Stable anisotropic capillary hypersurfaces in a wedge

1.   Introduction

We prove an existence and uniqueness result of stable equilibrium hypersurfaces in wedge-like domains in the $(n+1)$-dimensional Euclidean space ${\mathbb R}^{n+1}$ for anisotropic surface energy, which serve as a mathematical model of small crystals and small liquid crystals with anisotropy. Our result is new even when the special case where the surface energy is merely the surface area.

Let $\gamma:S^n \rightarrow {\mathbb R}_{ > 0}$ be a positive continuous function on the unit sphere $S^n = \{\nu \in {\mathbb R}^{n+1} \; ; \; \|\nu\| = 1\}$ in ${\mathbb R}^{n+1}$. Let $X$ be a closed piecewise-smooth hypersurface in ${\mathbb R}^{n+1}$. $X$ will be represented as a mapping $X:M\to {\mathbb R}^{n+1}$ from an $n$-dimensional oriented connected compact $C^\infty$ manifold $M$ into ${\mathbb R}^{n+1}$. Let $\nu$ be the unit normal vector field along $X|_{M\setminus S[X]}$, where $S[X]$ is the set of singular points of $X$. The anisotropic energy ${\mathcal F}_\gamma(X)$ of $X$ is defined as ${\mathcal F}_\gamma(X): = \int_{M\setminus S[X]} \gamma(\nu)\:dA$, where $dA$ is the $n$-dimensional volume form of $M$ induced by $X$. Such an energy was introduced by Gibbs (1839–1903) in order to model the shape of small crystals, and it is used as a mathematical model of anisotropic surface energy ^[19,20]. It is known that, for any positive number $V > 0$, among all closed piecewise-smooth hypersurfaces as above enclosing the same $(n+1)$-dimensional volume $V$, there exists a unique (up to translation in ${\mathbb R}^{n+1}$) minimizer $W_\gamma(V)$ of ${\mathcal F}_\gamma$ ^[17]. Each $W_\gamma(V)$ is homothetic to the so-called Wulff shape for $\gamma$ (the definition of the Wulff shape will be given in §2), which we will denote by $W_\gamma$. When $\gamma\equiv 1$, ${\mathcal F}_\gamma(X)$ is the usual $n$-dimensional volume of the hypersurface $X$ and $W_\gamma$ is the unit sphere $S^n$.

The Wulff shape $W_\gamma$ is not smooth in general. However, in this paper we assume that $W_\gamma$ is a smooth strictly convex hypersurface like the previous works that studied variational problems of anisotropic surface energies in differential geometry (cf. ^{[1,4,5,8,9,10,11,12,14,15]}).

Each equilibrium hypersurface $X$ of ${\mathcal F}_\gamma$ for variations that preserve the enclosed $(n+1)$-dimensional volume (we will call such a variation a volume-preserving variation) has constant anisotropic mean curvature. Here, the anisotropic mean curvature $\Lambda$ of a piecewise-$C^2$ hypersurface $X$ is defined at each regular point of $X$ as (cf. ^[8,15]) $\Lambda: = (1/n) \large(-{\rm div}_M D\gamma+nH\gamma\large)$, where $D\gamma$ is the gradient of $\gamma$ and $H$ is the mean curvature of $X$. If $\gamma\equiv 1$, $\Lambda = H$ holds.

