A partially overdetermined problem in domains with partial umbilical boundary in space forms

Jinyu Guo; Chao Xia

doi:10.1515/acv-2021-0090

Publicly Available Published by De Gruyter May 31, 2022

A partially overdetermined problem in domains with partial umbilical boundary in space forms

Jinyu Guo and Chao Xia

From the journal Advances in Calculus of Variations

https://doi.org/10.1515/acv-2021-0090

Abstract

In the first part of this paper, we consider a partially overdetermined mixed boundary value problem in space forms and generalize the main result in [11] to the case of general domains with partial umbilical boundary in space forms. Precisely, we prove that a partially overdetermined problem in a domain with partial umbilical boundary admits a solution if and only if the rest part of the boundary is also part of an umbilical hypersurface. In the second part of this paper, we prove a Heintze–Karcher–Ros-type inequality for embedded hypersurfaces with free boundary lying on a horosphere or an equidistant hypersurface in the hyperbolic space. As an application, we show an Alexandrov-type theorem for constant mean curvature hypersurfaces with free boundary in these settings.

Keywords: Overdetermined problem; space forms; Alexandrov theorem; free boundary hypersurfaces

MSC 2010: 35N25; 35J15; 53C40

1 Introduction

In a celebrated paper [26], Serrin initiated the study of the following overdetermined boundary value problem (BVP):

(1.1) { Δ ⁢ u = 1 in ⁢ Ω , u = 0 on ⁢ ∂ ⁡ Ω , ∂ ν ⁡ u = c on ⁢ ∂ ⁡ Ω ,

where Ω is an open, connected, bounded domain in ℝ n with smooth boundary ∂ ⁡ Ω , c ∈ ℝ is a constant and ν is the unit outward normal to ∂ ⁡ Ω . Serrin proved that if (1.1) admits a solution, then Ω must be a ball and the solution u is radially symmetric. Serrin’s proof is based on the moving plane method or Alexandrov reflection method, which has been invented by Alexandrov in order to prove the famous nowadays so-called Alexandrov soap bubble theorem [2]: any closed, embedded hypersurface of constant mean curvature (CMC) must be a round sphere.

In space forms^[1], Serrin’s symmetry result was proved in [13, 17] by the moving plane method. A special overdetermined problem in space forms has been considered by Qiu and Xia [22] by using Weinberger’s approach; see also [7]. We also mention that a corresponding result in the closed sphere case is no longer true; see, e.g., [8].

Serrin’s overdetermined BVP is closely related with closed CMC hypersurfaces. Analogously to closed CMC hypersurfaces, there are several rigidity results for free boundary CMC hypersurfaces in the Euclidean unit ball 𝔹 n . Here we use “free boundary” to mean a hypersurface which intersects 𝕊 n - 1 orthogonally. We refer to a recent survey paper [27] for details. In particular, the Alexandrov-type theorem says that a free boundary CMC hypersurface in a half ball must be a free boundary spherical cap. Motivated by this, we have proposed in [11] the study of a partially overdetermined BVP in a half ball. Precisely, let 𝔹 + n = { x ∈ 𝔹 n : x n > 0 } be the half Euclidean unit ball and let Ω ⊂ 𝔹 + n be an open bounded, connected domain with boundary ∂ ⁡ Ω = Σ ¯ ∪ T , where Σ ⊂ 𝔹 + n is a smooth open hypersurface and T ⊂ 𝕊 n - 1 meets Σ at a common ( n - 2 ) -dimensional submanifold Γ ⊂ 𝕊 n - 1 . We have considered the following partially overdetermined BVP in Ω:

(1.2) { Δ ⁢ u = 1 in ⁢ Ω ⊂ 𝔹 + n , u = 0 on ⁢ Σ ¯ , ∂ ν ⁡ u = c on ⁢ Σ ¯ , ∂ N ¯ ⁡ u = u on ⁢ T ,

where ν and N ¯ ⁢ ( x ) = x are the outward unit normal of Σ and T ⊂ 𝕊 n - 1 , respectively. We have proved the following result.

Theorem 1.1 ([11]).

Let Ω be as above. Assume (1.2) admits a weak solution

u ∈ W 0 1 , 2 ( Ω , Σ ) = { u ∈ W 1 , 2 ( Ω ) : u | Σ ¯ = 0 } ,

i.e.,

∫ Ω ( 〈 ∇ ⁡ u , ∇ ⁡ v 〉 + v ) ⁢ 𝑑 x - ∫ T u ⁢ v ⁢ 𝑑 A = 0 for all ⁢ v ∈ W 0 1 , 2 ⁢ ( Ω , Σ ) .

Assume further that u ∈ W 1 , ∞ ⁢ ( Ω ) ∩ W 2 , 2 ⁢ ( Ω ) . Then Ω must be of the form

Ω n ⁢ c ⁢ ( a ) := { x ∈ 𝔹 + n : | x - a ⁢ 1 + ( n ⁢ c ) 2 | 2 < ( n ⁢ c ) 2 } , a ∈ 𝕊 n - 1 ,

for some a ∈ S n - 1 , and

u ⁢ ( x ) = u a , n ⁢ c ⁢ ( x ) := 1 2 ⁢ n ⁢ ( | x - a ⁢ 1 + ( n ⁢ c ) 2 | 2 - ( n ⁢ c ) 2 ) .

We remark that ∂ ⁡ Ω n ⁢ c ⁢ ( a ) ∩ 𝔹 + n is a free boundary spherical cap. Thus, Theorem 1.1 gives a characterization of free boundary spherical caps by an overdetermined BVP, which can be regarded as Serrin’s analog for the setting of free boundary CMC hypersurfaces in a ball.

In this paper, we will generalize Theorem 1.1 into the setting of domains with partial umbilical boundary in space forms.

Let ( 𝕄 n ⁢ ( K ) , g ¯ ) be a complete simply-connected Riemann manifold with constant sectional curvature K. Up to homotheties, we may assume K = 0 , 1 , - 1 ; the case K = 0 corresponds to the case of the Euclidean space ℝ n , K = 1 is the unit sphere 𝕊 n with the round metric, and K = - 1 is the hyperbolic space ℍ n .

We recall some basic facts about umbilical hypersurfaces in 𝕄 n ⁢ ( K ) . It is well-known that an umbilical hypersurface in space forms has constant principal curvature κ ∈ ℝ . By a choice of orientation (or normal vector field N ¯ ), we may assume κ ∈ [ 0 , ∞ ) . It is also a well-known fact that in ℝ n and 𝕊 n geodesic spheres ( κ > 0 ) and totally geodesic hyperplanes ( κ = 0 ) are all complete umbilical hypersurfaces, while in ℍ n the family of all complete umbilical hypersurfaces includes geodesic spheres ( κ > 1 ) , totally geodesic hyperplanes ( κ = 0 ) , horospheres ( κ = 1 ) and equidistant hypersurfaces ( 0 < κ < 1 ) (see, e.g., [16]). We remark that unlike geodesic spheres, the horospheres and the equidistant hypersurfaces are non-compact umbilical hypersurfaces.

We use S K , κ to denote an umbilical hypersurface in 𝕄 n ⁢ ( K ) with principal curvature κ. This S K , κ divides 𝕄 n ⁢ ( K ) into two connected components. We use B K , κ int to denote the one component whose outward normal is given by the orientation N ¯ . The other one we denote by B K , κ ext . Let Ω ⊂ B K , κ int be a bounded, connected open domain whose boundary is ∂ ⁡ Ω = Σ ¯ ∪ T , where Σ ⊂ B K , κ int is a smooth open hypersurface and T ⊂ S K , κ meets Σ at a common ( n - 2 ) -dimensional submanifold Γ. We refer to Figures 1–3 in Section 2 for the corresponding domains for K = - 1 (hyperbolic space) and different values κ. Since the Euclidean case K = 0 has already been handled in [11], and the case κ = 0 in 𝕄 n ⁢ ( K ) has been considered in [6] (as a special case of a flat cone), in this paper we consider the hyperbolic case K = - 1 with κ > 0 and the spherical case K = 1 with κ > 0 .

We consider the following mixed BVP in Ω ⊂ B K , κ int :

(1.3) { Δ ¯ ⁢ u + n ⁢ K ⁢ u = 1 in ⁢ Ω , u = 0 on ⁢ Σ ¯ , ∂ N ¯ ⁡ u = κ ⁢ u on ⁢ T .

As we described above, N ¯ is the unit outward normal of B K , κ int .

If κ > 0 , for a general domain, there might not exist a solution to (1.3). Also, for a general domain, the maximum principle fails to hold. These are due to the fact that the Robin boundary condition on T has an unfavorable sign. In our case, we can show that there always exists a unique non-positive solution u ∈ C ∞ ⁢ ( Ω ¯ ∖ Γ ) ∩ C α ⁢ ( Ω ¯ ) to (1.3) for some α ∈ ( 0 , 1 ) ; see Proposition 3.3 below.

Remark 1.2.

For the other case Ω ⊆ B K , κ ext , - N ¯ plays the role of the unit outward normal of B K , κ ext along T. Hence, the Robin boundary condition becomes ∂ ( - N ¯ ) ⁡ u = - κ ⁢ u on T, which has a good sign, i.e., - κ < 0 , according to the classical elliptic PDE theory. The existence of the weak solution (1.3) follows directly from the Fredholm alternative theorem (see, for example, [9]).

In this paper, we study the following partially overdetermined BVP in Ω ⊂ B K , κ int (resp. B K , κ ext ):

(1.4) { Δ ¯ ⁢ u + n ⁢ K ⁢ u = 1 in ⁢ Ω , u = 0 on ⁢ Σ ¯ , ∂ ν ⁡ u = c on ⁢ Σ ¯ , ∂ N ¯ ⁡ u = κ ⁢ u on ⁢ T ,

where ν is the outward unit normal of Σ. Our main result is the following theorem.

Theorem 1.3.

Let Ω ⊂ B K , κ int (resp. B K , κ ext ). Assume the partially overdetermined BVP (1.4) admits a weak solution u ∈ W 0 1 , 2 ⁢ ( Ω , Σ ) , i.e.,

∫ Ω ( g ¯ ⁢ ( ∇ ¯ ⁢ u , ∇ ¯ ⁢ v ) + v - n ⁢ K ⁢ u ⁢ v ) ⁢ 𝑑 x - κ ⁢ ∫ T u ⁢ v ⁢ 𝑑 A = 0 for all ⁢ v ∈ W 0 1 , 2 ⁢ ( Ω , Σ ) ,

together with an additional boundary condition ∂ ν ⁡ u = c on Σ. Assume further that

(1.5) u ∈ W 1 , ∞ ⁢ ( Ω ) ∩ W 2 , 2 ⁢ ( Ω ) .

If S K , κ is a horosphere ( K = - 1 and κ = 1 ) or an equidistant hypersurface ( K = - 1 and 0 < κ < 1 ) in ℍ n , then Σ must be part of an umbilical hypersurface with principal curvature 1 / ( n ⁢ c ) which intersects S K , κ orthogonally.
If S K , κ is a geodesic sphere in ℍ n or 𝕊 + n , that is, K = - 1 and κ > 1 , or K = 1 and κ > 0 , then the same conclusion in (i) holds provided Ω ⊂ B K , κ int , + (resp. B K , κ ext , + ).

