Compact anisotropic stable hypersurfaces with free boundary in convex solid cones

Abstract

We consider a convex solid cone $\mathcal {C}\subset \mathbb {R}^{n+1}$ with vertex at the origin and boundary $\partial \mathcal {C}$ smooth away from 0. Our main result shows that a compact two-sided hypersurface $\Sigma$ immersed in $\mathcal {C}$ with free boundary in $\partial \mathcal {C}\setminus \{0\}$ and minimizing, up to second order, an anisotropic area functional under a volume constraint is contained in a Wulff-shape. The technique of proof also works for a non-smooth convex cone $\mathcal {C}$ provided the boundary of $\Sigma$ is away from the singular set of $\partial \mathcal {C}$.

1 Introduction

In this paper we study a variational problem for Euclidean hypersurfaces associated to an energy functional of anisotropic character. This means that the energy is computed by integrating an elliptic parametric function (or surface tension) which depends on the normal direction along the hypersurface. As it is explained in the introduction of Taylor [38] these functionals provide a mathematical model to study solid crystals. In the particular case of a constant surface tension we obtain the isotropic case, where the energy is proportional to the Euclidean area.

The surface tension that we consider in this work is the asymmetric norm given by the support function $h_K$ of a smooth strictly convex body $K\subset \mathbb {R}^{n+1}$ containing the origin in its interior. The corresponding anisotropic area of a two-sided hypersurface $\Sigma$ with unit normal N has the expression $A_K(\Sigma ):=\int _\Sigma h_K(N)\,d\Sigma$, where $d\Sigma$ is the Euclidean area element. By using the metric projection onto K it is possible to define also anisotropic counterparts to some of the classical notions in the extrinsic geometry of hypersurfaces, see Reilly [32] and Sect. 2 for a precise description. When K is the round unit ball about the origin we have $h_K(N)=1$, so that we recover the isotropic situation. We point out that the anisotropic geometry of hypersurfaces is typically introduced by means of a function $F:\mathbb {S}^n\rightarrow {\mathbb {R}}^+$ instead of a convex body K, see Remark 2.3 for more details.

It is well known that the round spheres uniquely solve the isoperimetric problem in $\mathbb {R}^{n+1}$. In our anisotropic setting, the strictly convex hypersurface $\partial K$ uniquely minimizes, up to translations and dilations centered at the origin, the anisotropic area computed with respect to the outer unit normal among hypersurfaces enclosing the same Euclidean volume. There are different proofs of this statement relying on analytical and geometric techniques, see Taylor [37, 38], Fonseca and Müller [13], Brothers and Morgan [3], Milman and Rotem [28], and Cabré et al. [4]. The minimizer $\partial K$ is usually referred to as the Wulff shape in honor of the crystallographer Georg Wulff, who first constructed the optimal crystal for a specified integrand [43].

As in the isotropic case, it is interesting to analyze the critical points (stationary hypersurfaces) and the second order minima (stable hypersurfaces) of the anisotropic area under a volume constraint. From the first variational formulas, a hypersurface $\Sigma$ is anisotropic stationary if and only if its anisotropic mean curvature $H_K$ defined in (2.6) is constant, see for instance Palmer [31] or Clarenz [7]. This motivated the generalization to the anisotropic context of some classical results for constant mean curvature hypersurfaces in $\mathbb {R}^{n+1}$. In this direction, an Alexandrov type theorem (resp. a Hopf type theorem) characterizing the Wulff shape among compact embedded hypersurfaces (resp. immersed spheres) with $H_K$ constant was obtained by Morgan [29] for $n=1$, and by He et al. [15] in higher dimension (resp. by He and Li [17], and Koiso and Palmer [25]). On the other hand, Palmer [31] and Winklmann [42] employed the second variation of the anisotropic area to show that, up to translation and homothety, the Wulff shape $\partial K$ is the unique compact, two-sided, anisotropic stable hypersurface immersed in $\mathbb {R}^{n+1}$. This is an anisotropic extension of a celebrated theorem of Barbosa and do Carmo [2] also discussed by Wente [41].

When we consider a smooth proper domain $\Omega$ instead of the whole space $\mathbb {R}^{n+1}$, we are naturally led to study the anisotropic partitioning problem. Here, we seek minimizers of the functional $A_K$ among hypersurfaces inside $\Omega$, possibly with non-empty boundary contained in $\partial \Omega$, and separating a fixed Euclidean volume. A critical point $\Sigma$ for this problem is called an anisotropic stationary hypersurface with free boundary in $\partial \Omega$. From the calculus of $A_K'(0)$, and reasoning as Koiso and Palmer [22], this is equivalent to that $H_K$ is constant on $\Sigma$ and the anisotropic normal $N_K$ defined in (2.2) is tangent to $\partial \Omega$ along $\partial \Sigma$. The more general case of anisotropic capillary hypersurfaces arises when both, $H_K$ and the angle between $N_K$ and the inner unit normal to $\partial \Omega$, are constant. These are critical points of an energy functional which involves not only $A_K(\Sigma )$ but also the wetting area of the set bounded by $\partial \Sigma$ in $\partial \Omega$. We remark that isotropic capillary hypersurfaces in convex domains of $\mathbb {R}^{n+1}$ have been extensively investigated, see Ros and Souam [34], Wang and Xia [39], and the references therein.

An anisotropic stationary hypersurface with free boundary in $\partial \Omega$ is stable if $A''_K(0)$ is nonnegative under variations preserving the volume separated by $\Sigma$ and the boundary of $\Omega$. The computation of $A_K''(0)$ and the subsequent analysis of anisotropic stable hypersurfaces have been treated in several previous works. In [22,23,24], Koiso and Palmer considered stable capillary surfaces when $\Omega$ is a slab of ${\mathbb {R}}^3$ and the Wulff shape is rotationally symmetric about a line orthogonal to $\partial \Omega$. In [19, 21], Koiso classified compact anisotropic stable capillary hypersurfaces disjoint from the edges in wedge-shaped domains of $\mathbb {R}^{n+1}$. Inside a smooth domain of revolution $\Omega \subset {\mathbb {R}}^3$, and for certain rotationally symmetric surface tensions, Barbosa and Silva [1] established that the totally geodesic disks orthogonal to the revolution axis are the unique compact stationary surfaces with free boundary in $\partial \Omega$ and meeting $\partial \Omega$ orthogonally. By assuming convexity of $\Omega$ they also discussed the stability of these surfaces. More recently, Jia et al. [18] have employed a Heintze-Karcher inequality and a Minkowski formula to prove that any compact, embedded, anisotropic capillary hypersurface inside an open half-space of $\mathbb {R}^{n+1}$ is part of a Wulff shape. In [14], besides the computation of a very general second variation formula, Guo and Xia have generalized the argument of Winklmann [42] after Barbosa and do Carmo [2] to show that a compact, two-sided, anisotropic stable capillary hypersurface immersed in a Euclidean half-space of $\mathbb {R}^{n+1}$ is a truncated Wulff shape.

In this paper we are interested in the stability question when the ambient domain is an open solid cone $\mathcal {C}\subset \mathbb {R}^{n+1}$ with vertex at the origin and boundary $\partial \mathcal {C}$ smooth away from 0. We do not require $\mathcal {C}$ to have any kind of symmetry. In the isotropic case it is known that round balls about the vertex provide the unique solutions to the partitioning problem in a (possibly non-smooth) convex cone $\mathcal {C}$, up to translations leaving $\mathcal {C}$ invariant, see Lions and Pacella [26], and Figalli and Indrei [12]. When $\partial \mathcal {C}\setminus \{0\}$ is smooth, Ritoré and the author [33] solved the problem by combining existence of minimizers with the classification of compact stable hypersurfaces with free boundary in $\partial \mathcal {C}$. A similar characterization result for homogenous weights in convex cones is due to Cañete and the author [5]. We must also mention the papers of Choe and Park [6], and Pacella and Tralli [30], where an Alexandrov type theorem for compact, embedded, stationary hypersurfaces in convex cones is proven.

In the anisotropic case it is easy to check that the truncated Wulff shape $\partial K\cap \overline{\mathcal {C}}$ is an anisotropic stationary hypersurface with free boundary in $\partial \mathcal {C}$, see Example 4.2. In [40], Weng derived an anisotropic version of the aforementioned result of Pacella and Tralli [30, Thm. 1.2]. By assuming an integral condition along $\partial \Sigma$ that involves the conormal vector it is shown that a compact embedded hypersurface $\Sigma$ in a smooth convex cone $\mathcal {C}$ with constant anisotropic mean curvature and boundary $\partial \Sigma \subset \partial \mathcal {C}\setminus \{0\}$ must be the truncated Wulff shape, up to translation and homothety. Also when the cone $\mathcal {C}$ is convex, results of Milman and Rotem [28], and Cabré et al. [4] for anisotropic area functionals with homogeneous weights entail that $\partial K\cap \overline{\mathcal {C}}$ minimizes the anisotropic area (computed with respect to the outer unit normal) for fixed volume. We remark that the techniques in [28] and [4] still hold under weaker regularity assumptions on the cone $\mathcal {C}$ and the surface tension. By extending arguments of Figalli and Indrei [12], the uniqueness of $\partial K\cap \overline{\mathcal {C}}$ as a solution to the partitioning problem in $\mathcal {C}$ was recently discussed by Dipierro et al. [11, Thm. 4.2]. The fact that $\partial K\cap \overline{\mathcal {C}}$ is a minimizer implies, in particular, the weaker property that it is an anisotropic stable hypersurface with free boundary in $\partial \mathcal {C}$. Our main result, Theorem 4.8, is the following uniqueness statement:

Up to a translation and a dilation centered at the vertex, any compact, connected, two-sided, anisotropic stable hypersurface immersed in a convex solid cone $\mathcal {C}$ with free boundary in $\partial \mathcal {C}\setminus \{0\}$ is part of the Wulff shape.

We must observe that, in general, we cannot expect $\partial K\cap \overline{\mathcal {C}}$ to be the only anisotropic stable hypersurface, up to translation and homothety, see Example 4.7. This motivates us to find additional conditions on the cone in order to deduce stronger rigidity consequences. With this idea in mind, in Corollary 4.11 we prove that a compact anisotropic stable hypersurface in a cone $\mathcal {C}$ over a smooth strictly convex domain of the sphere $\mathbb {S}^n$ must be a dilation of $\partial K\cap \overline{\mathcal {C}}$. On the other hand, for the case of a Euclidean half-space $\mathcal {H}$, recently treated by Guo and Xia [14], we can conclude that $\partial K\cap \overline{\mathcal {H}}$ is the only compact anisotropic stable hypersurface, up to translations along the boundary hyperplane.

