Variational formula and overdetermined problem for the principle eigenvalue of k-Hessian operator

J. Differential Equations 255 (2013) 4136–4148

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Journal of Differential Equations www.elsevier.com/locate/jde

Variational formula and overdetermined problem for the principle eigenvalue of k-Hessian operator ✩ Qiuyi Dai, Feilin Shi ∗ Department of Mathematics, Hunan Normal University, Changsha, Hunan 410081, PR China

a r t i c l e

i n f o

a b s t r a c t

Article history: Received 7 November 2012 Revised 28 July 2013 Available online 19 August 2013

We first deduce the first variational formula and some overdetermined problems for the principle eigenvalue of the k-Hessian operator, and then prove Serrin type symmetry result for our overdetermined problems. © 2013 Elsevier Inc. All rights reserved.

Keywords: k-Hessian operator Overdetermined problem The first variational formula Symmetry

1. Introduction Let Ω be a smooth bounded domain in R n with boundary of class C 2 . For a function u ∈ C 2 (Ω), we denote by λ( D 2 u ) = (λ1 , λ2 , . . . , λn ) the eigenvalues of the Hessian matrix D 2 u. The k-Hessian operator S k ( D 2 u ) is defined by





S k D 2 u = σk (λ) =



λi 1 · · · λi k

1i 1 <···
with σk (λ) being the k-th elementary symmetric function of λ = (λ1 , λ2 , . . . , λn ). A function u ∈ C 2 (Ω) ∩ C 0 (Ω) is said to be k-admissible if

  λ D2u ∈ Γ k ✩ This work is supported by NNSFC (No. 10971061 and No. 11271120), and by Hunan Provincial Innovation Foundation for Postgraduate (No. CX2011B198). Corresponding author. E-mail addresses: [email protected] (Q. Dai), [email protected] (F. Shi).

*

0022-0396/$ – see front matter © 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jde.2013.07.063

Q. Dai, F. Shi / J. Differential Equations 255 (2013) 4136–4148

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where Γk is an open symmetric convex cone in R n with vertex at the origin given by

   Γk = (λ1 , λ2 , . . . , λn ) ∈ R n  σ j (λ) > 0 ∀ j = 1, . . . , k , and Γ k is the closure of Γk . A domain Ω is called (k − 1)-convex if σk−1 (κ )  c 0 > 0 for some positive constant c 0 , where κ = (κ1 , κ2 , . . . , κn−1 ) denotes the principle curvature vector of ∂Ω with respect to its inner normal. When k = n it is equivalent to the usual convexity. The k-th mean curvature operator of ∂Ω is defined by

H k (∂Ω) = σk (κ1 , κ2 , . . . , κn−1 ). For convenience, we set H 0 (∂Ω) = 1 and H n (∂Ω) = 0. Obviously, H 1 (∂Ω) is the ordinary mean curvature of ∂Ω and H n−1 (∂Ω) is the Gaussian curvature of ∂Ω . Assume that Ω is a smooth (k − 1)-convex domain in R n . We consider the following problem

⎧  2  ⎨ S k D u = λ(−u )k , x ∈ Ω, u < 0, x ∈ Ω, ⎩ u = 0, x ∈ ∂Ω.

(1.1)

In 1994, X.J. Wang [36] proved that there exists a positive constant λk1 (Ω) such that problem (1.1)

has a unique smooth solution u 1 (x) up to multiplication by a positive number for λ = λk1 (Ω). More-

over, λk1 (Ω1 )  λk1 (Ω2 ) when Ω1 ⊆ Ω2 . This property of λk1 (Ω) is very similar to that of the first

eigenvalue for the linear elliptic operator Δu = S 1 ( D 2 u ). Hence, λk1 (Ω) is called by Wang the first

eigenvalue of S k ( D 2 u ). In this paper, we will call λk1 (Ω) the principle eigenvalue of S k ( D 2 u ) since the first eigenvalue of Laplacian operator is also called principle eigenvalue. It is also pointed out by Wang that λk1 (Ω) can be characterized by the following variational formula



2 Ω S k ( D u )u dx − λ1 (Ω) = inf k+1 dx u ∈Φ0k (Ω) Ω |u | k

where Φ0k (Ω) is the set of all k-admissible functions that vanish on the boundary.