Here B K , κ int , + means a half ball; see (2.3) below.

Remark 1.4.

The above umbilical hypersurface could be a horosphere, an equidistant hypersurface or a geodesic ball. We will give an example in Section A that Σ and T are parts of two orthogonal horospheres, for which the partially overdetermined BVP (1.4) still admits a solution.

We remark that we do not assume Σ meets S K , κ orthogonally a priori. Thus, it is impossible to use the Alexandrov reflection method as Ros and Souam did [25]. On the other hand, because of the lack of regularity of u on Γ, it is difficult to use the maximum principle as Weinberger did [29]. Higher-order regularity up to the interface Γ = Σ ¯ ∩ T ¯ is a subtle issue for mixed boundary value problems. A regularity result by Lieberman [15] shows that a weak solution u to (1.4) belongs to C ∞ ⁢ ( Ω ¯ ∖ Γ ) ∩ C α ⁢ ( Ω ¯ ) for some α ∈ ( 0 , 1 ) . The regularity assumption (1.5) is for technical reasons, that is, we will use an integration method which requires (1.5) to perform integration by parts.

Similar to the Euclidean case [11], we use a purely integral method to prove our theorem. The integration makes use of a non-negative weight function V, which is given by a multiplier of the divergence of a conformal Killing vector field X. In the case of geodesic spheres in ℍ n or 𝕊 + n , we use X defined by (2.4), which was found in [28]. In the case of horospheres or equidistant hypersurfaces in ℍ n , we use X defined by (2.5). The common feature of such conformal Killing vector fields is that they are parallel to the support hypersurfaces.

By using X, we get a Pohozaev-type identity with weight V; see Proposition 4.3. Then with the usual P-function P = | ∇ ¯ ⁢ u | 2 - 2 n ⁢ u + K ⁢ u 2 , we can show the identity

∫ Ω V ⁢ u ⁢ | ( ∇ ¯ 2 ⁢ u + K ⁢ u ⁢ g ) - 1 n ⁢ ( Δ ¯ ⁢ u + n ⁢ K ⁢ u ) ⁢ g ¯ | 2 ⁢ 𝑑 x = 0 .

Theorem 1.3 follows since the P-function is subharmonic.

In the second part of this paper, we will use the solution to (1.3) to study the Alexandrov-type theorem for embedded free boundary CMC hypersurfaces in ℍ n supported on a horosphere or an equidistant hypersurface.

It is nowadays a routine argument to combine a Minkowski-type formula and a sharp Heintze–Karcher–Ros-type inequality to prove the Alexandrov-type theorem; see, e.g., [14, 18, 21, 24, 28]. In the spirit of Wang and Xia [28], we shall first use the solution to (1.3) to prove the following Heintze–Karcher–Ros-type inequality for free boundary hypersurfaces in ℍ n supported on a horosphere or an equidistant hypersurface. The case of geodesic hyperplanes in space forms has been proved by Pyo; see [20, Theorem 4 and Theorem 10]. The case of geodesic spheres in space forms has been shown by Wang and Xia; see [28, Theorem 5.2 and Theorem 5.4].

Theorem 1.5.

Let H n be given by the half space model { x ∈ R + n : x n > 0 } with hyperbolic metric g ¯ = ( 1 / x n 2 ) ⁢ δ . Let Σ ⊂ H n be an embedded smooth hypersurface whose boundary ∂ ⁡ Σ lies on a support hypersurface S (that is, a horosphere or an equidistant hypersurface). Assume Σ intersects S orthogonally. Assume Σ has positive normalized mean curvature H 1 and let Ω be the enclosed domain by Σ and S. Then

(1.6) ∫ Σ 1 x n ⋅ H 1 ⁢ 𝑑 A ≥ ∫ Ω n x n ⁢ 𝑑 x .

Moreover, the above equality (1.6) holds if and only if Σ is part of an umbilical hypersurface which meets S orthogonally.

Using the above Heintze–Karcher–Ros-type inequality, we are able to reprove the Alexandrov-type theorem for free boundary constant mean curvature or constant higher-order mean curvature hypersurfaces in ℍ n supported by horospheres and equidistant hypersurfaces; see Theorem 5.4.

We remark that the Alexandrov-type theorem in this setting has been shown by López [16], using the classical Alexandrov reflection method; see also [30]. López was also able in [16] to handle the general capillary hypersurfaces, that is, constant mean curvature hypersurfaces with constant contact angle.

The rest of this paper is organized as follows. In Section 2, we review the conformal Killing vector fields X we shall use in each case and their properties. In Section 3, we study two kinds of eigenvalue problems in Ω in space forms and use them to prove the existence and uniqueness of the mixed BVP (1.3). In Section 4, we prove a weighted Pohozaev inequality and then Theorem 1.3. In Section 5, We prove Theorem 1.5 and the Alexandrov-type Theorem 5.4.

2 Conformal Killing vector fields in space forms

We first introduce the notations. Let us recall that S K , κ is an umbilical hypersurface in 𝕄 n ⁢ ( K ) with principal curvature κ ∈ [ 0 , ∞ ) .

2.1 Hyperbolic space ℍ n

Definition 2.1 ([16]).

In the hyperbolic space ℍ n , we call a support hypersurface a complete non-compact umbilical hypersurface, which means geodesic hyperplanes, horospheres and equidistant hypersurfaces.

A horosphere is a “sphere” whose center lies at ∂ ∞ ⁡ ℍ n . In the upper half-space model

(2.1) ℍ n = { x = ( x 1 , x 2 , … , x n ) ∈ ℝ + n : x n > 0 } , g ¯ = 1 x n 2 ⁢ δ ,

a horosphere, up to a hyperbolic isometry, is given by the horizontal plane

L ⁢ ( 1 ) = { x ∈ ℝ + n : x n = 1 }

By choosing N ¯ = - E n = ( 0 , … , 0 , - 1 ) , the principal curvature of a horosphere is given by κ = 1 . We remark that a horosphere is isometric to a Euclidean plane.

An equidistant hypersurface is a connected component of the set of points equidistant from a given hyperplane. In the half-space model, an equidistant hypersurface is given by a sloping Euclidean hyperplane Π which meets ∂ ∞ ⁡ ℍ n with angle θ through a point E n = ( 0 , 0 , … , 1 ) ∈ ℝ + n , say

Π = { x ∈ ℝ + n : x 1 ⁢ tan ⁡ θ + x n = 1 } ,

and by choosing N ¯ as the same direction as ( - tan ⁡ θ , 0 , … , 0 , - 1 ) , its principal curvature is κ = cos ⁡ θ ∈ ( 0 , 1 ) .

$Figure 1 S K , κ {S_{K,\kappa}} is a geodesic sphere with principal curvature κ = coth ⁡ R > 1 {\kappa=\coth R>1} , and the shaded area is B K , κ int , + {B_{K,\kappa}^{\mathrm{int},+}} .$

Figure 1

S K , κ is a geodesic sphere with principal curvature κ = coth ⁡ R > 1 , and the shaded area is B K , κ int , + .

Next, we clarify the unified notation we will use in each case.

Case 1. If S K , κ is a geodesic sphere of radius R, then κ = coth ⁡ R ∈ ( 1 , ∞ ) and let B K , κ int denote the geodesic ball enclosed by S K , κ . By using the Poincaré ball model

(2.2) 𝔹 n = { x ∈ ℝ n : | x | < 1 } , g ¯ = 4 ( 1 - | x | 2 ) 2 ⁢ δ ,

we have up to an hyperbolic isometry,

B K , κ int = { x ∈ 𝔹 n : | x | ≤ R ℝ := 1 - arccosh ⁡ R 1 + arccosh ⁡ R } .

Moreover, we let

(2.3) B K , κ int , + = { x ∈ B K , κ int : x n > 0 }

be a geodesic half ball; see Figure 1.
Case 2. If S K , κ is a support hypersurface, then κ ∈ [ 0 , 1 ] . By using the upper half-space model (2.1), we have, up to an hyperbolic isometry,

B K , κ int = { { x ∈ ℝ + n : x n > 1 } if ⁢ κ = 1 , { x ∈ ℝ + n : x 1 ⁢ tan ⁡ θ + x n > 1 } if ⁢ κ = cos ⁡ θ ∈ ( 0 , 1 ) ;

see Figures 2 and 3.

$Figure 2 S K , κ {S_{K,\kappa}} is a horosphere L ⁢ ( 1 ) {L(1)} with principal curvature κ = 1 {\kappa=1} , and the shaded area is B K , κ int {B_{K,\kappa}^{\mathrm{int}}} .$

Figure 2

S K , κ is a horosphere L ⁢ ( 1 ) with principal curvature κ = 1 , and the shaded area is B K , κ int .

$Figure 3 S K , κ {S_{K,\kappa}} is an equidistant hypersurface Π with principal curvature κ = cos ⁡ θ < 1 {\kappa=\cos\theta<1} , and the shaded area is B K , κ int {B_{K,\kappa}^{\mathrm{int}}} .$

Figure 3

S K , κ is an equidistant hypersurface Π with principal curvature κ = cos ⁡ θ < 1 , and the shaded area is B K , κ int .

Next, we introduce the conformal Killing vector field X in ℍ n and the weight V we will use later.

Case 1: κ > 1 . In this case, as before we use the Poincaré ball model (2.2). Set

(2.4) X := 2 1 - R ℝ 2 ⁢ [ x n ⁢ x - 1 2 ⁢ ( | x | 2 + R ℝ 2 ) ⁢ E n ] , V = 2 ⁢ x n 1 - | x | 2 .
Case 2: 0 < κ ≤ 1 . In this case, as before we use the upper half-space model (2.1). Set

(2.5) X := x - E n , V = 1 x n .

Proposition 2.2.

X is a conformal Killing vector field with L X ⁢ g ¯ = V ⁢ g ¯ , namely

(2.6) 1 2 ⁢ ( ∇ ¯ i ⁢ X j + ∇ ¯ j ⁢ X i ) = V ⁢ g ¯ i ⁢ j .
X ∣ S K , κ is a tangential vector field on S K , κ , i.e.,

(2.7) g ¯ ⁢ ( X , N ¯ ) = 0 on ⁢ S K , κ .

Proof.

We differentiate two cases.

Case 1: κ > 1 . See [28, Proposition 4.1].

Case 2: 0 < κ ≤ 1 . We choose an orthonormal basis { e i } i = 1 n in the upper half-space model:

e i = x n ⁢ E i , i = 1 , … , n .

where { E i } i = 1 n is the Euclidean orthonormal basis in ℝ n . Then

1 2 ⁢ ( ∇ ¯ i ⁢ X j + ∇ ¯ j ⁢ X i ) = 1 2 ⁢ ( D i ⁢ X j + D j ⁢ X i ) + X ⁢ ( - ln ⁡ x n ) ⁢ g ¯ i ⁢ j

= g ¯ i ⁢ j + ( 1 x n - 1 ) ⁢ g ¯ i ⁢ j

= 1 x n ⁢ g ¯ i ⁢ j ,

where D is the Levi-Civita connection in ℝ n . We use the relationship of ∇ ¯ and D, that is,

(2.8) ∇ ¯ Y ⁢ Z = D Y ⁢ Z + Y ⁢ ( - ln ⁡ x n ) ⁢ Z + Z ⁢ ( - ln ⁡ x n ) ⁢ Y - 〈 Y , Z 〉 ⁢ D ⁢ ( - ln ⁡ x n ) .