Our proof of Theorem 4.8 is different from the ones in [5, 33] and [14], which are mainly inspired by Barbosa and do Carmo [2, Thm. (1.3)]. Instead, we adapt to our setting the idea employed by Wente [41] to characterize the round spheres as the only compact, two-sided, stable hypersurfaces immersed in $\mathbb {R}^{n+1}$. Wente applied the stability inequality with the special volume-preserving deformation obtained by parallel hypersurfaces dilated to keep the volume constant. After computing the second variation formulas for this particular variation, he observed that the area decreases unless the hypersurface is a round sphere. By using dilations of anisotropic parallel hypersurfaces, Palmer [31] proved the uniqueness of the Wulff shape as a compact, two-sided, anisotropic stable hypersurface immersed in $\mathbb {R}^{n+1}$. The variation of a hypersurface $\Sigma$ by anisotropic parallels is defined as $\psi _t(p):=p+t\,N_K(p)$, where $p\in \Sigma$, $t\in {\mathbb {R}}$ and $N_K$ is the anisotropic normal introduced in (2.2). With a similar argument, Koiso [19, Thm. 4], [21, Thm. 1] was able to classify compact anisotropic stable hypersurfaces disjoint from the edges in a wedge-shape domain $\Omega$ of $\mathbb {R}^{n+1}$. Moreover, after a suitable translation, the same variation allows to treat the capillary case. We remark that such a deformation is possible because $\partial \Omega$ consists of finitely many pieces of hyperplanes.

In a convex cone $\mathcal {C}$ different from a half-space, Palmer’s deformation does not work in general. Indeed, even though $N_K$ is tangent to $\partial \mathcal {C}$ along $\partial \Sigma$, we cannot ensure that $\partial \mathcal {C}$ contains a small anisotropic normal segment $p+t\,N_K(p)$ centered at any point $p\in \partial \Sigma$. To solve this difficulty we replace the anisotropic parallels deformation of $\Sigma$ with a variation $\Sigma _t:=\psi _t(\Sigma )$ associated to the one-parameter group $\{\psi _t\}_{t\in {\mathbb {R}}}$ of a vector field X on $\mathbb {R}^{n+1}$ such that $X(0)=0$, X is tangent on $\partial \mathcal {C}\setminus \{0\}$ and $X_{|\Sigma }=N_K$. After this, as $\partial \mathcal {C}$ is invariant under dilations centered at 0, we can apply to $\Sigma _t$ a suitable dilation in order to construct a deformation that preserves both, $\partial \mathcal {C}$ and the volume of $\Sigma$. To finish the proof we compute the second derivative of the anisotropic area for this variation, and we discover that it is strictly negative unless $\Sigma$ is contained in the Wulff shape (up to translation and homothety). In the isotropic case this is a new demonstration of the stability result by Ritoré and the author [33] when we consider immersed hypersurfaces without singularities.

We emphasize that, to prove Theorem 4.8, we only need to calculate $A_K''(0)$ for deformations associated to a smooth vector field X on $\mathbb {R}^{n+1}$ which is tangent on $\partial \mathcal {C}\setminus \{0\}$ and satisfies $X_{|\Sigma }=N_K$. This is done in Proposition 3.3 for arbitrary compact hypersurfaces with non-empty boundary in $\mathbb {R}^{n+1}$. As a difference with respect to the anisotropic parallels deformation, a boundary term involving the acceleration vector field Z appears. For anisotropic stationary hypersurfaces in $\mathcal {C}$ we see in Proposition 4.5 that this term is related to the extrinsic Euclidean geometry of $\partial \mathcal {C}$. As the formula for $A_K''(0)$ in Proposition 4.5 is valid for any smooth domain $\Omega$, this shows that the stability condition is more restrictive when $\Omega$ is convex. It is worth mentioning that, in most of the aforementioned works about anisotropic capillary hypersurfaces in a Euclidean domain $\Omega$, this boundary term had no relevance since $\partial \Omega$ was contained in the union of finitely many hyperplanes.

Finally, we remark that the proof of Theorem 4.8 is still valid in non-smooth convex cones, provided the boundary $\partial \Sigma$ of an anisotropic stable hypersurface $\Sigma$ is inside a smooth open part of $\partial \mathcal {C}$.

The paper is organized into three sections besides this introduction. In Sect. 2 we review some basic facts about the anisotropic geometry of hypersurfaces. In Sect. 3 we compute the second derivative of the anisotropic area for certain deformations of a compact hypersurface with non-empty boundary. Finally, in Sect. 4 we prove our classification result of anisotropic stable hypersurfaces in convex solid cones.

2 Preliminaries

In this section we gather some definitions and results that will be needed throughout this work. Starting with a Euclidean convex body we will use its support function and metric projection to introduce the anisotropic area and notions of anisotropic extrinsic geometry for two-sided hypersurfaces. The relation between this point of view and previous approaches is shown in Remark 2.3.

By a convex body (about the origin) we mean a compact convex set $K\subset \mathbb {R}^{n+1}$ containing 0 in its interior. The associated support function $h_K:\mathbb {R}^{n+1}\rightarrow {\mathbb {R}}$ is given by

$$\begin{aligned} h_K(w):=\max \{\big \langle u,w\big \rangle ;\,u\in K\}, \end{aligned}$$

where $\big \langle \cdot ,\cdot \big \rangle$ is the standard scalar product in $\mathbb {R}^{n+1}$. This defines an asymmetric norm in $\mathbb {R}^{n+1}$ (which is a norm when K is centrally symmetric about 0), see [35, Sect. 1.7.1]. For any $w\ne 0$, the supporting hyperplane of K with exterior normal w is the set

$$\begin{aligned} \Pi _w:=\{p\in \mathbb {R}^{n+1};\,\big \langle p,w\big \rangle =h_K(w)\}. \end{aligned}$$

The corresponding support set is $\Pi _w\cap K$. This set is not empty because K is compact. If $\Pi _w\cap K$ is a single point p, then $h_K$ is differentiable at w, and its gradient satisfies $(\nabla h_K)(w)=p$, see [35, Cor. 1.7.3].

Henceforth, we suppose that K is a strictly convex body with smooth boundary $\partial K$. Thus, for any $w\ne 0$, there is a point $\pi _K(w)\in \partial K$ for which $\Pi _w\cap K=\{\pi _K(w)\}$. This provides a map $\pi _K:\mathbb {R}^{n+1}_*\rightarrow \partial K$, which is called the K-projection, and verifies these identities

$$\begin{aligned} \begin{aligned} \big \langle \pi _K(w),w\big \rangle&=h_K(w),\quad \text {for any } w\ne 0, \\ (\nabla h_K)(w)&=\pi _K(w), \quad \text {for any } w\ne 0. \end{aligned} \end{aligned}$$

(2.1)

On the other hand, if $\eta _K$ is the outer unit normal on $\partial K$, then $(\eta _K\circ \pi _K)(w)=w$ for any $w\in \mathbb {S}^n$. This equality and the fact that $\pi _K(\lambda \,w)=\pi _K(w)$ for any $w\ne 0$ and $\lambda >0$ entail that $\pi _K(w)=\eta _K^{-1}(w/|w|)$ for any $w\ne 0$ (here $|\cdot |$ stands for the Euclidean norm). As $\eta _K:\partial K\rightarrow \mathbb {S}^n$ is a diffeomorphism, the support function $h_K$ is $C^\infty$ on $\mathbb {R}^{n+1}_*$. Observe that, when K is the round unit ball about 0, then $h_K(w)=|w|$ and $\pi _K(w)=w/|w|$ for any $w\ne 0$, whereas $\eta _K(p)=p$ for any $p\in \partial K=\mathbb {S}^n$.

Let $\varphi _0:\Sigma \rightarrow \mathbb {R}^{n+1}$ be a smooth two-sided immersed hypersurface, possibly with smooth boundary $\partial \Sigma$. Most of the time we will omit the map $\varphi _0$. We will also identify any set $S\subseteq \Sigma$ with $\varphi _0(S)$, and the tangent space $T_p\Sigma$ at a point $p\in \Sigma$ with $(d\varphi _0)_p(T_p\Sigma )$. For a fixed smooth unit normal vector field N on $\Sigma$, the associated shape operator at p is the endomorphism $B_p:T_p\Sigma \rightarrow T_p\Sigma$ introduced by $B_p(w):=-D_wN$. Here D denotes the Levi-Civita connection for the Euclidean metric. It is well known that $B_p$ is self-adjoint with respect to the metric induced by the scalar product in $\mathbb {R}^{n+1}$.

The anisotropic Gauss map or anisotropic normal in $\Sigma$ is the map $N_K:\Sigma \rightarrow \partial K$ given by

$$\begin{aligned} N_K:=\pi _K\circ N. \end{aligned}$$

(2.2)

By setting $\varphi _K:=h_K(N)$ we infer from (2.1) that

$$\begin{aligned} \big \langle N_K,N\big \rangle =\varphi _K\quad \text {on } \Sigma . \end{aligned}$$

(2.3)

When $\partial \Sigma \ne \emptyset$ we introduce the anisotropic conormal by setting

$$\begin{aligned} \nu _K:=\varphi _K\,\nu -\big \langle N_K,\nu \big \rangle \,N, \end{aligned}$$

(2.4)

where $\nu$ is the inner conormal along $\partial \Sigma$. Note that $\nu _K$ is a normal vector to $\partial \Sigma$ with $\big \langle N_K,\nu _K\big \rangle =0$.