As in the case k = 1, it is of interest to minimize or maximize λk1 (Ω) under some constraints

C (Ω) = c . In this paper we mainly stress two cases in which C (Ω) is the volume V (Ω) of Ω , or the mean width M (Ω) of Ω . One important issue in these extremum problems is to determine the optimal shape of the critical domains which achieve the extremum values. To this end, it is asked to consider the first and second variation of λk1 (Ω). It is worth pointing out that the first variational formula will also be

important in solving the Minkowski problem for λk1 (Ω) (see [7,22]). The aims of this paper are twofold. One is to derive the first variational formula and some related overdetermined problems for λk1 (Ω), the other is to consider the validity of Serrin type result for the overdetermined problems we deduced above. The rest of this paper is arranged as follows: Section 2 includes some preliminaries. The first variational formula of the principle eigenvalue λk1 (Ω) is deduced in Section 3. Overdetermined problems and Serrin type symmetry result are presented in the final Section 4.

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2. Preliminaries This section collects some preliminaries needed in this paper. At first, we give some facts about k-Hessian operator. For k ∈ {1, 2, . . . , n} and u ∈ C 2 (Ω), the k-Hessian operator acting on u is defined by







S k D 2 u = σk (λ1 , . . . , λn ) =

λi 1 . . . λi k .

1i 1 <···
It is well known that S k ( D 2 u ) is the sum of all k × k principle minors of the Hessian matrix D 2 u for function u. Let ij



Sk D 2u =

∂ Sk ( D 2 u) . ∂ ui j

Then Euler identity for homogeneous functions implies that





Sk D 2u =

1 k

ij



Sk D 2u ui j .

ij

Since S k ( D 2 u ) is of divergence free, S k ( D 2 u ) can also be rewritten as the following divergence form





Sk D 2u =

1 k

ij



Sk D 2u ui j =

1 k

ij

 

Sk D 2 u ui j ,

where u j = ∂∂xu . Moreover, for any 1  k  n and any real number t, the following pointwise identity j ¯ u (x) = t } of function u (see [2,10,31]). holds on the level surface {x ∈ Ω: ij

S k u i u j = | Du |k+1 H k−1 .

(2.1)

Secondly, we collect some materials needed in the study of our overdetermined problem. Let K 0n denote the family of compact convex subsets in R n with a nonempty interior. For any K ∈ K 0n , the support function of K is defined by





h K ( v ) := sup x, v   x ∈ K



∀v ∈ Rn.

For K , L ∈ K 0n , the Minkowski addition K + L of K and L is defined by (see [8,33,34])

K + L = {x + y : x ∈ K ; y ∈ L }. According to [34], K + L can also be formally defined as the convex body K + L such that h K + L = h K + h L . For real number t ∈ R and convex body K ∈ K 0n , the scalar multiplication t K of K is defined by

t K = {tx: x ∈ K }. It is pointed out by Jerison in [22] that for any Ω, L ∈ K 0n , the algebraic sum

Ω + t L = {x + t y : x ∈ Ω and y ∈ L } gives a region whose boundary is the variation of ∂Ω by the distance th L (νΩ (x)) in the unit outer normal direction νΩ (x) at x ∈ ∂Ω .

Q. Dai, F. Shi / J. Differential Equations 255 (2013) 4136–4148

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For K ∈ K 0n , we denote by M ( K ) the mean width of K which is defined by

M(K ) =



2

ωn

h K ( v ) dH n−1

sn−1

with ωn being the area of unit sphere sn−1 in R n , and H n−1 the (n − 1)-dimensional Hausdorff measure. Obviously, M ( K ) is a linear functional with respect to the Minkowski addition. The following propositions are important in the proof of our Serrin type symmetry result.

¯ is a solution of the problem Proposition 2.1 (Pohožaev-type identity). (See [2,23].) If u ∈ C 2 (Ω) ∩ C (Ω)

⎧   ⎪ S k D 2 u = λ(−u )k , x ∈ Ω, ⎪ ⎪ ⎪ ⎪ x ∈ Ω, ⎪ u < 0, ⎨ u = 0, x ∈ ∂Ω,  ⎪ ⎪ ⎪ k + 1 ⎪ (−u ) dx = 1 ⎪ ⎪ ⎩

(2.2)

Ω

in a C 2 domain Ω ∈ R n , then

λk1 (Ω) =

1 2k







x · νΩ (x) | Du |k+1 H k−1

(2.3)

∂Ω

where νΩ (x) is the unit outer normal vector field on ∂Ω . Proposition 2.2. (See [16].) Let F : K 0n → R + be a Brunn–Minkowski functional, whose definition was given in [16], of degree α . If K ∈ K 0n is a stationary domain for the functional 1

Ψ (K ) =

in the sense that

d Ψ ((1 − t ) K dt

F α (K ) M(K )

+ t L )|t =0+ = 0 for any L ∈ K 0n , then K is a ball.