Here 〈 ⋅ , ⋅ 〉 is the Euclidean inner product.

On the other hand,

g ¯ ⁢ ( X , N ¯ ) = 1 x n ⁢ 〈 x - E n , N δ 〉 = 0 on ⁢ S K , κ ,

where N δ is the outward normal to a support hypersurface with respect to the Euclidean metric δ. ∎

Proposition 2.3.

V satisfies the following properties:

(2.9) ∇ ¯ 2 ⁢ V = - K ⁢ V ⁢ g ¯ ,

(2.10) ∂ N ¯ ⁡ V = κ ⁢ V on ⁢ S K , κ ,

where N ¯ is the unit outward normal of B K , κ int .

Proof.

We differentiate two cases.

Case 1: κ > 1 . See [28, Proposition 4.2].

Case 2: 0 < κ ≤ 1 . For (2.9), we take normal coordinates { e i } i = 1 n at p such that ∇ ¯ e i ⁢ e j ∣ p = 0 . Then

∇ ¯ e i ⁢ ∇ ¯ e j ⁢ V = ∇ ¯ e i ⁢ ∇ ¯ e j ⁢ ( 1 x n )

= ∇ ¯ e i ⁢ ( ∇ ¯ e j ⁢ ( 1 x n ) )

= - e i ⁢ ( g ¯ ⁢ ( E n , e j ) )

= - g ¯ ⁢ ( ∇ ¯ e i ⁢ E n , e j )

= E n ⁢ ( ln ⁡ x n ) ⁢ g ¯ i ⁢ j

= 1 x n ⁢ g ¯ i ⁢ j ,

where we use formula (2.8).

For (2.10), we compute

∂ N ¯ ⁡ V = ∂ N ¯ ⁡ ( 1 x n ) = - 1 x n 2 ⁢ 〈 E n , N ¯ 〉 = - 1 x n 2 ⁢ 〈 E n , x n ⁢ N δ 〉 = 1 x n ⁢ cos ⁡ θ .

Here we use the fact that 〈 E n , N δ 〉 = - cos ⁡ θ on the support hypersurface S K , κ . ∎

2.2 Spherical space 𝕊 n

In this subsection, we sketch the necessary modifications in the case that the ambient space is the spherical space 𝕊 n . We use the model

( ℝ n , g ¯ 𝕊 = 4 ( 1 + | x | 2 ) 2 ⁢ δ )

to represent 𝕊 n ∖ { 𝒮 } , the unit sphere without the south pole. Therefore, if S K , κ is a geodesic sphere of radius R, then in the above model

S K , κ = { x ∈ ℝ n : | x | = R ℝ := 1 - cos ⁡ R 1 + cos ⁡ R } .

Then κ = cot ⁡ R > 0 for R < π 2 . Let B K , κ int be a geodesic ball enclosed by S K , κ , and let B K , κ int , + be the geodesic half ball given by

(2.11) B K , κ int , + = { x ∈ B K , κ int : x n > 0 } .

Let X be the vector field

X = 2 1 + R ℝ 2 ⁢ [ x n ⁢ x - 1 2 ⁢ ( | x | 2 + R ℝ 2 ) ⁢ E n ] , V = 2 ⁢ x n 1 + | x | 2 .

It has been shown in [28] that X and V also satisfy Propositions 2.2 and 2.3.

3 Mixed BVP in space forms

From this section on, let Ω be a bounded, connected open domain in B K , κ int whose boundary ∂ ⁡ Ω consists of two parts Σ ¯ and T = ∂ ⁡ Ω ∖ Σ ¯ , where T ⊂ S K , κ is smooth and meets Σ at a common ( n - 2 ) -dimensional submanifold Γ. If S K , κ is a geodesic sphere, then we assume further that

Ω ⊂ B K , κ int , + .

For notation simplicity and unification, in the following sections we use Ω ⊂ B K , κ int to indicate that Ω ⊂ B K , κ int , + in the case that S K , κ is a geodesic sphere.

We consider the following two kinds of eigenvalue problems in Ω.

Mixed Robin--Dirichlet eigenvalue problem.

Consider

(3.1) { Δ ¯ ⁢ u = - λ ⁢ u in ⁢ Ω , u = 0 on ⁢ Σ ¯ , ∂ N ¯ ⁡ u = κ ⁢ u on ⁢ T .

The first Robin–Dirichlet eigenvalue can be variationally characterized by

(3.2) λ 1 = inf 0 ≠ u ∈ W 0 1 , 2 ⁢ ( Ω , Σ ) ⁡ ∫ Ω g ¯ ⁢ ( ∇ ¯ ⁢ u , ∇ ¯ ⁢ u ) ⁢ 𝑑 x - κ ⁢ ∫ T u 2 ⁢ 𝑑 A ∫ Ω u 2 ⁢ 𝑑 x .

Mixed Steklov--Dirichlet eigenvalue problem (see, e.g., [1, 4]).

Consider

(3.3) { Δ ¯ ⁢ u + n ⁢ K ⁢ u = 0 in ⁢ Ω , u = 0 on ⁢ Σ ¯ , ∂ N ¯ ⁡ u = μ ⁢ κ ⁢ u on ⁢ T .

The mixed Steklov–Dirichlet eigenvalues can be considered as the eigenvalues of the Dirichlet-to-Neumann map

ℒ : L 2 ⁢ ( T ) → L 2 ⁢ ( T ) ,

u ↦ 1 κ ⁢ ∂ N ¯ ⁡ u ^ ,

where u ^ ∈ W 0 1 , 2 ⁢ ( Ω , Σ ) is the extension of u to Ω satisfying Δ ¯ ⁢ u ^ + n ⁢ K ⁢ u ^ = 0 in Ω and u ^ = 0 on Σ. According to the spectral theory for compact, symmetric linear operators, ℒ has a discrete spectrum { μ i } i = 1 ∞ (see, e.g., [1, 4]):

0 < μ 1 ≤ μ 2 ≤ ⋯ → + ∞ .

The first eigenvalue μ 1 can be variationally characterized by

μ 1 = inf 0 ≠ u ∈ W 0 1 , 2 ⁢ ( Ω , Σ ) ⁡ ∫ Ω g ¯ ⁢ ( ∇ ¯ ⁢ u , ∇ ¯ ⁢ u ) ⁢ 𝑑 x - n ⁢ K ⁢ ∫ Ω u 2 ⁢ 𝑑 x κ ⁢ ∫ T u 2 ⁢ 𝑑 A .

In our case, we have the following proposition.

Proposition 3.1.

If S K , κ is a geodesic sphere in H n or S n and Ω ⊆ B K , κ int , + , then the following assertions hold:

λ 1 ⁢ ( Ω ) ≥ n ⁢ K and λ 1 = n ⁢ K if and only if Ω = B K , κ int , + .
μ 1 ⁢ ( Ω ) ≥ 1 and μ 1 = 1 if and only if Ω = B K , κ int , + .

Proof.

We proceed exactly as in [11]. If Ω = B K , κ int , + , one checks that u = V ≥ 0 indeed solves (3.1) with λ = n ⁢ K and (3.3) with μ = 1 . Since u = V is a non-negative solution, it must be the first eigenfunction, and hence λ 1 ⁢ ( B K , κ int , + ) = n ⁢ K and μ 1 ⁢ ( B K , κ int , + ) = 1 .

On the other hand, for Ω ⊂ B K , κ int , + , by the variational characterization and a standard argument of doing zero extension, one sees

λ 1 ⁢ ( Ω ) ≥ λ 1 ⁢ ( B K , κ int , + ) = n ⁢ K and μ 1 ⁢ ( Ω ) ≥ μ 1 ⁢ ( B K , κ int , + ) = 1 .

If Ω ⫋ B K , κ int , + , then the Aronszajn unique continuity theorem [3] implies

λ 1 ⁢ ( Ω ) > λ 1 ⁢ ( B K , κ int , + ) = n ⁢ K .

In fact, we extend the first Robin–Dirichlet eigenfunction u in Ω to u ~ in B K , κ int , + by defining u ~ = 0 outside Ω ¯ . Then u ~ is the first Robin–Dirichlet eigenfunction in B K , κ int , + by its variational characterization (3.2). However, the Aronszajn unique continuity theorem^[2] would imply that u = 0 is identically zero on B K , κ int , + . This is a contradiction to the fact that u is the first eigenfunction in Ω.

For μ 1 , it has been proved in [4, Proposition 3.1.1] that μ 1 ⁢ ( Ω ) > μ 1 ⁢ ( B K , κ int , + ) = 1 . ∎

Proposition 3.2.

If S K , κ is a horosphere or an equidistant hypersurface in H n and Ω ⊆ B K , κ int , then

(3.4) λ 1 ⁢ ( Ω ) > - n 𝑎𝑛𝑑 μ 1 ⁢ ( Ω ) > 1 .

Proof.

We first take an orthonormal basis { e i } i = 1 n in the upper half space model

e i = x n ⁢ E i , i = 1 , … , n .

By using the divergence theorem, we get

(3.5) ∫ Ω div g ¯ ⁡ ( u 2 ⁢ e n ) ⁢ 𝑑 x = ∫ ∂ ⁡ Ω u 2 ⁢ g ¯ ⁢ ( e n , ν ) ⁢ 𝑑 A = ∫ T u 2 ⁢ g ¯ ⁢ ( e n , N ¯ ) ⁢ 𝑑 A = - cos ⁡ θ ⁢ ∫ T u 2 ⁢ 𝑑 A ,

where we also use u = 0 on Σ and the fact that 〈 N ¯ , E n 〉 = - x n ⁢ cos ⁡ θ on T and θ ∈ [ 0 , π 2 ) .

On the other hand,

(3.6) ∫ Ω div g ¯ ⁡ ( u 2 ⁢ e n ) ⁢ 𝑑 x = ∫ Ω e n ⁢ ( u 2 ) ⁢ 𝑑 x + ∫ Ω u 2 ⁢ div g ¯ ⁡ ( e n ) ⁢ 𝑑 x = ∫ Ω e n ⁢ ( u 2 ) ⁢ 𝑑 x - ( n - 1 ) ⁢ ∫ Ω u 2 ⁢ 𝑑 x .