For any $p\in \Sigma$ we have $T_{N_K(p)}(\partial K)=[\eta _K(N_K(p))]^\bot =N(p)^\bot =T_p\Sigma$, so that the differential $(dN_K)_p$ is an endomorphism of $T_p\Sigma$. The anisotropic shape operator at p is defined by

$$\begin{aligned} (B_K)_p:=-(dN_K)_p=(d\pi _K)_{N(p)}\circ B_p. \end{aligned}$$

(2.5)

The anisotropic mean curvature at p is the number

$$\begin{aligned} H_K(p):=\frac{\text {tr}((B_K)_p)}{n}=-\frac{({{\,\textrm{div}\,}}_\Sigma N_K)(p)}{n}, \end{aligned}$$

(2.6)

where ${{\,\textrm{div}\,}}_\Sigma$ is the divergence relative to $\Sigma$ and $\text {tr}(f)$ is the trace of an endomorphism $f:T_p\Sigma \rightarrow T_p\Sigma$. It is known, see for instance Palmer [31, p. 3666], that

$$\begin{aligned} \text {tr}\big ((B_K)_p^2\big )\geqslant nH_K(p)^2, \quad \text {for any } p\in \Sigma , \end{aligned}$$

(2.7)

and equality holds if and only p is an anisotropic umbilical point, i.e., $(B_K)_p$ is a multiple of the identity map in $T_p\Sigma$. This fact has a short proof that we reproduce here for the sake of completeness. Since $\nabla h_K=\pi _K$ in $\mathbb {R}^{n+1}_*$ and K is a strictly convex body, then $(d\pi _K)_{N(p)}$ can be represented as a positive definite symmetric matrix of order n. Thus, there exist an orthonormal basis $\{e_1,\ldots ,e_n\}$ of $T_p\Sigma$ and $\{\lambda _1,\ldots ,\lambda _n\}\subset {\mathbb {R}}^+$ such that $(d\pi _K)_{N(p)}(e_j)=\lambda _j\,e_j$ for any $j=1,\ldots ,n$. From (2.5) we get $(B_K)_p(e_i)=\sum _{j=1}^n\sigma _{ij}\,\lambda _j\,e_j$, where $\sigma _{ij}:=\big \langle B_p(e_i),e_j\big \rangle$. By using the Cauchy-Schwarz inequality in ${\mathbb {R}}^n$ with the vectors $(\sigma _{11}\lambda _1,\ldots ,\sigma _{nn}\lambda _n)$ and $(1,\ldots ,1)$, we obtain

$$\begin{aligned}&\text {tr}(B_K^2)-nH_K^2=\sum _{i,j=1}^n\sigma ^2_{ij}\,\lambda _i\,\lambda _j-\frac{1}{n}\, \left( \sum _{i=1}^n\sigma _{ii}\,\lambda _i\right) ^2\\&\quad \geqslant \sum _{i,j=1}^n\sigma ^2_{ij}\, \lambda _i\,\lambda _j-\sum _{i=1}^n\sigma ^2_{ii}\,\lambda ^2_i=\sum _{i\ne j}\sigma ^2_{ij}\,\lambda _i\,\lambda _j\geqslant 0. \end{aligned}$$

Equality holds if and only if $\sigma _{ij}=0$ for any $i\ne j$ and $\sigma _{ii}\,\lambda _i=\sigma _{jj}\,\lambda _j$ for any $i,j=1,\ldots ,n$. By denoting $\alpha :=\sigma _{ii}\,\lambda _i$, this is equivalent to $(B_K)_p(w)=\alpha \,w$ for any $w\in T_p\Sigma$, as desired.

Examples 2.1

• (i). If K is the round unit ball about 0, then $N_K=N$, $B_K=B$ and $H_K$ equals the Euclidean mean curvature of $\Sigma$.

• (ii). For a hyperplane $\Sigma \subset \mathbb {R}^{n+1}$ oriented with a unit normal N, the map $N_K$ is constant, so that $B_K=0$ and $H_K=0$ in $\Sigma$.

• (iii). Consider the hypersurface $\Sigma =\partial K$ with outer unit normal $N=\eta _K$. Then $N_K(p)=p$, $(B_K)_p(w)=-w$ and $H_K(p)=-1$, for any $p\in \Sigma$ and $w\in T_p\Sigma$. Observe that, if K is not centrally symmetric about 0, and we take the inner normal $N=-\eta _K$ on $\Sigma$, then similar equalities need not hold and $H_K$ may be nonconstant. So, the role of the chosen unit normal N is very important for the computations.

The previous examples show that $\partial K$ and Euclidean hyperplanes are anisotropic umbilical hypersurfaces, i.e., all of its points are anisotropic umbilical. Next, we provide a converse statement after Palmer [31, p. 3666] and Clarenz [8, p. 358].

Proposition 2.2

Let $\Sigma$ be a two-sided, connected, anisotropic umbilical hypersurface immersed in $\mathbb {R}^{n+1}$, and such that $H_K$ is constant. If $H_K=0$ then $\Sigma$ is contained inside a hyperplane. If $H_K\ne 0$ then, up to a translation and a dilation centered at 0, we have $\Sigma \subset \partial K$.

Proof

For any $p\in \Sigma$, there is $\alpha (p)\in {\mathbb {R}}$ such that $(B_K)_p(w)=\alpha (p)\,w$ for any $w\in T_p\Sigma$. By definition (2.6) it is clear that $\alpha (p)=H_K$ for any $p\in \Sigma$. By Eq. (2.5) we deduce that the differential of $N_K+H_K\,\text {Id}$ vanishes on $\Sigma$. In case $H_K=0$ this implies that $N_K$ is a constant vector n, and so $\Sigma$ is within a hyperplane with unit normal $\eta _K(n)$. When $H_K\ne 0$ we can find $c\in \mathbb {R}^{n+1}$ satisfying

$$\begin{aligned} p=\frac{c}{H_K}-\frac{1}{H_K}\,N_K(p), \quad \text {for any }p\in \Sigma . \end{aligned}$$

From here we infer that $\Sigma \subseteq (c/H_K)-(1/H_K)\,(\partial K)$, as we claimed. $\square$

We finish this section with the notions of anisotropic area and algebraic volume for a two-sided hypersurface $\Sigma$. For a fixed smooth unit normal N on $\Sigma$, the anisotropic area is given by

$$\begin{aligned} A_K(\Sigma ):=\int _\Sigma \varphi _K\,d\Sigma =\int _\Sigma h_K(N)\,d\Sigma , \end{aligned}$$

where $d\Sigma$ denotes the area element of $\Sigma$. This coincides with the Euclidean area of $\Sigma$ when K is the round unit ball about 0. When K is not centrally symmetric about 0 the value of $A_K(\Sigma )$ may depend on the normal N that we consider on $\Sigma$. On the other hand, for a compact hypersurface $\Sigma$, we follow Barbosa and do Carmo [2, Eq. (2.2)] to define

$$\begin{aligned} V(\Sigma ):=\frac{1}{n+1}\int _{\Sigma }\big \langle p,N(p)\big \rangle \,d\Sigma . \end{aligned}$$

(2.8)

When $\Sigma$ is embedded and $\Sigma '\subset \Sigma$ is a sufficiently small open set, we can apply the divergence theorem to the position vector field $X(p)=p$ in the cone $\mathcal {C}$ over $\Sigma '$ with vertex at 0 to conclude that $|V(\Sigma ')|$ equals the Lebesgue measure of $\mathcal {C}$. If $\Sigma$ is immersed then it is possible that $V(\Sigma )=0$.

Consider a dilation $\delta _\lambda (p):=\lambda \,p$ with $\lambda >0$ and $p\in \mathbb {R}^{n+1}$. A unit normal $N_\lambda$ on the hypersurface $\delta _\lambda (\Sigma )$ is determined by $N_\lambda (\delta _\lambda (p)):=N(p)$ for any $p\in \Sigma$. Therefore, the change of variables formula and the fact that the Jacobian of the diffeomorphism $\delta _{\lambda |\Sigma }:\Sigma \rightarrow \delta _\lambda (\Sigma )$ equals $\lambda ^n$, entail that

$$\begin{aligned} \begin{aligned} A_K(\delta _\lambda (\Sigma ))&=\lambda ^n\,A_K(\Sigma ), \\ V(\delta _\lambda (\Sigma ))&=\lambda ^{n+1}\,V(\Sigma ). \end{aligned} \end{aligned}$$

(2.9)

These identities will play a relevant role in the proof of our main result in Sect. 4.

Remark 2.3

It is usual to introduce the anisotropic area $A_F$ associated to an arbitrary function $F:\mathbb {S}^n\rightarrow {\mathbb {R}}^+$. This has the expression $A_F(\Sigma ):=\int _\Sigma F(N)\,d\Sigma$. Convexity assumptions on F are necessary to deduce fine properties for $A_F$ and for the F-Laplacian, see for instance Maggi [27, Ch. 20] or Palmer [31]. Other ellipticity conditions for more general integrands, including equivalences between them and first variation formulas for the corresponding energies are found in De Rosa and Kolasiński [10], and De Lellis et al. [9]. By extending F as a 1-homogeneous function we get an asymmetric norm $\Phi$ in $\mathbb {R}^{n+1}$. The Wulff shape for $\Phi$ is the convex body $\mathcal {W}_\Phi$ supported by $\Phi$, see [27, Eq. (20.8)]. As we remembered in the Introduction, $\mathcal {W}_\Phi$ minimizes the anisotropic perimeter among sets of the same Euclidean volume. In our context we have $F=h_{K|\mathbb {S}^n}$, $\Phi =h_K$ and $\mathcal {W}_\Phi =K$. So, the initial convex body K is the optimal shape for the anisotropic area $A_K$ defined from its support function $h_K$. In Palmer [31, Sect. 1] the Wulff hypersurface of F is the strictly convex hypersurface defined as $\phi (\mathbb {S}^n)$, where $\phi (w):=F(w)\,w+(\nabla _{\mathbb {S}^n}F)(w)$. When $F=h_{K|\mathbb {S}^n}$ it follows from Eq. (2.1) that $\phi =\pi _K$ on $\mathbb {S}^n$, so that the corresponding Wulff hypersurface $\phi (\mathbb {S}^n)$ equals $\partial K$.

3 A second variation formula for the anisotropic area

In this section we compute the second derivative of the anisotropic area for certain deformations of an immersed hypersurface with non-empty boundary. The resulting formula will be employed in the proof of our main result in Theorem 4.8. We begin with some preliminary definitions.

A flow (of diffeomorphisms) in $\mathbb {R}^{n+1}$ is a smooth map $\phi :\mathbb {R}^{n+1}\times {\mathbb {R}}\rightarrow \mathbb {R}^{n+1}$ such that $\phi (p,0)=p$ for any $p\in \mathbb {R}^{n+1}$, and the map $\phi _t:\mathbb {R}^{n+1}\rightarrow \mathbb {R}^{n+1}$ defined by $\phi _t(p):=\phi (p,t)$ is a diffeomorphism for any $t\in {\mathbb {R}}$. Usually we will denote a flow by $\{\phi _t\}_{t\in {\mathbb {R}}}$. The associated velocity vector field is given by

$$\begin{aligned} X(p):=\frac{d}{dt}\bigg |_{t=0}\phi _t(p),\quad \text {for any }p\in \mathbb {R}^{n+1}. \end{aligned}$$

When Y is a smooth complete vector field on $\mathbb {R}^{n+1}$, the corresponding one-parameter group of diffeomorphisms $\{\phi _t\}_{t\in {\mathbb {R}}}$ is a flow with velocity Y.

For a hypersurface $\Sigma$ immersed in $\mathbb {R}^{n+1}$ the variation of $\Sigma$ induced by a flow $\{\phi _t\}_{t\in {\mathbb {R}}}$ is the family $\{\Sigma _t\}_{t\in {\mathbb {R}}}$, where $\Sigma _t:=\phi _t(\Sigma )$ for any $t\in {\mathbb {R}}$. The flow is compactly supported on $\Sigma$ if there is a compact set $C\subseteq \Sigma$ such that $\phi _t(p)=p$ for any $p\in \Sigma \setminus C$ and $t\in {\mathbb {R}}$. If $\Sigma$ is a two-sided hypersurface with unit normal N then, along the variation $\{\Sigma _t\}_{t\in {\mathbb {R}}}$, we can find a smooth vector field $\overline{N}$ whose restriction to $\Sigma _t$ provides a unit normal $N_t$ with $N_0=N$. For a fixed smooth strictly convex body $K\subset \mathbb {R}^{n+1}$ with support function $h_K$, the anisotropic area functional for the variation $\{\Sigma _t\}_{t\in {\mathbb {R}}}$ is the function

$$\begin{aligned} A_K(t):=A_K(\Sigma _t)=\int _{\Sigma _t}h_K(N_t)\,d\Sigma _t=\int _\Sigma h_K(\overline{N}\circ \phi _t)\,\text {Jac}\,\phi _t\,d\Sigma , \end{aligned}$$

(3.1)

where $\text {Jac}\,\phi _t$ is the Jacobian of the diffeomorphism $\phi _{t|\Sigma }:\Sigma \rightarrow \Sigma _t$.