3. Variational formula of λk1 (Ω) Let I be the identity map, and V (x) = ( v 1 (x), . . . , v n (x)) be a smooth vector field on R n . Setting φt = I + t V and Ωt = φt (Ω), we denote by λk1 (t ) the principle eigenvalue λk1 (Ωt ) of k-Hessian operator on Ωt with Dirichlet boundary condition and by u (t , x) the corresponding eigenfunction with the normalization

 Ωt

  u (t , x)k+1 (x) dx = 1.

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Then, we have

⎧    k ⎪ S k D 2 u (t , x) = λk1 (t ) −u (t , x) , x ∈ Ωt , ⎪ ⎪ ⎪ ⎪ x ∈ Ωt , ⎪ ⎨ u (t , x) < 0, u (t , x) = 0, x ∈ ∂Ωt ,  ⎪ ⎪ ⎪ k + 1 ⎪ u (t , x) dx = 1. ⎪ ⎪ ⎩

(3.1)

Ωt

Moreover, by the regularity of u (t , x) and the domain Ωt , we have λk1 (t ) → λk1 (Ω) and u (t , x) → u (x) as t → 0. ∂ f (t ,x) ∂ f (0,x) For any differentiable function f (t , x) we write f = ∂ t |t =0+ = ∂ t , f = f (0, x), and denote by (λk1 (Ω)) the first variation of λk1 (Ω), that is

   d k  λ1 (Ω) = λ1 (Ωt ) 

k

dt

t = 0+

  = λ1 ( I + t V )Ω  . dt t = 0+ d

k



Then, by differentiating with respect to t at t = 0 on the both side of (3.1), we obtain

⎧     ⎪ S D 2 u =  λk1 (Ω) (−u )k + λk1 (Ω)k(−u )k−1 (−u ) , ⎪ ⎨ k (−u )k (−u ) dx = 0. ⎪ ⎪ ⎩

x ∈ Ω, (3.2)

Ω



Since Ω |u |k+1 dx = 1, multiplying both sides of the first equation in (3.2) by −u and then integrating on Ω , it follows that



 1  λ1 (Ω) =

 

k

k

 ij ∂ Sk ∂ ui ij (0, x) S k u j + (0, x)u i u j dx. ∂t ∂t

(3.3)

Ω

The main result of this section can be stated as Theorem 3.1 (The first variational formula). Let Ω be a smooth bounded convex domain in R n . And suppose that u ∈ C 3 (Ω) ∩ C (Ω). Then, we have

   λk1 (Ω) = −



( V · νΩ )| Du |k+1 H k−1 dH n−1 ,

(3.4)

∂Ω

where νΩ is the unit outer normal vector field on ∂Ω . To prove this, we prove the following lemma first. Lemma 3.2. Let Ω be a smooth bounded convex domain in R n with boundary of class C 2 . And suppose that u ∈ C 3 (Ω) ∩ C (Ω). Then we have

 Ω

ij   ∂ S k (0, x) u i u j dx = (k − 1) λk1 (Ω) . ∂t

(3.5)

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Proof. On one hand,





S k D 2 u =

∂ S k ∂ u i j (0, x) ij · = S k · u i j . ∂ ui j ∂t

(3.6)

On the other hand,





S k D 2 u =



1 k



ij

Sk ui j

ij

=

1 ∂ S k (0, x) 1 ij u i j + S k u i j . k ∂t k

(3.7)

It follows from (3.2), (3.6) and (3.7) that ij     ∂ S k (0, x) u i j = (k − 1)  λk1 (Ω) (−u )k + λk1 (Ω)k(−u )k−1 (−u ) . ∂t ij

∂ Sk ( D 2 u) ∂ xi

Combining (3.2), (3.8) with



ij

∂ S k (0, x) u i u j dx = ∂t

Ω

 

= 0 we show that ij

∂ S k (0, x) ui ∂t

 (−u ) dx j

Ω

  =

ij

∂ S k (0, x) ∂t

Ω

 =

 ui + j

 ij ∂ S k (0, x) u i j (−u ) dx ∂t

ij

∂ S k (0, x) u i j (−u ) dx ∂t

Ω



=

    (k − 1)  λk1 (Ω) (−u )k + λk1 (Ω)k(−u )k−1 (−u ) (−u ) dx

Ω

  = (k − 1) λk1 (Ω) .