Combining (3.5) with (3.6), we have

cos ⁡ θ ⁢ ∫ T u 2 ⁢ 𝑑 A = ∫ Ω ( n - 1 ) ⁢ u 2 - e n ⁢ ( u 2 ) ⁢ d ⁢ x

= ∫ Ω ( n - 1 ) ⁢ u 2 - 2 ⁢ u ⁢ e n ⁢ ( u ) ⁢ d ⁢ x

≤ ∫ Ω ( n - 1 ) ⁢ u 2 + 2 ⁢ | u | ⁢ | ∇ ¯ ⁢ u | ⁢ d ⁢ x

≤ ∫ Ω ( n - 1 ) ⁢ u 2 + | u | 2 + g ¯ ⁢ ( ∇ ¯ ⁢ u , ∇ ¯ ⁢ u ) ⁢ d ⁢ x

= ∫ Ω n ⁢ u 2 + g ¯ ⁢ ( ∇ ¯ ⁢ u , ∇ ¯ ⁢ u ) ⁢ d ⁢ x .

Since 0 ≠ u ∈ W 0 1 , 2 ⁢ ( Ω , Σ ) , we know that the above equality is strict, namely

cos ⁡ θ ⁢ ∫ T u 2 ⁢ 𝑑 A < ∫ Ω n ⁢ u 2 + g ¯ ⁢ ( ∇ ¯ ⁢ u , ∇ ¯ ⁢ u ) ⁢ d ⁢ x .

Recall that κ = cos ⁡ θ . Therefore, we complete this proof by taking the infimum for u. ∎

Using Proposition 3.1 (ii) and Proposition 3.2, we show the existence and uniqueness of the mixed BVP (1.3).

Proposition 3.3.

Let f ∈ C ∞ ⁢ ( Ω ) , q ∈ C ∞ ⁢ ( T ) and Ω ⫋ B K , κ int . Then the mixed BVP

(3.7) { Δ ¯ ⁢ u + n ⁢ K ⁢ u = f in ⁢ Ω , u = 0 on ⁢ Σ ¯ , ∂ N ¯ ⁡ u = κ ⁢ u + q on ⁢ T ,

admits a unique weak solution u ∈ W 0 1 , 2 ⁢ ( Ω , Σ ) . Moreover, u ∈ C ∞ ⁢ ( Ω ¯ ∖ Γ ) ∩ C α ⁢ ( Ω ¯ ) for some α ∈ ( 0 , 1 ) .

Proof.

The weak solution to (3.7) is defined to be u ∈ W 0 1 , 2 ⁢ ( Ω , Σ ) such that

(3.8) B ⁢ [ u , v ] := ∫ Ω g ¯ ⁢ ( ∇ ¯ ⁢ u , ∇ ¯ ⁢ v ) ⁢ 𝑑 x - κ ⁢ ∫ T u ⁢ v ⁢ 𝑑 A - n ⁢ K ⁢ ∫ Ω u ⁢ v ⁢ 𝑑 x for all ⁢ v ∈ W 0 1 , 2 ⁢ ( Ω , Σ ) ,

and

(3.9) B ⁢ [ u , v ] = ∫ Ω - f ⁢ v ⁢ d ⁢ x + ∫ T q ⁢ v ⁢ 𝑑 A for all ⁢ v ∈ W 0 1 , 2 ⁢ ( Ω , Σ ) .

From Proposition 3.1 (ii) and Proposition 3.2, we know 1 - 1 μ 1 > 0 . There holds

(3.10) κ ⁢ ∫ T u 2 ⁢ 𝑑 A ≤ 1 μ 1 ⁢ ( ∫ Ω g ¯ ⁢ ( ∇ ¯ ⁢ u , ∇ ¯ ⁢ u ) ⁢ 𝑑 x - n ⁢ K ⁢ ∫ Ω u 2 ⁢ 𝑑 A ) .

By using (3.8), (3.10), Proposition 3.1 (i) and Proposition 3.2, there exists a positive constant β such that

B ⁢ [ u , u ] ≥ ( 1 - 1 μ 1 ) ⁢ ( ∫ Ω g ¯ ⁢ ( ∇ ¯ ⁢ u , ∇ ¯ ⁢ u ) ⁢ 𝑑 x - n ⁢ K ⁢ ∫ Ω u 2 ⁢ 𝑑 x ) ≥ β ⁢ ∥ u ∥ W 0 1 , 2 ⁢ ( Ω , Σ ) 2 .

Thus, B ⁢ [ u , v ] is coercive on W 0 1 , 2 ⁢ ( Ω , Σ ) . The standard Lax–Milgram theorem holds for the weak formulation to (3.7). Therefore, (3.7) admits a unique weak solution u ∈ W 0 1 , 2 ⁢ ( Ω , Σ ) .

The regularity u ∈ C ∞ ⁢ ( Ω ¯ ∖ Γ ) follows from the classical regularity theory for elliptic equations, and u ∈ C α ⁢ ( Ω ¯ ) has been proved by Lieberman in [15, Theorem 2]. Note that the global wedge condition in [15, Theorem 2] is satisfied for the domain Ω whose boundary parts Σ and T meet at a common in a smooth ( n - 2 ) -dimensional manifold; see [15, p. 426]. ∎

Proposition 3.4.

Let u be the unique solution to (3.7) with f ≥ 0 and q ≤ 0 . Then either u ≡ 0 in Ω, or u < 0 in Ω ∪ T .

Proof.

Since the Robin boundary condition has an unfavorable sign, we cannot use the maximum principle directly. Since u + = max ⁡ { u , 0 } ∈ W 0 1 , 2 ⁢ ( Ω , Σ ) , we can use it as a test function in the weak formulation (3.9) to get

∫ Ω - f ⁢ u + ⁢ d ⁢ x + ∫ T q ⁢ u + ⁢ 𝑑 A = ∫ Ω g ¯ ⁢ ( ∇ ¯ ⁢ u + , ∇ ¯ ⁢ u + ) ⁢ 𝑑 x - κ ⁢ ∫ T ( u + ) 2 ⁢ 𝑑 A - n ⁢ K ⁢ ∫ Ω ( u + ) 2 ⁢ 𝑑 x .

Since f ≥ 0 and q ≤ 0 , we have

∫ Ω - f ⁢ u + ⁢ d ⁢ x + ∫ T q ⁢ u + ⁢ 𝑑 A ≤ 0 .

On the other hand, it follows from Proposition 3.1 (i) and (3.4) that

∫ Ω g ¯ ⁢ ( ∇ ¯ ⁢ u + , ∇ ¯ ⁢ u + ) ⁢ 𝑑 x - κ ⁢ ∫ T ( u + ) 2 ⁢ 𝑑 A - n ⁢ K ⁢ ∫ Ω ( u + ) 2 ⁢ 𝑑 x ≥ ( λ 1 - n ⁢ K ) ⁢ ∫ Ω ( u + ) 2 ⁢ 𝑑 x ≥ 0 .

From above, we conclude that u + ≡ 0 , which means u ≤ 0 in Ω. Finally, by the strong maximum principle, we get either u ≡ 0 in Ω, or u < 0 in Ω ∪ T . ∎

Proposition 3.5.

Let e T be a tangent vector field to T. Let u be the unique solution to (1.3). Then

(3.11) ∇ ¯ 2 ⁢ u ⁢ ( N ¯ , e T ) = 0 on ⁢ T .

Proof.

By differentiating the equation ∂ N ¯ ⁡ u = κ ⁢ u with respect to e T , we get

κ ⁢ ∇ ¯ e T ⁢ ( u ) = e T ⁢ ( g ¯ ⁢ ( ∇ ¯ ⁢ u , N ¯ ) )

= ∇ ¯ 2 ⁢ u ⁢ ( N ¯ , e T ) + g ¯ ⁢ ( ∇ ¯ ⁢ u , ∇ ¯ e T ⁢ N ¯ )

= ∇ ¯ 2 ⁢ u ⁢ ( N ¯ , e T ) + h S K , κ ⁢ ( ∇ ¯ ⁢ u , e T )

= ∇ ¯ 2 ⁢ u ⁢ ( N ¯ , e T ) + κ ⁢ g ¯ ⁢ ( ∇ ¯ ⁢ u , e T ) .

Here we use the fact that S K , κ is an umbilical hypersurface with principal curvature κ. The assertion (3.11) follows. ∎

4 Partially overdetermined BVP in space forms

In this section, we will use a method totally based on integral identities and inequalities to prove Theorem 1.3. The proof follows closely our previous paper [11]. The main ingredients are Propositions 2.2 and 2.3.

First, we introduce the function P as follows:

P := g ¯ ⁢ ( ∇ ¯ ⁢ u , ∇ ¯ ⁢ u ) - 2 n ⁢ u + K ⁢ u 2 .

Proposition 4.1.

It holds

Δ ¯ ⁢ P = 2 ⁢ | ( ∇ ¯ 2 ⁢ u + K ⁢ u ⁢ g ) - 1 n ⁢ ( Δ ¯ ⁢ u + n ⁢ K ⁢ u ) ⁢ g ¯ | 2 ≥ 0 in ⁢ Ω .

Proof.

By direct computation and using Δ ¯ ⁢ u + n ⁢ K ⁢ u = 1 , we obtain

Δ ¯ ⁢ P ⁢ ( x ) = 2 ⁢ | ∇ ¯ 2 ⁢ u | 2 + 2 ⁢ g ¯ ⁢ ( ∇ ¯ ⁢ u , ∇ ¯ ⁢ Δ ¯ ⁢ u ) + 2 ⁢ Ric ¯ ⁢ ( ∇ ¯ ⁢ u , ∇ ¯ ⁢ u ) - 2 n ⁢ Δ ¯ ⁢ u + 2 ⁢ K ⁢ ( g ¯ ⁢ ( ∇ ¯ ⁢ u , ∇ ¯ ⁢ u ) + u ⁢ Δ ¯ ⁢ u )

= 2 ⁢ | ( ∇ ¯ 2 ⁢ u + K ⁢ u ⁢ g ) - 1 n ⁢ ( Δ ¯ ⁢ u + n ⁢ K ⁢ u ) ⁢ g ¯ | 2 ≥ 0 ,

as desired. ∎

Due to the lack of regularity, we need the following formula of integration by parts; see [19, Lemma 2.1] (the original statement [19, Lemma 2.1] is for a sector-like domain in a cone, nevertheless the proof is applicable in our case). We remark that a general version of the integration-by-parts formula for Lipschitz domains has been stated in some classical book by Grisvard [10, Theorem 1.5.3.1]. However, it seems not enough for our purpose.

Proposition 4.2 ([19, Lemma 2.1]).

Let F : Ω → R n be a vector field such that

F ∈ C 1 ⁢ ( Ω ∪ Σ ∪ T ) ∩ L 2 ⁢ ( Ω ) 𝑎𝑛𝑑 div ⁡ ( F ) ∈ L 1 ⁢ ( Ω ) .

Then

∫ Ω div ⁡ ( F ) ⁢ 𝑑 x = ∫ Σ g ¯ ⁢ ( F , ν ) ⁢ 𝑑 A + ∫ T g ¯ ⁢ ( F , N ¯ ) ⁢ 𝑑 A .

We first prove a Pohozaev-type identity for (1.4).

Proposition 4.3.

Let u be the unique solution to (1.4). Then we have

(4.1) ∫ Ω V ⁢ ( P - c 2 ) ⁢ 𝑑 x = 0 .

Proof.