Our objective is to provide a useful expression of $A_K''(0)$ for some variations. For that we must compute $A_K'(0)$ first. We need some preliminary calculations that we gather below.

Lemma 3.1

Let $K\subset \mathbb {R}^{n+1}$ be a smooth strictly convex body with support function $h_K$. For a two-sided hypersurface $\Sigma$ immersed in $\mathbb {R}^{n+1}$ with unit normal N, the function $\varphi _K:=h_K(N)$ satisfies

$$\begin{aligned} \nabla _\Sigma \varphi _K=-B(N_K^\top ), \end{aligned}$$

(3.2)

where $\nabla _\Sigma$ is the gradient relative to $\Sigma$ and $N_K^\top$ is the tangent projection of the anisotropic normal $N_K$. Moreover, for any smooth vector field U with compact support on $\Sigma$ and normal component $u:=\big \langle U,N\big \rangle$, we have

$$\begin{aligned} \int _\Sigma {{\,\textrm{div}\,}}_\Sigma (\varphi _K\,U)\,d\Sigma =-\int _\Sigma nH_K\,u\,d\Sigma +\int _\Sigma \big \langle N_K,\nabla _\Sigma u\big \rangle \,d\Sigma -\int _{\partial \Sigma }\big \langle U,\nu _K\big \rangle \,d(\partial \Sigma ), \end{aligned}$$

(3.3)

where $H_K$ is the anisotropic mean curvature and $\nu _K$ is the anisotropic conormal.

Proof

Take a point $p\in \Sigma$ and a vector $w\in T_p\Sigma$. By using (2.1), (2.2) and that the shape operator $B_p$ is a self-adjoint endomorphism of $T_p\Sigma$, we obtain

$$\begin{aligned} \begin{aligned} \big \langle (\nabla _\Sigma \varphi _K)(p),w\big \rangle&=\big \langle (\nabla h_K)(N(p)),D_wN\big \rangle =-\big \langle \pi _K(N(p)),B_p(w)\big \rangle \\&=-\big \langle N_K(p),B_p(w)\big \rangle =-\big \langle B_p(N^\top _K(p)),w\big \rangle . \end{aligned} \end{aligned}$$

(3.4)

This implies (3.2). Let us prove (3.3). For any $w\in \mathbb {R}^{n+1}$ we denote by $w^\top$ and $w^\bot$ the projections of w with respect to $T_p\Sigma$ and $(T_p\Sigma )^\bot$, respectively. For a smooth vector field U on $\Sigma$, note that

$$\begin{aligned} \varphi _K\,U=\varphi _K\,U^\top +\varphi _K\,u\,N=\varphi _K\,U^\top +u\,N_K^\bot =\varphi _K\,U^\top +u\,N_K-u\,N_K^\top , \end{aligned}$$

where in the second equality we have employed (2.3). By taking divergences relative to $\Sigma$ and having in mind (2.6), we get

$$\begin{aligned} {{\,\textrm{div}\,}}_\Sigma (\varphi _K\,U)={{\,\textrm{div}\,}}_\Sigma (\varphi _K\,U^\top )-nH_K\,u+\big \langle N_K,\nabla _\Sigma u\big \rangle -{{\,\textrm{div}\,}}_\Sigma (u\,N_K^\top ). \end{aligned}$$

From here the desired formula follows by using (2.4) and the divergence theorem on $\Sigma$. $\square$

Next, we compute the first variation of $A_K$. This was previously derived by many authors, see for instance Clarenz [7, Sect. 1], or Koiso and Palmer [22, Proof of Prop. 3.1]. As in Koiso [20, Lem. 9.1], our formula holds for arbitrary deformations of a Euclidean hypersurface with non-empty boundary. We include a short proof that will be helpful in the subsequent calculus of $A_K''(0)$.

Proposition 3.2

Let $K\subset \mathbb {R}^{n+1}$ be a smooth strictly convex body, $\Sigma$ a two-sided immersed hypersurface with boundary, and $\{\phi _t\}_{t\in {\mathbb {R}}}$ a flow in $\mathbb {R}^{n+1}$ with compact support on $\Sigma$. Then, we have

$$\begin{aligned} A_K'(0)=-\int _\Sigma nH_K\,u\,d\Sigma -\int _{\partial \Sigma }\big \langle X,\nu _K\big \rangle \,d(\partial \Sigma ), \end{aligned}$$

where $H_K$ is the anisotropic mean curvature, $u:=\big \langle X,N\big \rangle$ is the normal component of the velocity vector field X, and $\nu _K$ is the anisotropic conormal along $\partial \Sigma$.

Proof

Recall that we denote $\varphi _K:=h_K(N)$. For a fixed point $p\in \Sigma$, we define

$$\begin{aligned} h_p(t):=h_K(N_t\circ \phi _t)(p), \quad j_p(t):=(\text {Jac}\,\phi _t)(p), \quad \text {for any }t\in {\mathbb {R}}. \end{aligned}$$

(3.5)

By differentiating under the integral sign in (3.1) and taking into account that $\phi _0=\text {Id}$, we obtain

$$\begin{aligned} A_K'(0)=\int _\Sigma \big (h_p'(0)+\varphi _K(p)\,j_p'(0)\big )\,d\Sigma . \end{aligned}$$

(3.6)

On the one hand it is well known, see Simon [36, §9], that

$$\begin{aligned} j_p'(0)=\frac{d}{dt}\bigg |_{t=0}(\text {Jac}\,\phi _t)(p)=({{\,\textrm{div}\,}}_\Sigma X)(p). \end{aligned}$$

(3.7)

On the other hand, from Eqs. (2.1) and (2.2), it follows that

$$\begin{aligned} h_p'(0)=\big \langle (\nabla h_K)(N(p)),D_{X(p)}\overline{N}\big \rangle =\big \langle N_K(p),D_{X(p)}\overline{N}\big \rangle . \end{aligned}$$

The computation of $D_{X(p)}\overline{N}$ is found in [34, Lem. 4.1(1)]. We get

$$\begin{aligned} D_{X(p)}\overline{N}=\frac{d}{dt}\bigg |_{t=0}(N_t\circ \phi _t)(p)=-(\nabla _\Sigma u)(p)-B_p(X^\top (p)), \end{aligned}$$

(3.8)

where $X^\top (p)$ is the projection of X(p) onto $T_p\Sigma$. By substituting this information into the previous expression for $h_p'(0)$ and having in mind the third equality in (3.4), we arrive at

$$\begin{aligned} h_p'(0)=-\big \langle N_K,\nabla _\Sigma u\big \rangle (p)+\big \langle \nabla _\Sigma \varphi _K,X^\top \big \rangle (p). \end{aligned}$$

(3.9)

Thus, the integrand in (3.6) is the evaluation at p of the function

$$\begin{aligned} -\big \langle N_K,\nabla _\Sigma u\big \rangle +\big \langle \nabla _\Sigma \varphi _K,X^\top \big \rangle +\varphi _K\,{{\,\textrm{div}\,}}_\Sigma X=-\big \langle N_K,\nabla _\Sigma u\big \rangle +{{\,\textrm{div}\,}}_\Sigma (\varphi _K\,X). \end{aligned}$$

The proof finishes by applying the formula in (3.3) with $U=X$. $\square$

Second variation formulas for the anisotropic area under different hypotheses on $\Sigma$ and the deformation can be found in Koiso and Palmer [22, Prop. 3.3], [23, Prop. 3.3], Barbosa and Silva [1, Prop. 3], and Guo and Xia [14, Prop. 3.5]. In this work we only need to compute $A_K''(0)$ when we move a hypersurface $\Sigma$ with non-empty boundary by means of some special flows.

Proposition 3.3

Let $K\subset \mathbb {R}^{n+1}$ be a smooth strictly convex body, $\Sigma$ a compact two-sided immersed hypersurface with boundary, and X a smooth complete vector field on $\mathbb {R}^{n+1}$ such that $X_{|\Sigma }=N_K$. Then, for the one-parameter group of diffeomorphisms $\{\phi _t\}_{t\in {\mathbb {R}}}$ associated to X, we have

$$\begin{aligned} A_K''(0)=\int _\Sigma \left( n^2H_K^2-tr (B_K^2)\right) \varphi _K\,d\Sigma -\int _\Sigma n H_K\,v\,d\Sigma -\int _{\partial \Sigma }\big \langle Z,\nu _K\big \rangle \,d(\partial \Sigma ), \end{aligned}$$

where $H_K$ is the anisotropic mean curvature, $B_K$ is the anisotropic shape operator, $\nu _K$ is the anisotropic conormal and $v:=\big \langle Z,N\big \rangle$ is the normal component of the vector field $Z:=D_XX$.

Proof

For any $w\in \mathbb {R}^{n+1}$ the notations $w^\top$ and $w^\bot$ will stand for the projections of w onto $T\Sigma$ and $(T\Sigma )^\bot$, respectively. For a given point $p\in \Sigma$ we define $h_p(t)$ and $j_p(t)$ as in (3.5). By differentiating under the integral sign twice in (3.1) and taking into account that $\phi _0=\text {Id}$, we get

$$\begin{aligned} A_K''(0)=\int _\Sigma \big (h_p''(0)+2h_p'(0)\,j_p'(0)+\varphi _K(p)\,j_p''(0)\big )\,d\Sigma . \end{aligned}$$

(3.10)

Let us compute all the derivatives in the integrand above.