Let $\Omega$ be a wedge-shaped domain in ${\mathbb R}^{n+1}$ bounded by $k$ hyperplanes $\Pi_1, \cdots, \Pi_k$ ($k \le n+1$) such that the intersection $\Pi_1 \cap \cdots \cap \Pi_k$ includes the origin $0$ of ${\mathbb R}^{n+1}$ (Figure 1). Denote by $\tilde{N}_j$ the unit normal to $\Pi_j$ which points outward from $\Omega$. We assume that $\tilde{N}_1, \cdots, \tilde{N}_k$ are linearly independent. We call each $\Pi_i \cap \Pi_j$ ($i \ne j$) an edge of $\Omega$. Let $\omega_j$ ($j = 1, \cdots, k$) be non-negative constants. Let $M$ be an $n$-dimensional oriented connected compact $C^\infty$ manifold with boundary $\partial M = \sigma_1\cup \cdots \cup \sigma_k$, where each $\sigma_j$ is topologically $S^{n-1}$. Consider any $C^\infty$-immersion $X:(M, \sigma_1, \cdots, \sigma_k)\to (\overline\Omega, \Pi_1, \cdots, \Pi_k)$ of which the restriction $X|_{\partial M}$ to $\partial M$ is an embedding. Set $C_j = X(\sigma_j)$, and let $D_j = D_j(X)\subset\Pi_j$ be the $n$-dimensional domain bounded by $C_j$ ($j = 1, \cdots, k$). We assume that the unit normal $\nu$ to $X$ points outward from the $(n+1)$-dimensional domain bounded by $X(M)\cup (\cup_{j = 1}^k D_j)$ near each $C_j$. We define the wetting energy ${\mathcal W}(X)$ of $X$ as

where $\mathcal{H}^n(D_j)$ is the $n$-dimensional Hausdorff measure of $D_j$. Then, we define the total energy $E(X) = E_\gamma(X)$ of $X$ by

Note that $X(M)\cup D_1\cup \cdots \cup D_k$ is an oriented closed piecewise smooth hypersurface without boundary (possibly with self-intersection). We denote by $V(X)$ the oriented volume enclosed by $X(M)\cup D_1\cup \cdots \cup D_k$ which is represented as

We call a critical point of $E$ for volume-preserving variations an anisotropic capillary hypersurface (or, simply, a capillary hypersurface). A capillary hypersurface is said to be stable if the second variation of $E$ is nonnegative for all volume-preserving variations of $X$. Otherwise, it is said to be unstable. In this paper, we prove the following two results on the uniqueness and the existence of stable capillary hypersurfaces.

Theorem 1. Let $X:(M, \sigma_1, \cdots, \sigma_k)\to (\overline\Omega, \Pi_1, \cdots, \Pi_k)$ be a compact oriented immersed anisotropic capillary hypersurface that is disjoint from any edge of $\Omega$, having embedded boundary and satisfying $X(\partial M)\cap\Pi_j = \partial D_j$ for a nonempty bounded domain $D_j$ in $\Pi_j$. If $X$ is stable (and all $D_1, \cdots, D_k$ are convex if $n\ge 3$), then $X(M)$ is (up to translation and homothety) a part of the Wulff shape $W_\gamma$. Conversely, if $X$ is an embedding onto a part of $W_\gamma$ (up to translation and homothety), then it is stable.

Theorem 2. There exists an anisotropic capillary hypersurface $X$ that is a part of the Wulff shape (up to translation and homothety) and that intersects $\Pi_j$ with more than two points if and only if $\omega_j < \gamma(\tilde{N}_j)$ holds.

As for previous works which are closely related to Theorem 1, we have the followings. First, Theorem 1 is a generalization of the main result of ^[3], where we proved the uniqueness result similar to Theorem 1 for isotropic capillary hypersurfaces in a wedge in ${\mathbb R}^{n+1}$ with $k = 2$; here, isotropic means that $\gamma\equiv 1$. ^[13] also studies the isotropic case for general $k$, but it does not prove that parts of the Wulff shape are stable. As for the anisotropic case, we studied the existence and uniqueness of stable anisotropic capillary surfaces between two parallel planes $\Pi_1$ and $\Pi_2$ in ${\mathbb R}^3$ ^[9,10,11]. There, stable anisotropic capillary surfaces are not necessarily parts of the Wulff shape (up to translation and homothety). This suggests that the assumption of the linear independence of $\tilde{N}_1, \cdots, \tilde{N}_k$ cannot be removed in our Theorem 1. Finally we mention that Theorem 1 was announced in ^[7]. Moreover there an outline of the proof of Theorem 1 for $n = 2$ and $k = 2$ was given.