2

Proof of Theorem 3.1. Since

 k

Ωt

=







S k D 2 u (t , y ) −u (t , y ) dy

λ1 (t ) = 1



ij

S k (t , y )u i (t , y )u j (t , y ) dy

k Ωt

=

1



k



 

 



Ω

we have

  1  λk1 (Ω) =



ij S k t , φt (x) u i t , φt (x) u j t , φt (x) det D φt (x) dx,



k

d dt

Ω



 

ij S k t , φt (x) 

t = 0+

u i (0, x)u j (0, x) dx

(3.8)

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d

+

dt

u i t , φt (x) 

Ω



 



d

+

dt

u j t , φt (x) 

S k (0, x)u j (0, x) dx ij

t = 0+

Ω



ij

t = 0+

S k (0, x)u i (0, x) dx

  det D φt (x) 

d

ij

S k (0, x)u i (0, x)u j (0, x)

+

dt

t = 0+

Ω

=

1 k

 dx

( I 1 + I 2 + I 3 + I 4 ).

By Green’s formula, we can compute I 1 as



d

I1 =

dt

 



ij S k t , φt (x) 

Ω

  = Ω

  = Ω

u i (0, x)u j (0, x) dx

t = 0+

 ij ij ∂ S (0, x) ∂ Sk (0, x) + k v h u i (0, x)u j (0, x) dx ∂t ∂ xh   ij ∂ Sk i j ∂(u i u j v h ) ij (0, x)u i (0, x)u j (0, x) − S k + S k u i u j v h νh dH n−1 ∂t ∂ xh ∂Ω



ij ∂ Sk

=

∂t Ω



+

 (0, x)u i (0, x)u j (0, x) −



ij

ij

S k u i u j divx V − 2 Ω

S k u ih u j v h Ω

( V · νΩ )| Du |k+1 H k−1 dH n−1 .

∂Ω

The identity (2.1) is used in the above last equality. I 2 , I 3 and I 4 can be simply computed as

 I2 = I3 =

d dt

Ω

 

=

  u i t , φt (x)  

ij

t = 0+

S k (0, x)u j (0, x) dx

 ∂ ui ij (0, x) + u ih v h S k (0, x)u j (0, x) dx, ∂t

Ω

and

 I4 =

ij S k (0, x)u i (0, x)u j (0, x)

  det D φt (x)  dt d

Ω



ij

S k (0, x)u i (0, x)u j (0, x) divx V dx.

= Ω

By making use of Lemma 3.2 and (3.3), we have

t = 0+

Q. Dai, F. Shi / J. Differential Equations 255 (2013) 4136–4148

4143

  1  λk1 (Ω) = ( I 1 + I 2 + I 3 + I 4 ) k

=

1



k

ij

∂S ∂ ui ij 2 (0, x) S k u j + k (0, x)u i u j + ∂t ∂t

Ω

 k +1

( V · νΩ )| Du |

H k−1 dH

n−1



∂Ω

  1 = 2 λk1 (Ω) −





ij

∂ Sk 1 (0, x)u i u j + ∂t k

k

Ω

( V · νΩ )| Du |k+1 H k−1 dH n−1

∂Ω





 1   1 = 2 λk1 (Ω) − (k − 1) λk1 (Ω) + k

k

( V · νΩ )| Du |k+1 H k−1 dH n−1 ,

∂Ω

which implies that

   λk1 (Ω) = −



( V · νΩ )| Du |k+1 H k−1 dH n−1 .