First of all, we remark that by regularity in Proposition 3.3, u ∈ C ∞ ⁢ ( Ω ∪ Σ ∪ T ) , that is, u is smooth away from the corner Γ. Moreover, due to our assumption u ∈ W 1 , ∞ ⁢ ( Ω ) ∩ W 2 , 2 ⁢ ( Ω ) , Proposition 4.2 can be applied in all of the following integration by parts formulas.

Now, we consider the following differential identity:

div ⁡ ( u ⁢ X - g ¯ ⁢ ( X , ∇ ¯ ⁢ u ) ⁢ ∇ ¯ ⁢ u ) = g ¯ ⁢ ( X , ∇ ¯ ⁢ u ) + u ⁢ div ⁡ X - ∇ ¯ ⁢ X ⁢ ( ∇ ¯ ⁢ u , ∇ ¯ ⁢ u ) - 1 2 ⁢ g ¯ ⁢ ( X , ∇ ¯ ⁢ g ¯ ⁢ ( ∇ ¯ ⁢ u , ∇ ¯ ⁢ u ) ) - g ¯ ⁢ ( X , ∇ ¯ ⁢ u ) ⁢ Δ ¯ ⁢ u

= n ⁢ V ⁢ u - V ⁢ g ¯ ⁢ ( ∇ ¯ ⁢ u , ∇ ¯ ⁢ u ) - 1 2 ⁢ g ¯ ⁢ ( X , ∇ ¯ ⁢ g ¯ ⁢ ( ∇ ¯ ⁢ u , ∇ ¯ ⁢ u ) ) + n ⁢ K ⁢ g ¯ ⁢ ( X , ∇ ¯ ⁢ ( 1 2 ⁢ u 2 ) ) ,

where we use the equation Δ ¯ ⁢ u + n ⁢ K ⁢ u = 1 and (2.6).

Integrating by parts and using (2.7) and the boundary conditions (1.4), we see that

- c 2 ⁢ ∫ Σ g ¯ ⁢ ( X , ν ) ⁢ 𝑑 A - ∫ T g ¯ ⁢ ( X , ∇ ¯ ⁢ u ) ⁢ u N ¯ ⁢ 𝑑 A

= ∫ Ω ( n ⁢ V ⁢ u - V ⁢ g ¯ ⁢ ( ∇ ¯ ⁢ u , ∇ ¯ ⁢ u ) + 1 2 ⁢ g ¯ ⁢ ( ∇ ¯ ⁢ u , ∇ ¯ ⁢ u ) ⁢ div ⁡ X - n ⁢ K 2 ⁢ u 2 ⁢ div ⁡ X ) ⁢ 𝑑 x - c 2 2 ⁢ ∫ Σ g ¯ ⁢ ( X , ν ) ⁢ 𝑑 A .

It follows that

(4.2) ∫ Ω ( n ⁢ V ⁢ u + ( n 2 - 1 ) ⁢ V ⁢ g ¯ ⁢ ( ∇ ¯ ⁢ u , ∇ ¯ ⁢ u ) - n 2 ⁢ K 2 ⁢ V ⁢ u 2 ) ⁢ 𝑑 x = - c 2 2 ⁢ ∫ Σ g ¯ ⁢ ( X , ν ) ⁢ 𝑑 A - κ ⁢ ∫ T g ¯ ⁢ ( X , ∇ ¯ ⁢ u ) ⁢ u ⁢ 𝑑 A .

Using (2.6) and (2.7) yields

(4.3) ∫ Σ 1 2 ⁢ c 2 ⁢ g ¯ ⁢ ( X , ν ) ⁢ 𝑑 A = 1 2 ⁢ c 2 ⁢ ( ∫ Ω div ⁡ X ⁢ d ⁢ x - ∫ T g ¯ ⁢ ( X , N ¯ ) ⁢ 𝑑 A ) = n 2 ⁢ c 2 ⁢ ∫ Ω V ⁢ 𝑑 x ,

κ ⁢ ∫ T g ¯ ⁢ ( X , ∇ ¯ ⁢ u ) ⁢ u ⁢ 𝑑 A = κ ⁢ ∫ T g ¯ ⁢ ( X T , ∇ ¯ ⁢ ( 1 2 ⁢ u 2 ) ) ⁢ 𝑑 A

= κ 2 ⁢ ∫ Γ u 2 ⁢ g ¯ ⁢ ( X T , μ ) ⁢ 𝑑 s - κ 2 ⁢ ∫ T u 2 ⁢ div T ⁡ X T ⁢ d ⁢ A

= κ ⁢ ( 1 - n ) 2 ⁢ ∫ T V ⁢ u 2 ⁢ 𝑑 A .

In the last equality, we also used u = 0 on Γ and div T ⁡ X T = ( n - 1 ) ⁢ V .

To achieve (4.1), we do a further integration by parts and apply (2.9) and (2.10) to get

∫ Ω V ⁢ g ¯ ⁢ ( ∇ ¯ ⁢ u , ∇ ¯ ⁢ u ) ⁢ 𝑑 x = ∫ T V ⁢ u ⁢ ( u N ¯ ) ⁢ 𝑑 A - ∫ Ω ( g ¯ ⁢ ( ∇ ¯ ⁢ V , ∇ ¯ ⁢ ( 1 2 ⁢ u 2 ) ) + V ⁢ u ⁢ Δ ¯ ⁢ u ) ⁢ 𝑑 x

= κ ⁢ ∫ T V ⁢ u 2 ⁢ 𝑑 A - 1 2 ⁢ ∫ T ( V ) N ¯ ⁢ u 2 ⁢ 𝑑 A + ∫ Ω ( 1 2 ⁢ u 2 ⁢ Δ ¯ ⁢ V - V ⁢ u ⁢ Δ ¯ ⁢ u ) ⁢ 𝑑 x

= κ ⁢ ∫ T V ⁢ u 2 ⁢ 𝑑 A - κ 2 ⁢ ∫ T V ⁢ u 2 ⁢ 𝑑 A + ∫ Ω 1 2 ⁢ u 2 ⁢ ( - n ⁢ K ⁢ V ) - V ⁢ u ⁢ ( 1 - n ⁢ K ⁢ u ) ⁢ d ⁢ x

= κ 2 ⁢ ∫ T V ⁢ u 2 ⁢ 𝑑 A + n ⁢ K 2 ⁢ ∫ Ω V ⁢ u 2 ⁢ 𝑑 x - ∫ Ω V ⁢ u ⁢ 𝑑 x .

It follows that

(4.4) κ 2 ⁢ ∫ T V ⁢ u 2 ⁢ 𝑑 A = ∫ Ω ( V ⁢ g ¯ ⁢ ( ∇ ¯ ⁢ u , ∇ ¯ ⁢ u ) - n ⁢ K 2 ⁢ V ⁢ u 2 + V ⁢ u ) ⁢ 𝑑 x .

Substituting (4.3)–(4.4) into (4.2), we arrive at (4.1). ∎

Proposition 4.4.

Let u be the unique solution to (1.4) such that u ∈ W 1 , ∞ ⁢ ( Ω ) ∩ W 2 , 2 ⁢ ( Ω ) . Then

(4.5) ∫ Ω V ⁢ u ⁢ | ( ∇ ¯ 2 ⁢ u + K ⁢ u ⁢ g ) - 1 n ⁢ ( Δ ¯ ⁢ u + n ⁢ K ⁢ u ) ⁢ g ¯ | 2 ⁢ 𝑑 x = 0 .

Proof.

Since u ∈ W 1 , ∞ ⁢ ( Ω ) ∩ W 2 , 2 ⁢ ( Ω ) , we obtain

(4.6) Δ ⁢ P = 2 ⁢ | ( ∇ ¯ 2 ⁢ u + K ⁢ u ⁢ g ) - 1 n ⁢ ( Δ ¯ ⁢ u + n ⁢ K ⁢ u ) ⁢ g ¯ | 2 ∈ L 1 ⁢ ( Ω ) .

It follows that

div ⁡ ( V ⁢ u ⁢ ∇ ¯ ⁢ P - P ⁢ ∇ ¯ ⁢ ( V ⁢ u ) ) ∈ L 1 ⁢ ( Ω ) and ( V ⁢ u ⁢ ∇ ¯ ⁢ P - P ⁢ ∇ ¯ ⁢ ( V ⁢ u ) ) ∈ L 2 ⁢ ( Ω ) .

Firstly, we consider the following differential identity:

div ⁡ ( V ⁢ u ⁢ ∇ ¯ ⁢ P - P ⁢ ∇ ¯ ⁢ ( V ⁢ u ) ) + c 2 ⁢ div ⁡ ( V ⁢ ∇ ¯ ⁢ u - u ⁢ ∇ ¯ ⁢ V )

= V ⁢ u ⁢ Δ ¯ ⁢ P - P ⁢ Δ ¯ ⁢ ( V ⁢ u ) + c 2 ⁢ ( V ⁢ Δ ¯ ⁢ u - u ⁢ Δ ¯ ⁢ V )

= V ⁢ u ⁢ Δ ¯ ⁢ P - 2 ⁢ P ⁢ g ¯ ⁢ ( ∇ ¯ ⁢ V , ∇ ¯ ⁢ u ) - ( P + c 2 ) ⁢ u ⁢ Δ ¯ ⁢ V + ( c 2 - P ) ⁢ V ⁢ Δ ¯ ⁢ u

= V ⁢ u ⁢ Δ ¯ ⁢ P - 2 ⁢ P ⁢ g ¯ ⁢ ( ∇ ¯ ⁢ V , ∇ ¯ ⁢ u ) - ( P + c 2 ) ⁢ u ⁢ ( - n ⁢ K ⁢ V ) + ( c 2 - P ) ⁢ V ⁢ ( 1 - n ⁢ K ⁢ u )

(4.7) = V ⁢ u ⁢ Δ ¯ ⁢ P - 2 ⁢ P ⁢ g ¯ ⁢ ( ∇ ¯ ⁢ V , ∇ ¯ ⁢ u ) + 2 ⁢ n ⁢ K ⁢ P ⁢ u ⁢ V - ( P - c 2 ) ⁢ V ,

where we use the fact that ∇ ¯ 2 ⁢ V = - K ⁢ V ⁢ g ¯ and the equation Δ ¯ ⁢ u + n ⁢ K ⁢ u = 1 in Ω.

Applying the divergence theorem in (4.7) and the boundary conditions (1.4), we have

∫ Σ ( - P + c 2 ) ⁢ V ⁢ u ν ⁢ 𝑑 A + ∫ T ( V ⁢ u ⁢ P N ¯ - P ⁢ ( V ⁢ u ) N ¯ ) ⁢ 𝑑 A = ∫ Ω ( V ⁢ u ⁢ Δ ¯ ⁢ P - 2 ⁢ P ⁢ g ¯ ⁢ ( ∇ ¯ ⁢ V , ∇ ¯ ⁢ u ) + 2 ⁢ n ⁢ K ⁢ P ⁢ u ⁢ V - ( P - c 2 ) ⁢ V ) ⁢ 𝑑 x

(4.8) = ∫ Ω ( V ⁢ u ⁢ Δ ¯ ⁢ P - 2 ⁢ P ⁢ g ¯ ⁢ ( ∇ ¯ ⁢ V , ∇ ¯ ⁢ u ) + 2 ⁢ n ⁢ K ⁢ P ⁢ u ⁢ V ) ⁢ 𝑑 x ,

where in the last line we used the Pohozaev-type identity (4.1).