This article is organized as follows. In Section 2 we give preliminary contents. We give the definition and a representation of the Wulff shape. We also give the definitions of various anisotropic curvatures for hypersurfaces. Moreover we recall the definition of anisotropic parallel hypersurfaces and a Steiner-type integral formula for these hypersurfaces. In Section 3, we give the first variation formulas and the Euler–Lagrange equations for our variational problems. Also the proof of Theorem 2 is given. Sections 4, 6 and 7 will be devoted to proving the uniqueness part of Theorem 1. The existence part that is the last statement of Theorem 1 will be proved in Section 5.

2.   Preliminaries

In this paper, we call the boundary $W_\gamma$ of the convex set $\tilde{W}[\gamma] : = \cap _{\nu \in S^n} \bigl\{ X \in {\mathbb R}^{n+1}\; ;\; \langle X, \nu\rangle \le \gamma(\nu) \bigr\}$ the Wulff shape for $\gamma$, where $\langle \, \ \rangle$ means the standard inner product in ${\mathbb R}^{n+1}$. In other literatures, $\tilde{W}[\gamma]$ is often called the Wulff shape.

From now on, any parallel translation of the Wulff shape $W_\gamma$ will be also called the Wulff shape, and it will be denoted also by $W_\gamma$, if it does not cause any confusion.

From now on, we assume that, for simplicity, $\gamma$ is of class $C^\infty$. We also assume that the homogeneous extension $\overline{\gamma}:{\mathbb R}^{n+1} \to {\mathbb R}_{\ge 0}$ of $\gamma$ that is defined as $\overline{\gamma}(rX) : = r\gamma(X)$ $(\forall X \in S^n, \ \forall r \ge 0)$ is a strictly convex function. In this case, we say that $\gamma$ is strictly convex, which is equivalent to the $n\times n$ matrix $D^2\gamma+\gamma\cdot I_n$ is positive definite at any point in $S^n$. Here, $D^2\gamma$ is the Hessian of $\gamma$ on $S^n$ and $I_n$ is the identity matrix of size $n$. The Wulff shape $W_\gamma$ is smooth and strictly convex (that is, each principal curvature of $W_\gamma$ with respect to the inward-pointing normal is positive at each point of $W_\gamma$) if and only if $\gamma$ is of class $C^2$ and strictly convex.

The Cahn–Hoffman map $\xi_\gamma:S^n \to {\mathbb R}^{n+1}$ for $\gamma$ is defined as $\xi_\gamma(\nu) = D\gamma|_\nu+\gamma(\nu)\nu$, ($\nu \in S^n$). Here, the tangent space $T_\nu(S^n)$ of $S^n$ at $\nu$ is naturally identified with the hyperplane in ${\mathbb R}^{n+1}$ which is tangent to $S^n$ at $\nu$. Because $\gamma$ is strictly convex, $\xi_\gamma$ gives an embedding onto $W_\gamma$. Moreover, the outward-pointing unit normal at a point $\xi_\gamma(\nu)$ to $W_\gamma$ coincides with $\nu$ (cf. ^[8]).

The Cahn-Hoffman field $\tilde{\xi}$ along $X$ for $\gamma$ is defined as $\tilde{\xi}: = \xi_\gamma\circ \nu:M\to {\mathbb R}^{n+1}$. Since the unit normal $\nu(p)$ of $X$ at $p \in M$ coincides with the unit normal of $\xi_\gamma$ at the point $\nu(p)$, we can identify $T_pM$ with $T_{\tilde{\xi}(p)}\xi_\gamma(S^n)$.

The linear map $S^\gamma_p:T_p M \to T_p M$ ($p \in M$) given by the $n\times n$ matrix $S^\gamma: = -d\tilde{\xi}$ is called the anisotropic shape operator of $X$. Various anisotropic curvatures of $X$ are defined as follows.

Definition 1 (anisotropic curvatures; cf. ^[5,15]). $\rm (i)$ The eigenvalues of $S^\gamma$ are called the anisotropic principal curvatures of $X$. We denote them by $k^\gamma_1, \cdots, k^\gamma_n$.