For the calculus of $j_p'(0)$ and $j_p''(0)$ we refer the reader to Simon [36, §9]. From (3.7), (2.6), and the fact that $X_{|\Sigma }=N_K$, we obtain

$$\begin{aligned} j_p'(0)=({{\,\textrm{div}\,}}_\Sigma X)(p)=({{\,\textrm{div}\,}}_\Sigma N_K)(p)=-nH_K(p). \end{aligned}$$

(3.11)

On the other hand, we have

$$\begin{aligned} j_p''(0)=({{\,\textrm{div}\,}}_\Sigma Z)(p)+({{\,\textrm{div}\,}}_\Sigma X)^2(p)+\sum _{i=1}^n|(D_{e_i}X)^\bot |^2-\sum _{i,j=1}^n\big \langle D_{e_i}X,e_j\big \rangle \,\big \langle D_{e_j}X,e_i\big \rangle , \end{aligned}$$

where $\{e_1,\ldots ,e_n\}$ is an orthonormal basis of $T_p\Sigma$. Equation (2.5) and the fact that $X_{|\Sigma }=N_K$ yield

$$\begin{aligned} D_{e_i}X=D_{e_i}N_K=-(B_K)_p(e_i), \end{aligned}$$

so that $(D_{e_i}X)^\bot =0$ for any $i=1,\ldots ,n$. It is also clear that

$$\begin{aligned} \sum _{i,j=1}^n\big \langle D_{e_i}X,e_j\big \rangle \,\big \langle D_{e_j}X,e_i\big \rangle = \text {tr}\big ((B_K)^2_p\big ). \end{aligned}$$

All this together with (3.11) shows that

$$\begin{aligned} j_p''(0)=\big (\!{{\,\textrm{div}\,}}_\Sigma Z+n^2H_K^2-\text {tr}(B_K^2)\big )(p). \end{aligned}$$

(3.12)

Next, we compute $h_p'(0)$ and $h_p''(0)$. Note that $\big \langle X,N\big \rangle =\big \langle N_K,N\big \rangle =\varphi _K$ by (2.3). Hence, Eq. (3.9) implies that

$$\begin{aligned} h_p'(0)=-\big \langle N_K,\nabla _\Sigma \varphi _K\big \rangle (p)+\big \langle \nabla _\Sigma \varphi _K,N_K^\top \big \rangle (p)=0. \end{aligned}$$

(3.13)

In order to calculate $h_p''(0)$ we need an expression for $h_p'(t)$. From the definition in (3.5) we deduce

$$\begin{aligned} h_p'(t)=\big \langle (\nabla h_K)((N_t\circ \phi _t)(p)),(D_{X}\overline{N})(\phi _t(p))\big \rangle =\big \langle \pi _K((N_t\circ \phi _t)(p)),(D_{X}\overline{N})(\phi _t(p))\big \rangle \end{aligned}$$

because of Eq. (2.1). This entails that

$$\begin{aligned} h_p''(0)=\big \langle (d\pi _K)_{N(p)}(D_{X(p)}\overline{N}), D_{X(p)}\overline{N}\big \rangle +\big \langle N_K(p),D_{X(p)}D_X\overline{N}\big \rangle . \end{aligned}$$

As $\big \langle X,N\big \rangle =\varphi _K$ on $\Sigma$, Eqs. (3.8) and (3.2) lead to

$$\begin{aligned} D_{X(p)}\overline{N}=-(\nabla _\Sigma \varphi _K)(p)-B_p(N_K^\top (p))=0. \end{aligned}$$

(3.14)

On the other hand, by Lemma 3.4 below we know that

$$\begin{aligned} D_{X(p)}D_X\overline{N}=-(\nabla _\Sigma v)(p)-B_p(Z^\top (p)), \end{aligned}$$

and so

$$\begin{aligned} h_p''(0)&=-\big \langle N_K,\nabla _\Sigma v\big \rangle (p)-\big \langle N_K(p),B_p(Z^\top (p))\big \rangle \nonumber \\&=-\big \langle N_K,\nabla _\Sigma v\big \rangle (p)+\big \langle \nabla _\Sigma \varphi _K,Z^\top \big \rangle (p), \end{aligned}$$

(3.15)

where we have employed the third equality in Eq. (3.4).

Now, by having in mind (3.15), (3.13) and (3.12), we conclude that the integrand in (3.10) is the evaluation at p of the function

$$\begin{aligned} -\big \langle N_K,\nabla _\Sigma v\big \rangle +{{\,\textrm{div}\,}}_\Sigma (\varphi _K\,Z)+\big (n^2H^2_K-\text {tr}(B_K^2)\big )\,\varphi _K. \end{aligned}$$

From here the proof finishes by applying the formula in (3.3) with $U=Z$. $\square$

Lemma 3.4

In the conditions of Proposition 3.3, for any $p\in \Sigma$, we have

$$\begin{aligned} D_{X(p)}D_X\overline{N}=-(\nabla _\Sigma v)(p)-B_p(Z^\top (p)). \end{aligned}$$

Proof

Take an orthonormal basis $\{e_1,\ldots ,e_n\}$ of $T_p\Sigma$. We use the flow $\{\phi _t\}_{t\in {\mathbb {R}}}$ associated to X to construct, for any $i=1,\ldots ,n$, a smooth vector field $E_i$ around p which is tangent on any $\Sigma _t\!\!:=\phi _t(\Sigma )$ while satisfying $E_i(p)=e_i$ and $[X,E_i]=0$ (here $[\cdot ,\cdot ]$ stands for the Lie bracket of vector fields in $\mathbb {R}^{n+1}$). It is clear that

$$\begin{aligned} D_{X(p)}D_X\overline{N}=\sum _{i=1}^n\big \langle D_{X(p)}D_X\overline{N},e_i\big \rangle \,e_i +\big \langle D_{X(p)}D_X\overline{N},N(p)\big \rangle \,N(p). \end{aligned}$$

(3.16)

We will compute the different terms in the previous equation.

Since $|\overline{N}|^2=1$ we get $\big \langle D_X\overline{N},\overline{N}\big \rangle =0$. By differentiating and applying Eq. (3.14), we deduce

$$\begin{aligned} 0=\big \langle D_{X(p)}D_X\overline{N},N(p)\big \rangle +|D_{X(p)}\overline{N}|^2=\big \langle D_{X(p)}D_X\overline{N},N(p)\big \rangle . \end{aligned}$$

(3.17)

On the other hand, we differentiate twice with respect to X in equality $\big \langle \overline{N},E_i\big \rangle =0$. By taking into account that $D_{X(p)}\overline{N}=0$ (see Eq. (3.14)) and $[X,E_i]=0$, we obtain

$$\begin{aligned} \big \langle D_{X(p)}D_X\overline{N},e_i\big \rangle =-\big \langle N(p),D_{X(p)}D_XE_i\big \rangle =-\big \langle N(p),D_{X(p)}D_{E_i}X\big \rangle . \end{aligned}$$

Since the Riemann curvature tensor vanishes for the standard metric in $\mathbb {R}^{n+1}$, we infer

$$\begin{aligned} 0=D_XD_{E_i}X-D_{E_i}D_XX-D_{[X,E_i]}X=D_XD_{E_i}X-D_{E_i}Z \end{aligned}$$

because $D_XX=Z$. This shows that $D_{X(p)}D_{E_i}X=D_{e_i}Z$. As a consequence

$$\begin{aligned} \big \langle D_{X(p)}D_X\overline{N},e_i\big \rangle&=-\big \langle N(p),D_{e_i}Z\big \rangle =-e_i\big (\big \langle Z,N\big \rangle \big )+\big \langle Z(p),D_{e_i}N\big \rangle \nonumber \\&=-\big \langle (\nabla _\Sigma v)(p),e_i\big \rangle -\big \langle Z^\top (p),B_p(e_i)\big \rangle \nonumber \\&=-\big \langle (\nabla _\Sigma v)(p),e_i\big \rangle -\big \langle B_p(Z^\top (p)),e_i\big \rangle . \end{aligned}$$

(3.18)

By substituting (3.17) and (3.18) into (3.16) the proof follows. $\square$

Remarks 3.5

• (i). If $K\subset \mathbb {R}^{n+1}$ is the round unit ball about 0, then the anisotropic area coincides with the Euclidean area and the obtained formulas are well known.

• (ii). It is interesting to observe that, unless K is centrally symmetric about 0, the formulas for $A_K'(0)$ and $A_K''(0)$ may depend on the unit normal vector N fixed on $\Sigma$.

• (iii). In Proposition 4.5 we will see that, under some extra conditions, the boundary integrand in the expression of $A_K''(0)$ has a geometric interpretation.

4 Anisotropic stable hypersurfaces in solid cones

In this section we consider a Euclidean solid cone (not necessarily convex) and study compact hypersurfaces immersed in the cone and minimizing the anisotropic area up to second order for deformations preserving the volume of the hypersurface and the boundary of the cone. In this situation we will show that the variational formulas computed in Sect. 3 can be slightly simplified. After that we will prove our main theorem, where we characterize these second order minima when the cone is convex. It is worth mentioning that, excluding this classification statement and the Minkowski-type formula in Eq. (4.5), all the results in this section still hold when we replace the cone with any smooth Euclidean open set.

We begin by introducing some notation and definitions. For a domain $\mathcal {D}\subset \mathbb {S}^n$ with smooth boundary, the solid cone over $\mathcal {D}$ is the set

$$\begin{aligned} \mathcal {C}:=\{\lambda \,p;\, \lambda >0,\, p\in \mathcal {D}\}. \end{aligned}$$

This is a domain of $\mathbb {R}^{n+1}$ with boundary $\partial \mathcal {C}$ smooth away from 0. We call $\xi$ the inner unit normal along $\partial \mathcal {C}\setminus \{0\}$. Note that $\mathcal {C}$ and $\partial \mathcal {C}$ are invariant under the dilations $\delta _\lambda$ centered at 0. It is also clear that $\mathcal {C}$ coincides with an open half-space when $\mathcal {D}$ is an open hemisphere.