2

∂Ω

4. Overdetermined problems and symmetry result Let Ω ∈ K 0n . For any L ∈ K 0n , Jerison pointed out in [22] that the algebraic sum

Ω + t L = {x + t y | x ∈ Ω and y ∈ L } is a region whose boundary is the variation of ∂Ω by the distance th L (νΩ (x)) in the unit outer normal direction νΩ (x) at x ∈ ∂Ω . Therefore, we can choose a vector field V (x) on R n such that

V |∂Ω = h L





νΩ (x) νΩ (x), and Ω + t L = ( I + t V )(Ω).

From this observation and Theorem 3.1 we get

d dt

  λk1 (Ω + t L )



t = 0+

=−

hL





νΩ (x) | Du |k+1 H k−1 dH n−1 (x).

(4.1)

∂Ω

If we consider the extremum problem for λk1 (Ω) with constraint of fixed volume V (Ω) = c, then the Lagrange functional of this extremum problem is

  £(Ω) = λk1 (Ω) + μ V (Ω) − c . Hence, a smooth extremum point Ω of the above extremum problem must satisfy the following equality for any L ∈ K 0n .

d dt

  £(Ω + t L )

=

t = 0+

d dt

  λk1 (Ω + t L )

+μ

t = 0+

d dt

 

V (Ω + t L )

= 0.

(4.2)

t = 0+

Since

d dt

 

V (Ω + t L )

t = 0+

 = ∂Ω

hL





νΩ (x) dH n−1 .

(4.3)

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By (4.1), (4.2) and (4.3), we deduce that

 hL







νΩ (x) | Du |k+1 H k−1 − μ dH n−1 (x) = 0, ∀ L ∈ K 0n .

(4.4)

∂Ω

By the arbitrariness of L, we have

| Du |k+1 H k−1 = c on ∂Ω,

(4.5)

for some constant c. This leads to the following overdetermined problem

⎧   ⎪ S k D 2 u = λk1 (Ω)(−u )k , ⎪ ⎪ ⎨ u < 0, ⎪ u = 0, ⎪ ⎪ ⎩ | Du |k+1 H k−1 = c ,

x ∈ Ω, x ∈ Ω,

(4.6)

x ∈ ∂Ω, x ∈ ∂Ω.

If we consider the extremum problem for λk1 (Ω) with constraint of fixed mean width M (Ω) = c, that is

M (Ω) =



2 wn

hΩ dy = c ,

(4.7)

S n−1

then, the corresponding Lagrange functional is

  £(Ω) = λk1 (Ω) + μ M (Ω) − c . A smooth extremum point Ω of this extremum problem must satisfy the following condition

d dt

  £(Ω + t L )

=

t = 0+

d dt

  λk1 (Ω + t L )

+μ

t = 0+

d dt

 

M (Ω + t L )

= 0,

t = 0+

∀ L ∈ K 0n .

(4.8)

Since M (Ω) is linear under Minkowski addition, it follows that

d dt

  M (Ω + t L )

t = 0+

=

2 wn

 h L ( y ) dH

n−1

( y) =

S n−1

2

 hL

wn





νΩ (x) G Ω (x) dH n−1 (x),

(4.9)

∂Ω

with G Ω (x) being the Gaussian curvature of ∂Ω at x. By (4.1), (4.8) and (4.9), we deduce that



   2μ k +1 h L νΩ (x) | Du | H k−1 − G Ω dH n−1 (x) = 0, 

wn

∀ L ∈ K 0n .

(4.10)

∂Ω

Therefore, by the arbitrariness of L, we have

| Du |k+1 H k−1 −

2μ wn

GΩ = 0

on ∂Ω.

(4.11)

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This leads to the following overdetermined problem

⎧   ⎪ S k D 2 u = λk1 (Ω)(−u )k , ⎪ ⎪ ⎨ u < 0, ⎪ u = 0, ⎪ ⎪ ⎩ | Du |k+1 H k−1 = cG Ω ,

x ∈ Ω, x ∈ Ω, x ∈ ∂Ω,

(4.12)

x ∈ ∂Ω.