(ii) Let $\sigma^\gamma_r$ be the elementary symmetric functions of $k^\gamma_1, \cdots, k^\gamma_n$:

Set $\sigma^\gamma_0: = 1$. $H^\gamma_r: = \sigma^\gamma_r/{}_nC_r$ is called the $r$th anisotropic mean curvature of $X$, where $_nC_r = \frac{n!}{k!(n-k)!}$.

(iii) $H^\gamma_1$ is called also the anisotropic mean curvature of $X$, and we often denote it by $\Lambda$; that is, $\Lambda = \frac{1}{n}\sum\limits_{i = 1}^n k^\gamma_i = -\frac{1}{n}{\rm trace}_M(d\tilde\xi)$. $

As we mentioned above, for the Cahn-Hoffman map $\xi_\gamma:S^n \to {\mathbb R}^{n+1}$, it is shown that the unit normal vector field $\nu_{\xi_\gamma}$ is given by $\xi_\gamma^{-1}$. Hence, the anisotropic shape operator of $\xi_\gamma$ is $S^\gamma = -d(\xi_\gamma\circ \nu_{\xi_\gamma}) = -d({\rm id}_{S^n}) = -I_n$. Therefore, the anisotropic principal curvatures of $\xi_\gamma$ are $-1$, and hence, each $r$th anisotropic mean curvature of $\xi_\gamma$ is $(-1)^r$. Particularly, the anisotropic mean curvature of $\xi_\gamma$ for the normal $\nu$ and that of $W_\gamma$ for the outward-pointing unit normal is $-1$ at any point. More generally, the anisotropic mean curvature of an immersion $X:M\to {\mathbb R}^{n+1}$ is the mean value of the ratios of the principal curvatures of the Wulff shape and the corresponding curvatures of $X$ which is explained as follows.

Remark 1 (cf. ^[8]). Let $X:M\to {\mathbb R}^{n+1}$ be an immersion. Take any point $p \in M$. We compute the anisotropic mean curvature $\Lambda(p)$ of $X$ at $p$. Let $\{e_1, \cdots, e_n\}$ be a locally defined frame on $S^{n}$ such that $(D^2\gamma+\gamma\cdot I_n)e_i = (1/\mu_i)e_i$, where $\mu_i$ are the principal curvatures of $\xi_\gamma$ with respect to $\nu$. Note that the basis $\{e_1, \cdots, e_n\}$ at $\nu(p)$ also serves as an orthogonal basis for the tangent hyperplane of $X$ at $p$. Let $(-w_{ij})$ be the matrix representing $d\nu$ with respect to this basis. Then

This with the definition of $\Lambda$ gives

$S^\gamma$ is not symmetric in general. However, we have the following good properties of the anisotropic curvatures.

Remark 2. $\rm (i)$ If $d\xi_\gamma = D^2 \gamma+\gamma \cdot I_n$ is positive definite at a point $\nu(p)$ ($p \in M$), then all of the anisotropic principal curvatures of $X$ at $p$ are real ^[4].

(ii) $k^\gamma_i$ is not a real value in general. However, each $H^\gamma_r$ is always a real valued function on $M$ ^[6].

At the end of this section, we recall a useful concept that is "anisotropic parallel hypersurface" and an important integral formula that is a generalization of the classical Steiner's formula. Anisotropic parallel hypersurface is a generalization of parallel hypersurface and is defined as follows.

Definition 2 (Anisotropic parallel hypersurface, cf. ^[15]). Let $X:M \to {\mathbb R}^{n+1}$ be an immersion. For any real number $t$, we call the map $X_t: = X+t\tilde{\xi}:M \to {\mathbb R}^{n+1}$ the anisotropic parallel deformation of $X$ of height $t$. If $X_t$ is an immersion, then we call it the anisotropic parallel hypersurface of $X$ of height $t$.

The anisotropic energy ${\mathcal F}_\gamma(X_t)$ of the anisotropic parallel hypersurface $X_t: = X+t\tilde{\xi}$ is a polynomial of $t$ with degree at the most $n$ as follows.