Let $\Sigma$ be a smooth, compact, two-sided hypersurface immersed in $\overline{\mathcal {C}}$ with smooth boundary $\partial \Sigma$ in $\partial \mathcal {C}\setminus \{0\}$. We suppose that $\Sigma \cap \partial \mathcal {C}=\partial \Sigma$, so that $\Sigma \setminus \partial \Sigma \subseteq \mathcal {C}$. We fix a smooth unit normal vector field N on $\Sigma$. The inner conormal vector of $\partial \Sigma$ in $\Sigma$ is represented by $\nu$. In the planar distribution $T(\partial \Sigma )^\bot$ we choose the orientation induced by $\{\nu ,N\}$. Thus, for any $p\in \partial \Sigma$, there is a unique $\mu (p)\in T_p(\partial \Sigma )^\bot$ such that $\{\xi (p),\mu (p)\}$ is a positively oriented orthonormal basis. Observe that $\mu$ is tangent to $\partial \mathcal {C}$ and normal to $\partial \Sigma$. It is easy to check that these equalities hold along $\partial \Sigma$

$$\begin{aligned} \begin{aligned} \nu&=(\cos \theta )\,\xi -(\sin \theta )\,\mu , \qquad \mu =-(\sin \theta )\,\nu +(\cos \theta )\,N, \\ N&=(\sin \theta )\,\xi +(\cos \theta )\,\mu ,\qquad \xi =(\cos \theta )\,\nu +(\sin \theta )\,N, \end{aligned} \end{aligned}$$

(4.1)

where $\theta$ is the oriented angle function between $\nu$ and $\xi$ in $T(\partial \Sigma )^\bot$. As a consequence, for a fixed smooth strictly convex body $K\subset \mathbb {R}^{n+1}$, the anisotropic normal $N_K$ on $\Sigma$ and the anisotropic conormal $\nu _K$ along $\partial \Sigma$ given in (2.2) and (2.4) verify

$$\begin{aligned} \big \langle N_K,\mu \big \rangle =\big \langle \nu _K,\xi \big \rangle ,\qquad \big \langle N_K,\xi \big \rangle =-\big \langle \nu _K,\mu \big \rangle . \end{aligned}$$

(4.2)

A flow of diffeomorphisms $\{\phi _t\}_{t\in {\mathbb {R}}}$ in $\mathbb {R}^{n+1}$ is admissible for $\mathcal {C}$ if $\phi _t(\partial \mathcal {C})=\partial \mathcal {C}$ and $\phi _t(0)=0$, for any $t\in {\mathbb {R}}$. The induced variation $\{\Sigma _t\}_{t\in {\mathbb {R}}}$ of $\Sigma$ satisfies $\partial \Sigma _t\subset \partial \mathcal {C}\setminus \{0\}$ and $\Sigma _t\cap \partial \mathcal {C}=\partial \Sigma _t$, for any $t\in {\mathbb {R}}$. Moreover, the velocity vector field X is tangent on $\partial \mathcal {C}\setminus \{0\}$ and vanishes at 0. We consider the anisotropic area functional $A_K(t)$ introduced in (3.1) and the volume functional V(t), which assigns to any $t\in {\mathbb {R}}$ the algebraic volume enclosed by $\Sigma _t$ as defined in (2.8). We computed $A_K'(0)$ in Proposition 3.2. On the other hand, it is well known ([2, Eq. (2.3)]) that

$$\begin{aligned} V'(0)=\int _\Sigma u\,d\Sigma , \end{aligned}$$

(4.3)

where $u:=\big \langle X,N\big \rangle$. We say that the flow preserves the volume of $\Sigma$ if V(t) is constant for any t small enough. This implies that $\int _\Sigma u\,d\Sigma =0$. Conversely, for any smooth function $u:\Sigma \rightarrow {\mathbb {R}}$ with $\int _\Sigma u\,d\Sigma =0$, there is a flow $\{\phi _t\}_{t\in {\mathbb {R}}}$ admissible for $\mathcal {C}$, preserving the volume of $\Sigma$, and such that $\big \langle X,N\big \rangle =u$ on $\Sigma$, see Barbosa and do Carmo [2, Lem. (2.4)].

A hypersurface $\Sigma$ in the previous conditions is anisotropic stationary (with respect to the fixed unit normal N) if $A_K'(0)=0$ for any admissible flow for $\mathcal {C}$ preserving the volume of $\Sigma$. We emphasize that this property may depend on the unit normal N when K is not centrally symmetric. Thus, $\Sigma$ could be anisotropic stationary with respect to N but not with respect to $-N$.

The first variational formulas in Proposition 3.2 and Eq. (4.3), together with the aforementioned construction of volume-preserving flows, lead to a characterization of anisotropic stationary hypersurfaces. Indeed, we can reason as Koiso and Palmer [22, Prop. 3.1] to deduce the next result, which generalizes the isotropic situation.

Proposition 4.1

A two-sided hypersurface $\Sigma$ immersed in $\overline{\mathcal {C}}$ with boundary $\partial \Sigma$ in $\partial \mathcal {C}\setminus \{0\}$ is anisotropic stationary if and only if the anisotropic mean curvature $H_K$ is constant on $\Sigma$ and $\big \langle N_K,\xi \big \rangle =0$ along $\partial \Sigma$.

Example 4.2

Take the hypersurface $\Sigma =\partial K\cap \overline{\mathcal {C}}$ with unit normal $N=\eta _K$ (the one pointing outside K). Note that $\partial K$ meets $\partial \mathcal {C}$ transversally because K is convex and contains 0 in its interior. Hence, $\Sigma$ is a compact hypersurface with boundary $\partial \Sigma$ such that $\Sigma \cap \partial \mathcal {C}=\partial \Sigma$. In Examples 2.1 (iii) we saw that $H_K(p)=-1$ and $N_K(p)=p$, for any $p\in \Sigma$. As $\partial \mathcal {C}$ is invariant under dilations centered at 0 then $N_K(p)\in T_p(\partial \mathcal {C})$ for any $p\in \partial \Sigma$ and so, $\big \langle N_K,\xi \big \rangle =0$. From the previous proposition we conclude that $\Sigma$ is anisotropic stationary.

In the isotropic case the orthogonality condition for a stationary hypersurface $\Sigma$ entails that $\nu =\xi$ along $\partial \Sigma$. In the next lemma we establish a similar equality for the anisotropic conormal $\nu _K$ that will be useful in future results.

Lemma 4.3

Let $\Sigma$ be a two-sided hypersurface immersed in $\overline{\mathcal {C}}$ with boundary $\partial \Sigma$ in $\partial \mathcal {C}\setminus \{0\}$. If $\big \langle N_K,\xi \big \rangle =0$ in $\partial \Sigma$, then $\cos \theta$ never vanishes and

$$\begin{aligned} \nu _K=\frac{\varphi _K}{\cos \theta }\,\xi \quad \text {along }\partial \Sigma , \end{aligned}$$

where $\varphi _K:=h_K(N)$ and $\theta$ is the oriented angle function between $\nu$ and $\xi$ in $T(\partial \Sigma )^\bot$.

Proof

Recall that $\nu _K:=\varphi _K\,\nu -\big \langle N_K,\nu \big \rangle \,N$, which is normal to $\partial \Sigma$. By Eq. (4.2) we have $\big \langle \nu _K,\mu \big \rangle =-\big \langle N_K,\xi \big \rangle =0$. This shows that $\nu _K$ is normal to $\partial \mathcal {C}$ and so, $\nu _K=\big \langle \nu _K,\xi \big \rangle \,\xi$ along $\partial \Sigma$.

Let us compute $\big \langle \nu _K,\xi \big \rangle$. Consider the projection $N_K^*$ of $N_K$ onto $T(\partial \Sigma )^\bot$. By using (4.2), equality $\big \langle N_K,\xi \big \rangle =0$, and that $\{\nu ,N\}$ and $\{\xi ,\mu \}$ are orthonormal basis of $T(\partial \Sigma )^\bot$, we obtain

$$\begin{aligned} \big \langle \nu _K,\xi \big \rangle ^2=\big \langle N_K,\mu \big \rangle ^2=|N_K^*|^2=\big \langle N_K,\nu \big \rangle ^2+\varphi _K^2, \end{aligned}$$

which is a positive number. Hence, $\big \langle \nu _K,\xi \big \rangle$ never vanishes along $\partial \Sigma$. On the other hand, by substituting the expression for $\xi$ in (4.1) into equality $\big \langle N_K,\xi \big \rangle =0$, we get

$$\begin{aligned} \sin \theta =-\frac{\big \langle N_K,\nu \big \rangle \,\cos \theta }{\varphi _K}\quad \text {along }\partial \Sigma . \end{aligned}$$

The definition of $\nu _K$ and the two previous relations lead to

$$\begin{aligned} \big \langle \nu _K,\xi \big \rangle =\varphi _K\,\cos \theta -\big \langle N_K,\nu \big \rangle \,\sin \theta =\frac{\varphi ^2_K +\big \langle N_K,\nu \big \rangle ^2}{\varphi _K}\,\cos \theta =\frac{\big \langle \nu _K,\xi \big \rangle ^2}{\varphi _K}\,\cos \theta , \end{aligned}$$

and so

$$\begin{aligned} \frac{\big \langle \nu _K,\xi \big \rangle }{\varphi _K}\,\cos \theta =1\quad \text {along }\partial \Sigma . \end{aligned}$$

This implies that $\cos \theta$ never vanishes and allows to deduce the announced expression for $\nu _K$. $\square$

Now, we can provide new expressions for the derivatives of the anisotropic area when we consider an anisotropic stationary hypersurface $\Sigma$ in $\mathcal {C}$. On the one hand, the boundary term in the formula for $A_K'(0)$ obtained in Proposition 3.2 vanishes for any flow $\{\phi _t\}_{t\in {\mathbb {R}}}$ admissible for $\mathcal {C}$. This comes from Proposition 4.1 and Lemma 4.3 since the velocity vector field X is tangent to $\partial \mathcal {C}\setminus \{0\}$. Hence

$$\begin{aligned} A_K'(0)=-nH_K\int _\Sigma u\,d\Sigma , \end{aligned}$$

(4.4)

where $u:=\big \langle X,N\big \rangle$ on $\Sigma$. Note that, for the flow $\phi _t(p):=e^t\,p$, the first equation in (2.9) implies that $A_K'(0)=nA_K(\Sigma )$. Thus, by having in mind (4.4) and Eq. (2.8), it follows that

$$\begin{aligned} A_K(\Sigma )=-(n+1)\,H_K\,V(\Sigma ). \end{aligned}$$

(4.5)

This identity is a Minkowski-type formula for compact anisotropic stationary hypersurfaces in $\mathcal {C}$.

Remark 4.4

We can use (2.8) to write (4.5) as

$$\begin{aligned} \int _\Sigma \big (\varphi _K+H_K\big \langle p,N(p)\big \rangle \big )\,d\Sigma =0. \end{aligned}$$

The proof of (4.5) implies that the previous identity is true for any compact hypersurface $\Sigma$ immersed in $\mathcal {C}$ with boundary $\partial \Sigma$ in $\partial \mathcal {C}\setminus \{0\}$ and satisfying $\big \langle N_K,\xi \big \rangle =0$ along $\partial \Sigma$. Similar formulas were previously obtained by He and Li [16, Thm. 1.1], and Jia et al. [18, Thm. 1.3].

On the other hand, the boundary integrand appearing in $A_K''(0)$, see Proposition 3.3, can be written in terms of the Euclidean extrinsic geometry of $\partial \mathcal {C}$. We prove this in the next proposition, where we also compute the second derivative of the volume for certain flows.