Remark 4.1. We can obtain many overdetermined problems for λk1 (Ω) other than (4.6) and (4.12) by considering extremum problems of λk1 (Ω) under other constraints such as fixed quermassintegral, or fixed perimeter. The study of overdetermined problems has a long history. Assume that Ω is a C 2 domain, J. Serrin [30] considered in 1971 the following overdetermined problem

⎧ x ∈ Ω, ⎨ u = 1, u = 0, x ∈ ∂Ω, ⎩ | Du | = constant, x ∈ ∂Ω,

(4.13)

and proved that the existence of a solution to problem (4.13) implies that the domain Ω must be a ball. This result was also extended by J. Serrin [30] to the case when | Du | is a non-decreasing regular function of the mean curvature of ∂Ω . The basic tool used by J. Serrin is the so-called moving planes method combining with a clever modification of the classical Hopf boundary lemma. From then on, there are many different methods to solve this problem (see [2,9,12,35]). Moreover, J. Prajapat and A.L. Vogel [27,24] generalized the Serrin’s result to a less regular domain. Extensions of Serrin’s result have also been given to other domains such as exterior domains (see [19,26,28,29]) and annular domains (see [1,17]). The generalization of Serrin’s result to different operators more general than Laplacian and possibly degenerated has been also studied in [4–6,11,13–15,20,18]. Recently, I. Fragalà [16] generalized the Serrin’s result to the Dirichlet Laplacian with an additional Neumann boundary condition involving Gauss curvature of the boundary. More information of Serrin type result with Neumann boundary condition involving boundary curvature can be found in [17,21]. However, much less has been done for the overdetermined problem of k-Hessian operator when 1 < k  n. So far, to our best knowledge, the only result in this aspect is given by B. Brandolini, C. Nitsch, P. Salani, C. Trombetti in [2]. In that paper they considered the following problem

  ⎧   2u = n , ⎪ ⎪ S D k ⎪ ⎪ k ⎨ u < 0, ⎪ ⎪ ⎪ ⎪ ⎩ u = 0, | Du | = 1,

x ∈ Ω, x ∈ Ω,

(4.14)

x ∈ ∂Ω, x ∈ ∂Ω

and proved that the existence of a solution to problem (4.14) implies that the domain Ω must be a ball. The main feature of the method used in [2] is that it is not explicitly depending on the maximum principle. However, this method is not applicable in the study of overdetermined problem (4.6) and (4.12). It is also worth pointing out that neither the moving plane method is applicable in the study of overdetermined problem (4.6) and (4.12) when 1 < k < n, due to the complexity of the Neumann boundary condition. In the sequel, we consider the validity of Serrin type result to overdetermined problem (4.6) and (4.12). At first, concerning overdetermined problem (4.6), we recall that B. Brandolini, C. Nitsch, C. Trombetti proved in [3] that

  λn1 (Ω)  λn1 Ω ∗

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for any ellipsoid Ω ∗ having the same volume as Ω , and the equality holds if and only if Ω is an ellipsoid. This result implies that Ω ∗ is a critical domain of the following extremum problem

  sup λn1 (Ω): Ω is convex and V (Ω) = c . Hence, Ω ∗ is a solution of overdetermined problem (4.6) when k = n. Summing-up, we have the following result. Theorem 4.2. If k = n, then overdetermined problem (4.6) has at least two type of solutions. One is a ball, and the other is an ellipsoid. Hence, Serrin type result is invalid for overdetermined problem (4.6) in the case k = n. Remark 4.3. The philosophy of Theorem 4.2 may root in the fact that the Monge–Ampère operator is invariant under the volume preserving affine transformation. This invariant property is not borne by k-Hessian operator when 1 < k < n. Hence, we would like to believe that Serrin type result is still valid for overdetermined problem (4.6) when 1 < k < n. Secondly, we use Proposition 2.1 and Proposition 2.2 to prove a Serrin type result for overdetermined problem (4.12). This is an adoption of the method used by I. Fragalà in [16]. The result we will prove can be stated as the following Theorem 4.4. Let Ω be a smooth bounded strictly convex domain in R n . If k = 2 and n = 3, or k = n, then the existence of a solution to the overdetermined problem (4.12) implies that Ω is a ball. Proof. Assume that u is a solution of problem (4.12). Multiplying both sides of the equation | Du |k+1 H k−1 = cG Ω by hΩ (νΩ (x)) and integrating over ∂Ω , we obtain

 k +1

| Du |

H k−1 dH

n−1

∂Ω

 =

cG Ω dH n−1 .

(4.15)

∂Ω

By (2.3), (4.7) and (4.15) we have

4k λk1 (Ω) . w n M (Ω)

c=

(4.16)

Therefore,

| Du |k+1 H k−1 = cG Ω =

4k λk1 (Ω) GΩ w n M (Ω)

on ∂Ω.