Fact 1 (Steiner-type formula ^[4]). Assume that $\gamma:S^n \to {\mathbb R}_{ > 0}$ is of class $C^\infty$. Let $X:M\rightarrow {\mathbb R}^{n+1}$ be an immersion. Consider anisotropic parallel hypersurfaces $X_t = X+t\tilde{\xi}:M \to {\mathbb R}^{n+1}$. Then, the $n$-dimensional volume form $dA_{X_t}$ and the anisotropic energy ${\mathcal F}_\gamma(X_t)$ of $X_t$ have the following representations.

The isotropic version of Fact 1 is known as the Weyl's tube formula ^[18]. The isotropic $2$-dimensional version is the well-known Steiner's formula.

3.   Euler–Lagrange equations

In order to obtain the Euler–Lagrange equations for our capillary problem, first we recall the first variation formula for the anisotropic surface energy ${\mathcal F}_\gamma$.

Proposition 1 (^[6]). Let $X_\epsilon:M\to {\mathbb R}^{n+1}$ ($\epsilon \in J: = (-\epsilon_0, \epsilon_0)$), be a smooth variation of $X$; that is, $\epsilon_0 > 0$ and $X_0 = X$. Set

Then, the first variation of the anisotropic energy $\mathcal F_\gamma$ is given as follows.

where $ds$ is the $(n-1)$-dimensional volume form of $\partial M$ induced by $X$, $N$ is the outward-pointing unit conormal along $\partial M$, $R$ is the $\pi/2$-rotation on the $(N, \nu)$-plane, and $p$ is the projection from ${\mathbb R}^{n+1}$ to the $(N, \nu)$-plane.

On the other hand, the first variation of the $(n+1)$-dimensional volume enclosed by the region between $X$ and $X_\epsilon$ is given by (^[2]) as

Similarly, the first variation of $\mathcal{H}^n(D_j)$ is given as follows.

where $\rho$ is the outward-pointing unit normal along $X|_{\sigma_j}:\sigma_j \to \Pi_j$, and $ds$ is the $(n-1)$-dimensional volume form of $X|_{\sigma_j}$.

The Eq (3.1) with (3.2), (3.3) gives the following Euler–Lagrange equations.

Proposition 2. An immersion $X:(M, \sigma_1, \cdots, \sigma_k)\to (\overline\Omega, \Pi_1, \cdots, \Pi_k)$ is a capillary hypersurface if and only if both of the following conditions (i) and (ii) hold.

(i) The anisotropic mean curvature $\Lambda$ of $X$ is constant on $M$.

(ii) $\langle \tilde{\xi}, \tilde{N}_j\rangle = \omega_j$ on $\sigma_j$ ($j = 1, \cdots, k$), where $\tilde{\xi}$ is the Cahn–Hoffman field along $X$.

Proof. Note that $X$ is a capillary hypersurface if and only if, for all volume-preserving variations $X_\epsilon:(M, \sigma_1, \cdots, \sigma_k)\to (\overline\Omega, \Pi_1, \cdots, \Pi_k)$, ($-\epsilon_0 < \epsilon < \epsilon_0$), $\delta E = 0$ holds. This is equivalent to the fact that, there exists a constant $\Lambda_0$ such that

holds for all variations $X_\epsilon:(M, \sigma_1, \cdots, \sigma_k)\to (\overline\Omega, \Pi_1, \cdots, \Pi_k)$, ($-\epsilon_0 < \epsilon < \epsilon_0, \ \epsilon_0 > 0$) of $X$.

First we assume that $X$ is a capillary hypersurface. By taking variations that preserve the boundary values $X|_{\partial M}$, we have from (3.4), (3.5) that

which implies

This proves (i). Next, take variations that preserve the boundary values $X|_{\sigma_2\cup\cdots \cup\sigma_k}$. Then we have from (3.4), (3.5), (3.6) that

holds for all variations $X_\epsilon$ satisfying $X_\epsilon(\sigma_1) \subset \Pi_1$. This means that

holds. Note that the $(N, \nu)$-plane is the same as the $(\tilde{N}_1, \rho)$-plane because both of them are the orthogonal compliment of the $(n-1)$-dimensional tangent space of $X(\sigma_1)$ in ${\mathbb R}^{n+1}$ at each point in $X(\sigma_1)$. Therefore, (3.9) is equivalent to

And (3.10) is equivalent to

which means that

Similarly, we have

which proves (ii).