Proposition 4.5

Let $\Sigma$ be a two-sided anisotropic stationary hypersurface immersed in $\overline{\mathcal {C}}$ with boundary $\partial \Sigma$ in $\partial \mathcal {C}\setminus \{0\}$. Consider a smooth complete vector field X on $\mathbb {R}^{n+1}$ such that $X_{|\Sigma }=N_K$, $X(0)=0$ and X is tangent to $\partial \mathcal {C}\setminus \{0\}$. Then, for the admissible flow $\{\phi _t\}_{t\in {\mathbb {R}}}$ defined by the one-parameter group of diffeomorphisms associated to X, we have

$$\begin{aligned} A_K''(0)&=\int _\Sigma \left( n^2H_K^2-tr (B_K^2)\right) \varphi _K\,d\Sigma \nonumber \\&\quad -\int _\Sigma n H_K\,v\,d\Sigma \nonumber \\&\quad -\int _{\partial \Sigma }\frac{II (N_K,N_K)}{\cos \theta }\,\varphi _K\,d(\partial \Sigma ), \end{aligned}$$

(4.6)

$$\begin{aligned} V''(0)&=\int _\Sigma (-nH_K\,\varphi _K+v)\,d\Sigma , \end{aligned}$$

(4.7)

where $B_K$ is the anisotropic shape operator of $\Sigma$, $II$ is the second fundamental form of $\partial \mathcal {C}\setminus \{0\}$ with respect to the inner normal, $\theta$ is the oriented angle function between $\nu$ and $\xi$ in $T(\partial \Sigma )^\bot$, and $v:=\big \langle Z,N\big \rangle$ is the normal component of $Z:=D_XX$.

Remark 4.6

Observe that $N_K$ is tangent to $\partial \mathcal {C}$ along $\partial \Sigma$ because $\Sigma$ is anisotropic stationary. This guarantees the existence of a vector field X in the conditions of the statement and that the term $\text {II}(N_K,N_K)$ is well defined. Note also that $\cos \theta$ never vanishes by Lemma 4.3.

Proof of Proposition 4.5

We first check that (4.6) holds. From Proposition 3.3 it suffices to see that

$$\begin{aligned} \big \langle Z,\nu _K\big \rangle =\frac{\text {II}(N_K,N_K)}{\cos \theta }\,\varphi _K\quad \text {along }\partial \Sigma . \end{aligned}$$

By Lemma 4.3 we know that

$$\begin{aligned} \big \langle Z,\nu _K\big \rangle =\frac{\varphi _K}{\cos \theta }\,\big \langle Z,\xi \big \rangle \quad \text {along }\partial \Sigma . \end{aligned}$$

It is clear that $\big \langle X,\xi \big \rangle =0$ along $\partial \mathcal {C}$ because X is tangent to $\partial \mathcal {C}\setminus \{0\}$ and $X(0)=0$. By differentiating with respect to X, we obtain

$$\begin{aligned} 0=\big \langle D_XX,\xi \big \rangle +\big \langle X,D_X\xi \big \rangle =\big \langle Z,\xi \big \rangle -\text {II}(X,X), \end{aligned}$$

so that $\big \langle Z,\xi \big \rangle =\text {II}(X,X)=\text {II}(N_K,N_K)$ along $\partial \Sigma$.

Now we compute $V''(0)$. As the family $\{\phi _t\}_{t\in {\mathbb {R}}}$ is a one-parameter group of diffeomorphisms, we can employ Eq. (4.3) to get

$$\begin{aligned} V'(t)=\int _{\Sigma _t}\big \langle X,N_t\big \rangle \,d\Sigma _t=\int _\Sigma \big (\big \langle X,N_t\big \rangle \circ \phi _t\big )\,\text {Jac}\,\phi _t\,d\Sigma . \end{aligned}$$

By differentiating and having in mind (3.8), (3.7), (3.14), (2.6) and (2.3), it follows that

$$\begin{aligned} V''(0)=\int _\Sigma \big (\big \langle D_XX,N\big \rangle +\big \langle X,D_X\overline{N}\big \rangle +\big \langle X,N\big \rangle \,{{\,\textrm{div}\,}}_\Sigma X\big )\,d\Sigma =\int _\Sigma (v-nH_K\,\varphi _K)\,d\Sigma . \end{aligned}$$

This completes the proof. $\square$

We now turn to the main result of the paper. This is a classification of anisotropic stable hypersurfaces in a Euclidean solid cone.

Let $\Sigma$ be a two-sided hypersurface immersed in $\overline{\mathcal {C}}$ with boundary $\partial \Sigma$ in $\partial \mathcal {C}\setminus \{0\}$. We fix a smooth unit normal vector field N on $\Sigma$. We say that $\Sigma$ is anisotropic stable (with respect to N) if $A_K'(0)=0$ and $A_K''(0)\geqslant 0$ for any flow admissible for $\mathcal {C}$ and preserving the volume of $\Sigma$. When $\mathcal {C}$ is convex, the hypersurface $\partial K\cap \overline{\mathcal {C}}$ is anisotropic stable with respect to the outer unit normal since it minimizes the anisotropic area among compact hypersurfaces in $\mathcal {C}$ separating the same volume, see [28, Cor. 1.2] and [4, Thm. 1.3]. So, it is natural to ask if this property characterizes $\partial K\cap \overline{\mathcal {C}}$ up to translations and dilations centered at 0. The next example shows that, in general, the answer is negative because other anisotropic stable hypersurfaces may appear.

Example 4.7

Let $\mathcal {C}$ be a convex cone different from a Euclidean half-space and such that $\partial \mathcal {C}$ contains a half-hyperplane P (any solid cone in ${\mathbb {R}}^2$ satisfies this property). Consider the open half-space $\mathcal {H}\subset \mathbb {R}^{n+1}$ with $P\subset \partial \mathcal {H}$ and $\mathcal {C}\subset \mathcal {H}$. Since $\mathcal {H}$ is a convex cone, we can apply the isoperimetric result in [28, Cor. 1.2] and [4, Thm. 1.3] to deduce that the truncated Wulff shape $\partial K\cap \overline{\mathcal {H}}$ minimizes the anisotropic area in $\mathcal {H}$ for fixed volume. Next, we apply to $\partial K\cap \overline{\mathcal {H}}$ a translation along P so that the resulting hypersurface $\Sigma$ is contained in $\overline{\mathcal {C}}$. Note that $\Sigma$ is anisotropic stable in $\mathcal {C}$ because $\partial K\cap \overline{\mathcal {H}}$ is anisotropic stable in $\mathcal {H}$. Finally, as $\mathcal {C}\ne \mathcal {H}$, then $\Sigma \ne p_0+\lambda \,(\partial K\cap \overline{\mathcal {C}})$ for any $p_0\in \mathbb {R}^{n+1}$ and $\lambda >0$.

This example illustrates that the optimal conclusion to be deduced for an anisotropic stable hypersurface $\Sigma$ in $\mathcal {C}$ is that $\Sigma \subset \partial K$, up to translation and homothety. We prove this fact in the next uniqueness statement under our regularity conditions for $\Sigma$, $\mathcal {C}$ and K.

Theorem 4.8

Let $\mathcal {C}\subset \mathbb {R}^{n+1}$ be a solid convex cone over a smooth domain of $\mathbb {S}^n$. Consider a compact, connected, two-sided hypersurface $\Sigma$ immersed in $\overline{\mathcal {C}}$ with smooth boundary $\partial \Sigma$ in $\partial \mathcal {C}\setminus \{0\}$ and such that $\Sigma \cap \partial \mathcal {C}=\partial \Sigma$. If $\Sigma$ is anisotropic stable for the area $A_K$ defined by a smooth strictly convex body $K\subset \mathbb {R}^{n+1}$ then, there is $p_0\in \mathbb {R}^{n+1}$ and $\lambda >0$ such that $\Sigma \subset p_0+\lambda \,(\partial K)$.

Proof

We will follow the idea explained in the Introduction. As $\Sigma$ is anisotropic stationary, Proposition 4.1 implies that $H_K$ is constant on $\Sigma$ and $\big \langle N_K,\xi \big \rangle =0$ along $\partial \Sigma$. So, we can find a smooth complete vector field X on $\mathbb {R}^{n+1}$ such that $X_{|\Sigma }=N_K$, $X(0)=0$ and X is tangent to $\partial \mathcal {C}\setminus \{0\}$. By Eq. (2.3) it is clear that $u=\varphi _K$ on $\Sigma$, where $u:=\big \langle X,N\big \rangle$ and $\varphi _K:=h_K(N)$. Let $\{\psi _t\}_{t\in {\mathbb {R}}}$ be the one-parameter group of diffeomorphisms associated to X. We take the functionals $A_K(t):=A_K(\Sigma _t)$ and $V(t):=V(\Sigma _t)$ associated to the variation $\Sigma _t:=\psi _t(\Sigma )$. From the Minkowski formula in Eq. (4.5) we have $H_K\ne 0$ and $V(\Sigma )\ne 0$. Hence, there is $\varepsilon >0$ such that V(t) has the same sign as $V(\Sigma )$ for any $t\in (-\varepsilon ,\varepsilon )$. Next, for any $t\in (\varepsilon ,\varepsilon )$, we apply to $\Sigma _t$ a dilation $\delta _{\lambda (t)}(p):=\lambda (t)\,p$ with $\lambda (t)>0$, so that the volume of $\delta _{\lambda (t)}(\Sigma _t)$ equals the volume of $\Sigma$. Since $V(\delta _{\lambda (t)}(\Sigma _t))=\lambda (t)^{n+1}\,V(t)$, see (2.9), we get

$$\begin{aligned} \lambda (t):=\left( \frac{V(\Sigma )}{V(t)}\right) ^{\frac{1}{n+1}}, \quad \text {for any } t\in (-\varepsilon ,\varepsilon ). \end{aligned}$$

(4.8)

In particular, $\lambda (0)=1$. We consider a smooth positive function on ${\mathbb {R}}$ extending $\lambda (t)$ for small values of t. We also denote this function by $\lambda (t)$. If we define

$$\begin{aligned} \phi _t:=\delta _{\lambda (t)}\circ \psi _t, \quad \text {for any }t\in {\mathbb {R}}, \end{aligned}$$

then we produce an admissible flow for $\mathcal {C}$ that preserves the volume of $\Sigma$. Hence, the anisotropic stability of $\Sigma$ entails that the functional $a_K(t):=A_K(\phi _t(\Sigma ))$ satisfies $a_K''(0)\geqslant 0$. To prove the theorem we need to compute $a_K''(0)$. By Eq. (2.9) we know that

$$\begin{aligned} a_K(t)=A_K(\delta _{\lambda (t)}(\Sigma _t))=\lambda (t)^n\,A_K(t). \end{aligned}$$

(4.9)

Thus, the calculus of $a_K''(0)$ relies on the values of $\lambda '(0)$, $\lambda ''(0)$, $A_K'(0)$ and $A_K''(0)$.