Note that P = c 2 on Σ. It follows from (4.8) that

(4.9) ∫ Ω V ⁢ u ⁢ Δ ¯ ⁢ P ⁢ 𝑑 x = ∫ Ω ( 2 ⁢ P ⁢ g ¯ ⁢ ( ∇ ¯ ⁢ V , ∇ ¯ ⁢ u ) - 2 ⁢ n ⁢ K ⁢ P ⁢ u ⁢ V ) ⁢ 𝑑 x + ∫ T ( V ⁢ u ⁢ P N ¯ - P ⁢ ( V ⁢ u ) N ¯ ) ⁢ 𝑑 A .

Now, we compute the first term of (4.9):

∫ Ω 2 ⁢ g ¯ ⁢ ( ∇ ¯ ⁢ V , ∇ ¯ ⁢ u ) ⁢ P ⁢ 𝑑 x = 2 ⁢ ∫ Ω g ¯ ⁢ ( ∇ ¯ ⁢ V , ∇ ¯ ⁢ u ) ⁢ ( g ¯ ⁢ ( ∇ ¯ ⁢ u , ∇ ¯ ⁢ u ) - 2 n ⁢ u + K ⁢ u 2 ) ⁢ 𝑑 x

= 2 ⁢ ∫ Ω g ¯ ⁢ ( ∇ ¯ ⁢ V , ∇ ¯ ⁢ u ) ⁢ g ¯ ⁢ ( ∇ ¯ ⁢ u , ∇ ¯ ⁢ u ) ⁢ 𝑑 x - 2 n ⁢ ∫ Ω g ¯ ⁢ ( ∇ ¯ ⁢ V , ∇ ¯ ⁢ u 2 ) ⁢ 𝑑 x + 2 ⁢ K ⁢ ∫ Ω g ¯ ⁢ ( ∇ ¯ ⁢ V , ∇ ¯ ⁢ u ) ⁢ u 2 ⁢ 𝑑 x

= 2 ⁢ ∫ T ( ∂ N ¯ ⁡ V ) ⁢ u ⁢ g ¯ ⁢ ( ∇ ¯ ⁢ u , ∇ ¯ ⁢ u ) ⁢ 𝑑 A - 2 ⁢ ∫ Ω ( Δ ¯ ⁢ V ⁢ g ¯ ⁢ ( ∇ ¯ ⁢ u , ∇ ¯ ⁢ u ) + 2 ⁢ ∇ ¯ 2 ⁢ u ⁢ ( ∇ ¯ ⁢ V , ∇ ¯ ⁢ u ) ) ⁢ u ⁢ 𝑑 x

- 2 n ⁢ ∫ T ( ∂ N ¯ ⁡ V ) ⁢ u 2 ⁢ 𝑑 A + 2 n ⁢ ∫ Ω Δ ¯ ⁢ V ⁢ u 2 ⁢ 𝑑 x + 2 ⁢ K ⁢ ∫ Ω g ¯ ⁢ ( ∇ ¯ ⁢ V , ∇ ¯ ⁢ u ) ⁢ u 2 ⁢ 𝑑 x

= 2 ⁢ κ ⁢ ∫ T V ⁢ u ⁢ g ¯ ⁢ ( ∇ ¯ ⁢ u , ∇ ¯ ⁢ u ) ⁢ 𝑑 A + 2 ⁢ n ⁢ K ⁢ ∫ Ω V ⁢ g ¯ ⁢ ( ∇ ¯ ⁢ u , ∇ ¯ ⁢ u ) ⁢ u ⁢ 𝑑 x - 2 ⁢ ∫ Ω ∇ ¯ 2 ⁢ u ⁢ ( ∇ ¯ ⁢ V , ∇ ¯ ⁢ ( u 2 ) ) ⁢ 𝑑 x

- 2 n ⁢ κ ⁢ ∫ T V ⁢ u 2 ⁢ 𝑑 A + 2 n ⁢ ∫ Ω ( - n ⁢ K ⁢ V ) ⁢ u 2 ⁢ 𝑑 x + 2 ⁢ K ⁢ ∫ Ω g ¯ ⁢ ( ∇ ¯ ⁢ V , ∇ ¯ ⁢ u ) ⁢ u 2 ⁢ 𝑑 x

= 2 ⁢ κ ⁢ ∫ T ( g ¯ ⁢ ( ∇ ¯ ⁢ u , ∇ ¯ ⁢ u ) - u n ) ⁢ u ⁢ V ⁢ 𝑑 A + 2 ⁢ K ⁢ ∫ Ω ( n ⁢ g ¯ ⁢ ( ∇ ¯ ⁢ u , ∇ ¯ ⁢ u ) - u ) ⁢ u ⁢ V ⁢ 𝑑 x

(4.10) - 2 ⁢ ∫ Ω ( ∇ ¯ 2 ⁢ u ⁢ ( ∇ ¯ ⁢ V , ∇ ¯ ⁢ ( u 2 ) ) - K ⁢ g ¯ ⁢ ( ∇ ¯ ⁢ V , ∇ ¯ ⁢ u ) ⁢ u 2 ) ⁢ 𝑑 x ,

where we have used the fact that ∇ ¯ 2 ⁢ V = - K ⁢ V ⁢ g ¯ and ∂ N ¯ ⁡ V = κ ⁢ V on T.

Now, we compute the last term of (4.10) by using ∇ ¯ 2 ⁢ V = - K ⁢ V ⁢ g ¯ , the Ricci identity and (3.11):

- 2 ⁢ ∫ Ω ∇ ¯ 2 ⁢ u ⁢ ( ∇ ¯ ⁢ V , ∇ ¯ ⁢ ( u 2 ) ) - K ⁢ g ¯ ⁢ ( ∇ ¯ ⁢ V , ∇ ¯ ⁢ u ) ⁢ u 2 ⁢ d ⁢ x

= - 2 ⁢ ∫ T u 2 ⁢ ∇ ¯ 2 ⁢ u ⁢ ( ∇ ¯ ⁢ V , N ¯ ) ⁢ 𝑑 A + 2 ⁢ ∫ Ω ( g ¯ ⁢ ( ∇ ¯ 2 ⁢ V , ∇ ¯ 2 ⁢ u ) + g ¯ ⁢ ( ∇ ¯ ⁢ V , ∇ ¯ ⁢ Δ ¯ ⁢ u ) + Ric ¯ ⁢ ( ∇ ¯ ⁢ V , ∇ ¯ ⁢ u ) ) ⁢ u 2 ⁢ 𝑑 x

+ 2 ⁢ K ⁢ ∫ Ω g ¯ ⁢ ( ∇ ¯ ⁢ V , ∇ ¯ ⁢ u ) ⁢ u 2 ⁢ 𝑑 x

= - 2 ⁢ ∫ T u 2 ⁢ ( V N ¯ ) ⁢ ∇ ¯ 2 ⁢ u ⁢ ( N ¯ , N ¯ ) ⁢ 𝑑 A + 2 ⁢ ∫ Ω ( - K ⁢ V ⁢ Δ ¯ ⁢ u + g ¯ ⁢ ( ∇ ¯ ⁢ V , ∇ ¯ ⁢ ( 1 - n ⁢ K ⁢ u ) ) + ( n - 1 ) ⁢ K ⁢ g ¯ ⁢ ( ∇ ¯ ⁢ V , ∇ ¯ ⁢ u ) ) ⁢ u 2 ⁢ 𝑑 x

+ 2 ⁢ K ⁢ ∫ Ω g ¯ ⁢ ( ∇ ¯ ⁢ V , ∇ ¯ ⁢ u ) ⁢ u 2 ⁢ 𝑑 x

= - 2 ⁢ κ ⁢ ∫ T u 2 ⁢ V ⁢ ∇ ¯ 2 ⁢ u ⁢ ( N ¯ , N ¯ ) ⁢ 𝑑 A - 2 ⁢ K ⁢ ∫ Ω V ⁢ ( Δ ¯ ⁢ u ) ⁢ u 2 ⁢ 𝑑 x

(4.11) = - 2 ⁢ κ ⁢ ∫ T u 2 ⁢ V ⁢ ∇ ¯ 2 ⁢ u ⁢ ( N ¯ , N ¯ ) ⁢ 𝑑 A - 2 ⁢ K ⁢ ∫ Ω V ⁢ ( 1 - n ⁢ K ⁢ u ) ⁢ u 2 ⁢ 𝑑 x .

Next, we compute the boundary term of (4.9):

∫ T V ⁢ u ⁢ P N ¯ - ( V ⁢ u ) N ¯ ⁢ P ⁢ d ⁢ A = ∫ T ( P N ¯ - 2 ⁢ κ ⁢ P ) ⁢ u ⁢ V ⁢ 𝑑 A

= ∫ T ( 2 ⁢ ∇ ¯ 2 ⁢ u ⁢ ( ∇ ¯ ⁢ u , N ¯ ) + 2 ⁢ κ ⁢ ( u n - g ¯ ⁢ ( ∇ ¯ ⁢ u , ∇ ¯ ⁢ u ) ) ) ⁢ u ⁢ V ⁢ 𝑑 A

(4.12) = ∫ T ( 2 ⁢ κ ⁢ u ⁢ ∇ ¯ 2 ⁢ u ⁢ ( N ¯ , N ¯ ) + 2 ⁢ κ ⁢ ( u n - g ¯ ⁢ ( ∇ ¯ ⁢ u , ∇ ¯ ⁢ u ) ) ) ⁢ u ⁢ V ⁢ 𝑑 A .

In the last equality, we used (3.11) and ∂ N ¯ ⁡ u = κ ⁢ u on T.

Substituting (4.10)–(4.12) into (4.9) and noticing (4.6), we get the conclusion (4.5). ∎

Proof of Theorem 1.3.

We note that in both cases we have V > 0 . In the case that S K , κ is a horosphere or an equidistant hypersurface, we have V = 1 x n > 0 . In the case that S K , κ is a geodesic sphere in ℍ n or 𝕊 + n , we have V > 0 in Ω since Ω ⊆ B K , κ int , + ; see (2.3) and (2.11).

From Propositions 3.4 and 4.1 as well as V > 0 in Ω, we have

V ⁢ u ⁢ | ( ∇ ¯ 2 ⁢ u + K ⁢ u ⁢ g ) - 1 n ⁢ ( Δ ¯ ⁢ u + n ⁢ K ⁢ u ) ⁢ g ¯ | 2 ≤ 0 in ⁢ Ω .

It follows from Proposition 4.4 that

V ⁢ u ⁢ | ( ∇ ¯ 2 ⁢ u + K ⁢ u ⁢ g ) - 1 n ⁢ ( Δ ¯ ⁢ u + n ⁢ K ⁢ u ) ⁢ g ¯ | 2 ≡ 0 in ⁢ Ω .

