiction. When c s 0, ur x j is a Dirichlet eigenfunction for Žy1. mDm on V, j s 1, . . . , n, corresponding to the eigenvalue l. The first part of the proof implies that l g lŽj m. ; j G 24 . Now we consider problem Ž1.1. ] Ž1.3. with m s 0 and V a bounded connected domain with smooth boundary. Clearly, the overdetermined eigenvalue problem has radial solutions on balls for infinitely many values of l. The problem is to prove the conjecture that the overspecification of data on the boundary forces the domain to be a ball. When m s 1 and c / 0, u is an eigenfunction corresponding to lŽ1. 1 . We can assume that u ) 0 in V. Then by the classical result of Serrin w18x, V is a ball. By Theorem 1.2 we can state: If
H V w dx s 0 for all w g C 2 Ž V . l C Ž V . satisfying Dw q lŽ1. 1 w s 0 in V, then V is a ball. The situation is quite different when m G 2, since the eigenfunctions corresponding to lŽ1m. are not necessarily of one sign. We refer to w11x for a detailed discussion of this problem. However, when m s 2, we have a partial result.
OVERDETERMINED EIGENVALUE PROBLEMS
403
THEOREM 6.1. Assume that n s m s 2 and that V g C 6, a for some a g Ž0, 1x. Let the assumptions of Theorem 2.2 be satisfied with L s D2 . If
H V
w n
ds s 0
for all w g C 4 Ž V . l C 1 Ž V . satisfying D2 w s lŽ2. 1 w ws0 then V is a ball.
in V , on V ,
Proof. By Theorem 1.2 and Corollaries 2.1 and 2.2 there exists u g C 4, a Ž V . such that D2 u s lŽ2. 1 u
in V ,
u)0 in V , u us s0 on V , n
2u n 2
s c Ž const..
on V .
Then Theorem 2.1 implies that V is a ball Žsee Remark 4..
ACKNOWLEDGMENT The author is grateful to the referee for his comments and suggestions on the first draft of this paper.
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