By using (4.3), (4.4), the fact that $u=\varphi _K$ on $\Sigma$, and that $A_K(\Sigma )=\int _\Sigma \varphi _K\,d\Sigma$, we obtain

$$\begin{aligned} V'(0)&=A_K(\Sigma ), \end{aligned}$$

(4.10)

$$\begin{aligned} A_K'(0)&=-nH_K A_K(\Sigma ). \end{aligned}$$

(4.11)

On the other hand, from the expression of $\lambda (t)$ in (4.8), we have

$$\begin{aligned} \lambda '(t)=-\frac{1}{n+1}\,V(\Sigma )^{\frac{1}{n+1}}\,V(t)^{\frac{-n-2}{n+1}}\,V'(t). \end{aligned}$$

(4.12)

Hence, from Eqs. (4.10) and (4.5), it follows that

$$\begin{aligned} \lambda '(0)=-\frac{1}{n+1}\,\frac{A_K(\Sigma )}{V(\Sigma )}=H_K. \end{aligned}$$

(4.13)

Now we compute $A_K''(0)$ and $\lambda ''(0)$. From (4.7) and (4.6), since $H_K$ is constant, we get

$$\begin{aligned} V''(0)&=-nH_KA_K(\Sigma )+\alpha , \end{aligned}$$

(4.14)

$$\begin{aligned} A_K''(0)&=n^2H_K^2\,A_K(\Sigma )-\int _\Sigma \text {tr}(B_K^2)\,\varphi _K\,d\Sigma -n H_K\,\alpha \nonumber \\&\quad -\int _{\partial \Sigma }\frac{\text {II}(N_K,N_K)}{\cos \theta }\,\varphi _K\,d(\partial \Sigma ), \end{aligned}$$

(4.15)

where $\alpha :=\int _\Sigma v\,d\Sigma$ and $v:=\big \langle D_XX,N\big \rangle$. Recall that $\text {II}$ is the second fundamental form of $\partial \mathcal {C}\setminus \{0\}$ with respect to the inner normal $\xi$. On the other hand, from (4.12) we deduce

$$\begin{aligned} \lambda ''(t)=\frac{-1}{n+1}\,V(\Sigma )^{\frac{1}{n+1}}\,\left( -\frac{n+2}{n+1}\,V(t)^ {\frac{-2n-3}{n+1}}\,V'(t)^{2}+V(t)^{\frac{-n-2}{n+1}}\,V''(t)\right) . \end{aligned}$$

By evaluating at $t=0$ and simplifying, Eqs. (4.10) and (4.14) give us

$$\begin{aligned} \lambda ''(0)=\frac{n+2}{(n+1)^2}\,\frac{A_K(\Sigma )^2}{V(\Sigma )^2}+\frac{n}{n+1}\, \frac{H_KA_K(\Sigma )}{V(\Sigma )}-\frac{1}{n+1}\,\frac{\alpha }{V(\Sigma )}. \end{aligned}$$

When we employ the identity (4.5) in the three summands above, we arrive at

$$\begin{aligned} \lambda ''(0)=2H_K^2+\frac{H_K}{A_K(\Sigma )}\,\alpha . \end{aligned}$$

(4.16)

Finally, we differentiate into Eq. (4.9) to infer

$$\begin{aligned} a_K'(t)=n\,\lambda (t)^{n-1}\,\lambda '(t)\,A_K(t)+\lambda (t)^{n}\,A_K'(t). \end{aligned}$$

As a consequence

$$\begin{aligned} a_K''(t)= & {} n\left\{ (n-1)\,\lambda (t)^{n-2}\,\lambda '(t)^{2}\,A_K(t)+\lambda (t)^{n-1}\, \lambda ''(t)\,A_K(t)+2\lambda (t)^{n-1}\,\lambda '(t)\,A_K'(t)\right\} \\{} & {} +\lambda (t)^n\,A_K''(t). \end{aligned}$$

By substituting above the expressions in (4.13), (4.16), (4.11), (4.15), and simplifying, we obtain

$$\begin{aligned} a_K''(0)=-\int _\Sigma \big (\text {tr}(B_K^2)-nH_K^2\big )\,\varphi _K\,d\Sigma -\int _{\partial \Sigma } \frac{\text {II}(N_K,N_K)}{\cos \theta }\,\varphi _K\,d(\partial \Sigma ). \end{aligned}$$

(4.17)

The first integrand in the previous formula is nonnegative by (2.7). The convexity of $\mathcal {C}$ implies that $\text {II}(N_K,N_K)\geqslant 0$ along $\partial \Sigma$. Let us see that $\cos \theta >0$ along $\partial \Sigma$. From the first equation in (4.1) we know that $\cos \theta =\big \langle \nu ,\xi \big \rangle$. Take a point $p\in \partial \Sigma$. Since $\nu (p)$ is the inner conormal of $\partial \Sigma$ in $\Sigma$, there is a smooth curve $\gamma :[0,\tau )\rightarrow \Sigma$ such that $\gamma (0)=p$ and $\gamma '(0)=\nu (p)$. As $\Sigma \subset \overline{\mathcal {C}}$ and $\mathcal {C}$ is convex, the function $f(t):=\big \langle \gamma (t)-p,\xi (p)\big \rangle$ satisfies $f(t)\geqslant 0$ for any $t\in [0,\tau )$ and $f(0)=0$. This gives us $\big \langle \nu (p),\xi (p)\big \rangle =f'(0)\geqslant 0$. Indeed we have $\big \langle \nu (p),\xi (p)\big \rangle >0$ because $\cos \theta$ never vanishes, see Lemma 4.3. All this together with the stability inequality $a_K''(0)\geqslant 0$ entails that $\text {tr}(B_K^2)=nH_K^2$ on $\Sigma$ (this means that $\Sigma$ is anisotropic umbilical) and

$$\begin{aligned} \text {II}(N_K,N_K)=0 \quad \text {along }\partial \Sigma . \end{aligned}$$

(4.18)

Since $H_K\ne 0$ (as a consequence of Eq. (4.5)) the proof finishes by invoking Proposition 2.2. $\square$

Remark 4.9

(Non-smooth cones) The proof of Theorem 4.8 is still valid when $\mathcal {C}$ is an arbitrary open convex cone (the base domain ${\mathcal {D}}\subset \mathbb {S}^n$ need not be smooth), provided the boundary $\partial \Sigma$ is contained in a smooth open portion of $\partial \mathcal {C}$. For instance, the conclusion holds for a compact anisotropic stable hypersurface $\Sigma$ disjoint from the edge of a domain $\mathcal {C}$ bounded by two transversal hyperplanes. In this direction, Koiso [19, Thm. 4], [21, Thm. 1] has characterized compact anisotropic stable capillary hypersurfaces in wedge-shaped domains of $\mathbb {R}^{n+1}$.

Remark 4.10

(Planar cones) For a solid cone $\mathcal {C}\subset {\mathbb {R}}^2$, the boundary $\partial \mathcal {C}$ is the union of two-closed half-lines leaving from 0. Thus, we have $\text {II}=0$ along $\partial \mathcal {C}\setminus \{0\}$ and the boundary term in (4.17) disappears. Hence, the conclusion of Theorem 4.8 remains valid even if the cone is not convex.

The situation described in Example 4.7 leads us to seek additional conditions on the cone in order to deduce stronger uniqueness conclusions. In this direction we can prove the next statement.

Corollary 4.11

Let $\mathcal {C}\subset \mathbb {R}^{n+1}$ be a solid cone over a smooth strictly convex domain of $\mathbb {S}^n$. Consider a compact, connected, two-sided hypersurface $\Sigma$ immersed in $\overline{\mathcal {C}}$ with smooth boundary $\partial \Sigma$ in $\partial \mathcal {C}\setminus \{0\}$ and such that $\Sigma \cap \partial \mathcal {C}=\partial \Sigma$. If $\Sigma$ is anisotropic stable for the area $A_K$ defined by a smooth strictly convex body $K\subset \mathbb {R}^{n+1}$, then $\Sigma =\lambda \,(\partial K\cap \overline{\mathcal {C}})$ for some $\lambda >0$.

Proof

From Theorem 4.8 we know that $\Sigma \subset p_0+\lambda \,(\partial K)$ for some $p_0\in \mathbb {R}^{n+1}$ and $\lambda >0$. If we define on $\Sigma$ the unit normal vector $N(p):=\eta _K\big ((p-p_0)/\lambda \big )$ then, by using (2.2) and equality $(\pi _K\circ \eta _K)(w)=w$ for any $w\in \partial K$, we get $N_K(p)=(p-p_0)/\lambda$ for any $p\in \Sigma$.

The fact that the base set ${\mathcal {D}}\subset \mathbb {S}^n$ of the cone $\mathcal {C}$ is a strictly convex domain means that the second fundamental form of $\partial {\mathcal {D}}$ as a hypersurface of $\mathbb {S}^n$ is always positive definite with respect to the inner unit normal. This implies that $\text {II}_p(w,w)>0$ for any $p\in \partial \Sigma$ and any vector $w\in T_p(\partial \mathcal {C})$ non-proportional to p. From the identity in (4.18) it follows that $(p-p_0)/\lambda$ is proportional to p, for any $p\in \partial \Sigma$. From here we get $p_0=0$, and this completes the proof. $\square$

A special example of convex cone is a half-space $\mathcal {H}\subset \mathbb {R}^{n+1}$ with $0\in \partial \mathcal {H}$. Since $\partial \mathcal {H}$ is smooth we do not need to assume $\partial \Sigma \subset \partial \mathcal {H}\setminus \{0\}$. Note that Corollary 4.11 cannot be applied in this situation because $\text {II}=0$ on $\partial \mathcal {C}$. However, in the next statement we deduce that $\partial K\cap \overline{\mathcal {H}}$ is the unique compact anisotropic stable hypersurface in $\mathcal {H}$, up to dilations about 0 and translations along $\partial \mathcal {H}$. A similar result for anisotropic stable capillary hypersurfaces in $\mathcal {H}$ has been given by Guo and Xia [14, Thm. 1.1] with a different proof.

Corollary 4.12

Let $\Sigma$ be a compact, connected, two-sided hypersurface immersed in an open half-space $\mathcal {H}\subset \mathbb {R}^{n+1}$ with smooth boundary $\partial \Sigma \subset \partial \mathcal {H}$ such that $\Sigma \cap \partial \mathcal {H}=\partial \Sigma$. If $\Sigma$ is anisotropic stable, then $\Sigma =p_0+\lambda \,(\partial K\cap \overline{\mathcal {H}})$ for some $p_0\in \partial \mathcal {H}$ and $\lambda >0$.

Proof

After applying Theorem 4.8 it remains to see that $p_0\in \partial \mathcal {H}$. By reasoning as in the proof of Corollary 4.11 we obtain $N_K(p)=(p-p_0)/\lambda$, for any $p\in \Sigma$. The orthogonality condition $\big \langle N_K,\xi \big \rangle =0$ along $\partial \Sigma$ entails that $p-p_0\in T_p(\partial \mathcal {H})$ for any $p\in \partial \Sigma$. As $\partial \mathcal {H}$ is a Euclidean hyperplane, the straight line starting from a point $p\in \partial \Sigma$ with $p\ne p_0$ and generated by the vector $p-p_0$ is entirely contained in $\partial \mathcal {H}$. This shows that $p_0\in \partial \mathcal {H}$, as we claimed. $\square$