Since u < 0 in Ω by Proposition 3.4, we see immediately that ∇ ¯ 2 ⁢ u is proportional to the metric g in Ω. Since Δ ¯ ⁢ u + n ⁢ K ⁢ u = 1 , we get

∇ ¯ 2 ⁢ u = ( 1 n - K ⁢ u ) ⁢ g ¯ .

From this, by restricting on Σ and using u = 0 on Σ, we get h i ⁢ j = 1 n ⁢ c ⁢ g i ⁢ j , which means Σ must be part of an umbilical hypersurface with principal curvature 1 n ⁢ c . ∎

5 Heintze–Karcher–Ros inequality and Alexandrov theorem

In this section, We shall first use the solution to (1.3) to prove the Heintze–Karcher–Ros-type inequality for free boundary hypersurfaces in ℍ n supported on a horosphere or an equidistant hypersurface.

Proof of Theorem 1.5.

The proof follows closely [28, Theorem 5.2]. Let Ω be a bounded connected domain enclosed by Σ and S K , κ whose boundary is ∂ ⁡ Ω = Σ ∪ T . Let u be a solution of the following mixed BVP:

(5.1) { Δ ¯ ⁢ u - n ⁢ u = 1 in ⁢ Ω , u = 0 on ⁢ Σ ¯ , ∂ N ¯ ⁡ u = κ ⁢ u on ⁢ T ,

where N ¯ is the unit outward normal of B K , κ int . The existence and regularity of u has been proved in Proposition 3.3.

Using (2.9), we have

(5.2) Δ ¯ ⁢ ( 1 x n ) - n x n = 0 , Δ ¯ ⁢ ( 1 x n ) ⁢ g ¯ - ∇ ¯ 2 ⁢ ( 1 x n ) + 1 x n ⁢ Ric ¯ = 0 .

Combining (5.1) and (5.2), we apply Green’s formula

∫ Ω 1 x n ⁢ 𝑑 x = ∫ Ω 1 x n ⁢ ( Δ ¯ ⁢ u - n ⁢ u ) - ( Δ ¯ ⁢ ( 1 x n ) - n x n ) ⁢ u ⁢ d ⁢ x

= ∫ ∂ ⁡ Ω 1 x n ⁢ u ν - ∂ ν ⁡ ( 1 x n ) ⁡ u ⁢ d ⁢ A

= ∫ Σ 1 x n ⁢ u ν ⁢ 𝑑 A + ∫ T 1 x n ⁢ u N ¯ - u ⁢ ∂ N ¯ ⁡ ( 1 x n ) ⁡ d ⁢ A

(5.3) = ∫ Σ 1 x n ⁢ u ν ⁢ 𝑑 A ,

where we use the fact (2.10).

Using Hölder’s inequality for the right-hand side of (5.3), we have

(5.4) ( ∫ Ω 1 x n ⁢ 𝑑 x ) 2 ≤ ∫ Σ 1 x n ⁢ H 1 ⁢ u ν 2 ⁢ 𝑑 A ⁢ ∫ Σ 1 x n ⋅ H 1 ⁢ 𝑑 A .

Applying the weighted Reilly-type formula in [14, 22] (see also [28, Theorem 5.1]), in our case with V = 1 x n , we see

n - 1 n ⁢ ∫ Ω 1 x n ⁢ 𝑑 x = ∫ Ω 1 x n ⁢ ( Δ ¯ ⁢ u - n ⁢ u ) 2 ⁢ 𝑑 x - 1 n ⁢ ∫ Ω 1 x n ⁢ ( Δ ¯ ⁢ u - n ⁢ u ) 2 ⁢ 𝑑 x

≥ ∫ Ω 1 x n ⁢ ( ( Δ ¯ ⁢ u - n ⁢ u ) 2 - | ∇ ¯ 2 ⁢ u - u ⁢ g ¯ | 2 ) ⁢ 𝑑 x

= ∫ Σ n - 1 x n ⁢ H 1 ⁢ u ν 2 ⁢ 𝑑 A + ∫ T ( h S K , κ - κ ⋅ g ¯ ) ⁢ ( ∇ ⁡ u - x n ⁢ ∇ ⁡ ( 1 x n ) ⁡ u , ∇ ⁡ u - x n ⁢ ∇ ⁡ ( 1 x n ) ⁡ u ) ⁢ 𝑑 A

(5.5) = ∫ Σ n - 1 x n ⁢ H 1 ⁢ u ν 2 ⁢ 𝑑 A ,

where we use (2.10) and S K , κ is an umbilical hypersurface with principal curvature κ.

Combining (5.4) and (5.5), we get (1.6).

If equality in (5.5) holds, we get ∇ ¯ 2 ⁢ u = ( 1 n + u ) ⁢ g ¯ in Ω. Since u = 0 on Σ, we know that Σ must be part of an umbilical hypersurface. ∎

Denote by h and H r the second fundamental form and normalized r-th mean curvature of Σ, respectively. Precisely, h ⁢ ( X , Y ) = g ¯ ⁢ ( ∇ ¯ X ⁢ ν , Y ) and H r := ( n - 1 r ) - 1 ⁢ S r , where S r is given by

S r = ∑ i 1 < i 2 < ⋯ < i r κ i 1 ⁢ κ i 2 ⁢ ⋯ ⁢ κ i r for all ⁢ r = 1 , … , n - 1 ,

where κ 1 , κ 2 , … , κ n - 1 are principal curvatures of Σ in ℍ n . As a convention, we define H 0 = 1 .

Let T r ⁢ ( h ) = ∂ ⁡ S r + 1 ∂ ⁡ h be the Newton transformation. We state the following properties of T r .

Lemma 5.1 ([23, 5]).

For each 0 ≤ r ≤ n - 2 , the following assertions hold:

The Newton tensor T r is divergence-free, i.e., div ⁡ T r = 0 .
trace ⁡ ( T r ) = ( n - 1 - r ) ⁢ S r .
trace ⁡ ( T r ⁢ h ) = ( r + 1 ) ⁢ S r + 1 .
trace ⁡ ( T r ⁢ h 2 ) = S 1 ⁢ S r + 1 - ( r + 2 ) ⁢ S r + 2 .

In the next proposition, we prove the Minkowski formulas for free boundary hypersurfaces in ℍ n supported on a support hypersurface.

Proposition 5.2.

It holds

(5.6) ∫ Σ H k - 1 x n ⁢ 𝑑 A = ∫ Σ H k ⁢ g ¯ ⁢ ( X , ν ) ⁢ 𝑑 A for all ⁢ k = 1 , … , n - 1 .

Proof.

Let X T be the tangential projection of X on Σ . We know that X T ⊥ N ¯ along ∂ ⁡ Σ by (2.7). Let { e α } α = 1 n - 1 be an orthonormal frame on Σ . From Proposition 2.2 (i), we have that

(5.7) 1 2 ⁢ [ ∇ ¯ α ⁢ ( X T ) β + ∇ ¯ β ⁢ ( X T ) α ] = 1 x n ⁢ g ¯ α ⁢ β - h α ⁢ β ⁢ g ¯ ⁢ ( X , ν ) .

Multiplying (5.7) by T k - 1 α ⁢ β ⁢ ( h ) and integrating by parts on Σ, from Lemma 5.1, we get

∫ Σ ( n - k ) ⁢ 1 x n ⁢ S k - 1 - k ⁢ S k ⁢ g ¯ ⁢ ( X , ν ) ⁢ d ⁢ A = ∫ Σ T k - 1 α ⁢ β ⁢ ∇ ¯ α ⁢ ( X T ) β ⁢ 𝑑 A

= ∫ Σ ∇ ¯ α ⁢ ( T k - 1 α ⁢ β ⁢ X T ) β ⁢ 𝑑 A

= ∫ ∂ ⁡ Σ T k - 1 ⁢ ( X T , N ¯ ) ⁢ 𝑑 s = 0 .

In the last line, we used the fact that S K , κ is an umbilical hypersurface, and N ¯ is a principal direction of h; it is also a principal direction of the Newton tensor T k - 1 of h, which implies that T k - 1 ⁢ ( X T , N ¯ ) = 0 . The above proposition is completed by the definition of H k . ∎

Now, we use the Minkowski formulas (5.6) to prove the Alexandrov-type theorem for free boundary CMC hypersurfaces in ℍ n supported by a support hypersurface.

Theorem 5.3.

Assume S K , κ is a horosphere or an equidistant hypersurface. Let x : Σ → H n be an embedded smooth CMC hypersurface into B K , κ int (or B K , κ ext ) whose boundary ∂ ⁡ Σ lies on S K , κ . Assume Σ meets S K , κ orthogonally. Then Σ must be part of an umbilical hypersurface.

Proof.

Consider the upper half-space model. In the case that T is a horosphere, let D R = { x ∈ ℝ + n : | x | < R } . In the case that T is an equidistant, let D R = { x ∈ ℝ + n : | x - b | < R } where b = ( 1 , 0 , … , 0 ) .

Since Σ is a compact hypersurface, we take R large enough (resp. small) such that Σ ⊆ D R (resp. D R ∩ Σ = Ω ) when Σ lies in B K , κ int (resp. B K , κ ext ). Let D R shrink (resp. expand) along the radial direction in the Euclidean sense, until it touches Σ at some point p at a first time. By our choice of D R , it does not intersect with T orthogonally. Since D R does not intersect with T orthogonally but Σ does, we see that p is an interior point of Σ. It follows that H 1 = H 1 ⁢ ( p ) ≥ 0 . If H 1 = 0 , then the maximum principle implies that Σ must be some totally geodesic, which is a contradiction since Σ is perpendicular to S K , κ by hypothesis. Therefore, H 1 is positive.

Let Ω be a bounded connected domain enclosed by Σ and S K , κ whose boundary is ∂ ⁡ Ω = Σ ∪ T . Using Proposition 2.2 (i) and (ii), we see

(5.8) ∫ Ω div ⁡ X ⁢ d ⁢ x = ∫ Ω n x n ⁢ 𝑑 x

and

∫ Ω div ⁡ X ⁢ d ⁢ x = ∫ Σ g ¯ ⁢ ( X , ν ) ⁢ 𝑑 A + ∫ T g ¯ ⁢ ( X , N ¯ ) ⁢ 𝑑 A

= ∫ Σ g ¯ ⁢ ( X , ν ) ⁢ 𝑑 A

= 1 H 1 ⁢ ∫ Σ H 1 ⁢ g ¯ ⁢ ( X , ν ) ⁢ 𝑑 A

= 1 H 1 ⁢ ∫ Σ 1 x n ⁢ 𝑑 A

(5.9) = ∫ Σ 1 x n ⋅ H 1 ⁢ 𝑑 A ,

where we also used Proposition 5.2 with k = 1 . Then the conclusion follows from (5.8), (5.9) and Theo- rem 1.5. ∎

Next, we use the method of Ros [24] and Koh and Lee [12] to prove the higher-order Alexandrov theorem for embedded free boundary CMC hypersurfaces in ℍ n supported by a support hypersurface.

Theorem 5.4.