(4.17)

Multiplying both sides of (4.17) by (h L − hΩ )(νΩ (x)), and integrating on ∂Ω , we get

 k +1

(h L − hΩ )(νΩ )| Du |

H k−1 dH

n−1

4k λk1 (Ω) = w n M (Ω)

∂Ω



(h L − hΩ )(νΩ )G Ω dH n−1 .

∂Ω

From Theorem 3.1 we have

d dt

  λk1 (Ω + t L )

t = 0+

 =− ∂Ω

h L (νΩ )| Du |k+1 H k−1 dH n−1 .

(4.18)

Q. Dai, F. Shi / J. Differential Equations 255 (2013) 4136–4148

4147

Thus, it follows from the (−2k)-homogeneity of λk1 and Proposition 2.1 that

d dt

   λ1 (1 − t )Ω + t L 



k

t = 0+

=−

(h L − hΩ )(νΩ )| Du |k+1 H k−1 dH n−1 .

∂Ω

By a simple computation, we obtain

  − 1  λk1 (Ω)−(2k+1)/2k 2 k λ1 (1 − t )Ω + t L = (h L − hΩ )(νΩ )| Du |k+1 H k−1 dH n−1 . (4.19)  + dt 2k t =0 d

k



∂Ω

Furthermore, we can compute

d dt



 

M (1 − t )Ω + t L 

=

t = 0+

2



wn

(h L − hΩ )(νΩ )G Ω dH n−1 .

(4.20)

∂Ω

Combining (4.18), (4.19) with (4.20), we have 1



d λk1 ((1 − t )Ω + t L )− 2k   = 0. dt M ((1 − t )Ω + t L ) t =0+ Hence, Ω is a stationary domain for the functional 1

λk1 (Ω)− 2k . M (Ω) Since λk1 (Ω) is a Brunn–Minkowski functional of degree −4 in the case k = 2 and n = 3 (see [25]), and is a Brunn–Minkowski functional of degree −2n in the case k = n (see [32]), Proposition 2.2 implies that Ω is a ball. 2 References [1] G. Alessandrini, A symmetry theorem for condensers, Math. Methods Appl. Sci. 15 (1992) 315–320. [2] B. Brandolini, C. Nitsch, P. Salani, C. Trombetti, Serrin-type overdetermined problems: an alternative proof, Arch. Ration. Mech. Anal. 190 (2008) 267–280. [3] B. Brandolini, C. Nitsch, C. Trombetti, New isoperimetric estimates for solutions to Monge–Ampère equations, Ann. Inst. H. Poincaré Anal. Non Linéaire 26 (2009) 1265–1275. [4] F. Brock, A. Henrot, A symmetry result for an overdetermined elliptic problem using continuous rearrangement and domain derivative, Rend. Circ. Mat. Palermo 51 (2002) 375–390. [5] G. Buttazzo, B. Kawohl, Overdetermined boundary value problems for the ∞-Laplacian, Int. Math. Res. Not. IMRN (2011) 237–247. [6] A. Cianchi, P. Salani, Overdetermined anisotropic elliptic problems, Math. Ann. 345 (2009) 859–881. [7] A. Colesanti, M. Fimiani, The Minkowski problem for the torsional rigidity, Indiana Univ. Math. J. 59 (2010) 1013–1039. [8] A. Colesanti, P. Cuoghi, P. Salani, Brunn–Minkowski inequalities for two functionals involving the p-Laplace operator, Appl. Anal. 85 (2006) 45–66. [9] M. Choulli, A. Henrot, Use of the domain derivative to prove symmetry results in partial differential equations, Math. Nachr. 192 (1998) 91–103. [10] R.C. Reilly, On the Hessian of a function and the curvatures of its graph, Michigan Math. J. 20 (1973) 373–383. [11] L. Damascelli, F. Pacella, Monotonicity and symmetry results for p-Laplace equations and applications, Adv. Differential Equations 5 (2000) 1179–1200. [12] L.E. Payne, P.W. Schaefer, Duality theorems in some overdetermined boundary value problems, Math. Methods Appl. Sci. 11 (1989) 805–819. [13] A. Farina, B. Kawohl, Remarks on an overdetermined boundary value problem, Calc. Var. Partial Differential Equations 89 (2008) 351–357.

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