Assume S K , κ is a horosphere or an equidistant hypersurface. Let x : Σ → H n be an isometric proper immersion smooth hypersurface into B K , κ int (or B K , κ ext ) whose boundary ∂ ⁡ Σ lies on S K , κ . Assume Σ meets S K , κ orthogonally.

If x is an embedding and has nonzero constant higher-order mean curvatures H k , 1 ≤ k ≤ n - 1 , then Σ is part of an umbilical hypersurface.
If x has nonzero constant curvature quotient, i.e.,

H k H l = const , H l > 0 , 1 ≤ l < k ≤ n - 1 ,

then Σ is part of an umbilical hypersurface.

Proof.

Since H k is a constant, arguing as at the beginning of the proof of Theorem 5.3, we get H k > 0 by the compactness of Σ. The principal curvature are continuous functions. Therefore, we can choose a connected component such that H 1 , … , H k - 1 are all positive at any point. According to [24] and the Newton–MacLaurin inequality, we have for each 1 ≤ r ≤ k ,

(5.10) 0 ≤ H r 1 r ≤ H r - 1 1 r - 1 ≤ ⋯ ≤ H 1 ,

(5.11) 0 ≤ H r H r - 1 ≤ H r - 1 H r - 2 ≤ ⋯ ≤ H 1 H 0 = H 1 ,

with the equality holding only at umbilical points on Σ. Here H 0 = 1 by convention. It follows from (5.10) that

(5.12) 1 H 1 ≤ H k - 1 k .

Using Theorem 1.5 and (5.12), we have

(5.13) ∫ Ω n x n ⁢ 𝑑 x ≤ ∫ Σ 1 x n ⋅ H 1 ⁢ 𝑑 A ≤ ∫ Σ H k - 1 k ⋅ 1 x n ⁢ 𝑑 A .

On the other hand, by Proposition 2.2 (i) and (ii), we have

(5.14) H k ⁢ ∫ Ω div ⁡ X ⁢ d ⁢ x = H k ⁢ ∫ Ω n x n ⁢ 𝑑 x

and

H k ⁢ ∫ Ω div ⁡ X ⁢ d ⁢ x = H k ⁢ ∫ Σ g ¯ ⁢ ( X , ν ) ⁢ 𝑑 A

= ∫ Σ H k ⁢ g ¯ ⁢ ( X , ν ) ⁢ 𝑑 A

= ∫ Σ H k - 1 x n ⁢ 𝑑 A

(5.15) ≥ ∫ Σ H k k - 1 k x n ⁢ 𝑑 A ,

where we have used (5.6) and (5.10).

Combining (5.13)–(5.15), we get that equality holds in (5.10) on Σ. Therefore, Σ is part of an umbilical hypersurface. The proof of (i) is finished.

Arguing as at the beginning of the proof of Theorem 5.3, there is a point p on Σ such that all principal curvatures are positive. Therefore, H k and H l are positive at p. Since α = H k H l is constant and H l is positive on Σ, then H k > 0 on Σ and α > 0 .

By the Newton–MacLaurin inequality, we have

(5.16) 0 < α = H k H l ≤ H k - 1 H l - 1 .

Using Proposition 5.2 and H k = α ⁢ H l , we have

∫ Σ H k - 1 x n ⁢ 𝑑 A = ∫ Σ H k ⁢ g ¯ ⁢ ( X , ν ) ⁢ 𝑑 A

= ∫ Σ α ⁢ H l ⁢ g ¯ ⁢ ( X , ν ) ⁢ 𝑑 A

= α ⁢ ∫ Σ H l ⁢ g ¯ ⁢ ( X , ν ) ⁢ 𝑑 A

(5.17) = α ⁢ ∫ Σ H l - 1 x n ⁢ 𝑑 A ,

where in the last line we have used (5.6) again. Combining (5.16) with (5.17), we get

α = H k H l = H k - 1 H l - 1 = const on ⁢ Σ .

Proceeding inductively and taking p = k - l , we see that

H p + 1 H 1 = H p H 0 = H p .

By using (5.11), we have

H p + 1 H p = H p H p - 1 = ⋯ = H 1 H 0 = H 1 .

Therefore, Σ is part of an umbilical hypersurface. The proof of (ii) is completed. ∎

Communicated by Guofang Wang

Funding source: National Natural Science Foundation of China

Award Identifier / Grant number: 11871406

Funding statement: This work is supported by NSFC (Grant No. 11871406).

A An example for BVP with horospheres as boundary

In this section, we give an example that the partial overdetermined problem (1.4) admits a solution if the domain is bounded by two orthogonal horospheres.

We use the Poincaré ball model (2.2). Let L 1 and L 2 be two horospheres as follows:

(A.1) L 1 := { x ∈ 𝔹 n ∣ | x ′ | 2 + ( x n - 1 2 ) 2 = 1 4 } ,

(A.2) L 2 := { x ∈ 𝔹 n ∣ | x ′ | 2 + ( x n + 1 3 ) 2 = 4 9 } .

Then L 1 and L 2 are mutually orthogonal; see Figure 4.

$Figure 4 L 1 {L_{1}} and L 2 {L_{2}} are two horospheres with principal curvature κ = 1 {\kappa=1} , and the shaded area is B K , κ int {B_{K,\kappa}^{\mathrm{int}}} .$

Figure 4

L 1 and L 2 are two horospheres with principal curvature κ = 1 , and the shaded area is B K , κ int .

The domain Ω is bounded by Σ ¯ and T, that is, ∂ ⁡ Ω = Σ ¯ ∪ T , where Σ ⊆ L 1 and T ⊆ L 2 .

Let

V 0 := 1 + | x | 2 1 - | x | 2 , V = 2 ⁢ x n 1 - | x | 2 .

It is direct to see

(A.3) | x | 2 = V 0 - 1 V 0 + 1 , x n = V V 0 + 1 .

One has the following proposition.

Proposition A.1 ([28, Proposition 4.2]).

It holds

∇ ¯ 2 ⁢ V 0 = V 0 ⁢ g ¯ , ∇ ¯ 2 ⁢ V = V ⁢ g ¯ .

Proposition A.2 ([28, Proposition 4.3]).

For any tangential vector field Z on H n ,

(A.4) ∇ ¯ Z ⁢ V 0 = g ¯ ⁢ ( x , Z ) ,

(A.5) ∇ ¯ Z ⁢ V = e - ω ⁢ g ¯ ⁢ ( Z , E n ) + e - 2 ⁢ ω ⁢ g ¯ ⁢ ( x , E n ) ⁢ g ¯ ⁢ ( x , Z ) ,

where e 2 ⁢ ω = 4 ( 1 - | x | 2 ) 2 .

Let

u := 1 n ⁢ ( V 0 - V - 1 ) .

By Proposition A.1, we have

Δ ¯ ⁢ u - n ⁢ u = 1 in ⁢ Ω .

Combining (A.3) with (A.1), we get

u = 0 on ⁢ Σ ⊂ L 1 .

Since Σ ⊆ L 1 and g ¯ = e 2 ⁢ ω ⁢ δ , the unit outward normal vector of Σ is

ν = 2 ⁢ e - ω ⁢ ( x ′ , x n - 1 2 ) .

Using (A.4) and (A.5), one checks that on Σ,

∇ ¯ ν ⁢ u = 1 n ⁢ ( ∇ ¯ ν ⁢ V 0 - ∇ ¯ ν ⁢ V )

= 1 n ⁢ g ¯ ⁢ ( x , ν ) ⁢ ( 1 - e - 2 ⁢ ω ⁢ g ¯ ⁢ ( x , E n ) ) - 1 n ⁢ e - ω ⁢ g ¯ ⁢ ( ν , E n )

= 2 n ⁢ e ω ⁢ ( | x | 2 - 1 2 ⁢ x n ) ⁢ ( 1 - x n ) - 2 n ⁢ ( x n - 1 2 )

= 1 n .

In the last line above, we used (A.1).

On the other hand, since T ⊆ L 2 and g ¯ = e 2 ⁢ ω ⁢ δ , the unit outward normal vector of T is

N ¯ = 3 2 ⁢ e - ω ⁢ ( x ′ , x n + 1 3 ) .

Using (A.2), we get

(A.6) u = 1 n ⁢ ( 1 + | x | 2 1 - | x | 2 - 2 ⁢ x n 1 - | x | 2 - 1 ) = 5 ⁢ | x | 2 - 1 n ⁢ ( 1 - | x | 2 ) on ⁢ T .

Using (A.4), (A.5) and (A.2), one checks that on T,

∇ ¯ N ¯ ⁢ u = 1 n ⁢ ( ∇ ¯ N ¯ ⁢ V 0 - ∇ ¯ N ¯ ⁢ V )

= 1 n ⁢ g ¯ ⁢ ( x , N ¯ ) ⁢ ( 1 - e - 2 ⁢ ω ⁢ g ¯ ⁢ ( x , E n ) ) - 1 n ⁢ e - ω ⁢ g ¯ ⁢ ( N ¯ , E n )

= 3 2 ⁢ n ⁢ e ω ⁢ ( | x | 2 + 1 3 ⁢ x n ) ⁢ ( 1 - x n ) - 3 2 ⁢ n ⁢ ( x n + 1 3 )

(A.7) = 5 ⁢ | x | 2 - 1 n ⁢ ( 1 - | x | 2 ) .

Combining (A.6) with (A.7), we get

∇ ¯ N ¯ ⁢ u = u on ⁢ T .

In summary, we see that u = 1 n ⁢ ( V 0 - V - 1 ) is a solution of the partially overdetermined BVP (1.4), but Σ is part of a horosphere.

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Received: 2021-11-04

Revised: 2022-01-23

Accepted: 2022-01-28

Published Online: 2022-05-31

Published in Print: 2024-01-01

A partially overdetermined problem in domains with partial umbilical boundary in space forms

Abstract

1 Introduction

Theorem 1.1 ([11]).

Remark 1.2.

Theorem 1.3.

Remark 1.4.

Theorem 1.5.

2 Conformal Killing vector fields in space forms

2.1 Hyperbolic space ℍ n

Definition 2.1 ([16]).

Proposition 2.2.

Proof.

Proposition 2.3.

Proof.

2.2 Spherical space 𝕊 n

3 Mixed BVP in space forms

Mixed Robin--Dirichlet eigenvalue problem.

Mixed Steklov--Dirichlet eigenvalue problem (see, e.g., [1, 4]).

Proposition 3.1.

Proof.

Proposition 3.2.

Proof.

Proposition 3.3.

Proof.

Proposition 3.4.

Proof.

Proposition 3.5.

Proof.

4 Partially overdetermined BVP in space forms

Proposition 4.1.

Proof.

Proposition 4.2 ([19, Lemma 2.1]).

Proposition 4.3.

Proof.

Proposition 4.4.

Proof.

Proof of Theorem 1.3.

5 Heintze–Karcher–Ros inequality and Alexandrov theorem

Proof of Theorem 1.5.

Lemma 5.1 ([23, 5]).

Proposition 5.2.

Proof.

Theorem 5.3.

Proof.

Theorem 5.4.

Proof.

A An example for BVP with horospheres as boundary

Proposition A.1 ([28, Proposition 4.2]).

Proposition A.2 ([28, Proposition 4.3]).

References

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