Next we assume that (i) and (ii) are satisfied. Then, by using the computations above, we know that $X$ is a capillary hypersurface.

Theorem 2 is given by Proposition 2.

Here we pose a new variational problem that will be used in the proof of a balancing formula (Lemma 2) in §4. Consider a more general class of surfaces than above:

For $X \in {\mathcal S}$, set $C_j: = X(\sigma_j)$ and denote by $D_j$ the $n$-dimensional domain bounded by $C_j$ in $\tilde{\Pi}_j$. Moreover, set

Denote by $dA$ the $n$-dimensional standard volume form on $S_X$. Note that each $\tilde{N}_j$ is a unit normal to $\tilde{\Pi}_j$. Define the algebraic volume $\overline{V}(X)$ enclosed by $S_X$ as

where we denoted the variable point in $D_j$ by $x$. And define the energy $\overline{\mathcal F}_\gamma(X)$ of $X$ as

Then, similarly to our capillary problem, we obtain the following first variation formulas.

Lemma 1.

Proof. (3.14) is a standard formula (cf. ^[2]). (3.15) is given in proposition 1. We will prove (3.16). Consider the orthogonal projection $p_j:\tilde{\Pi}_j \to \Pi_j$. Then,

Since

we obtain

which proves (3.16).

4.   Proof of the first half of Theorem 1: the uniqueness of the stable solution

In order to prove the uniqueness part of Theorem 1, we will show that any capillary hypersurface $X:(M, \sigma_1, \cdots, \sigma_k)\to (\overline\Omega, \Pi_1, \cdots, \Pi_k)$ is unstable unless it is a part of a rescaling of the Wulff shape.

In view of the condition (ii) in Proposition 2, it is useful to consider the anisotropic energy for $(n-1)$-dimensional closed hypersurfaces in $\Pi_j$. First, define hyperplanes $P_j$ in ${\mathbb R}^{n+1}$ ($j = 1, \cdots, k$) by

Then, set the followings:

Assume that $\omega_j\ge 0$ is sufficiently small so that $\hat{W}_j$ includes at least two distinct points. Then, $\hat{W}_j$ is a strictly convex closed $(n-1)$-dimensional $C^\infty$ hypersurface in the $n$-dimensional linear space $P_j$. We regard the point $\hat{O}_j$ as the origin of $P_j$. Denote by $\hat{\gamma}_j:S^{n-1} \to {\mathbb R}_{ > 0}$ the support function of $\hat{W}_j$, that is, for any $\rho\in S^{n-1}$, $\hat{\gamma}_j(\rho)$ is the distance between the origin $\hat{O}_j$ and the tangent space of $\hat{W}_j$ at the uniquely-determined point $w\in \hat{W}_j$ such that the outward-pointing unit normal to $\hat{W}_j$ at $p$ coincides with $\rho$. Then, $\hat{W}_j$ is the Wulff shape for $\hat{\gamma}_j$. For later use, we denote by $\hat{\xi}_j$ the Cahn–Hoffman map for $\hat{\gamma}_j$.

Now, let $\chi:S^{n-1} \to \Pi_j$ be a $C^\infty$ embedding with outward unit normal $\rho = \rho_j$. Define the anisotropic energy of $\chi$ by

where $ds$ is the $(n-1)$-dimensional volume form of $\chi$.

From now on, we assume that

is a capillary hypersurface. Set

Denote by $\rho$ the outward-pointing unit normal to $\chi_j$ in the hyperplane $\Pi_j$. $X$ has the following property, which we call the balancing formula that is a generalization of the balancing formula for the isotropic case ^[3].

Lemma 2.

Proof. Let $u$ be a constant vector in ${\mathbb R}^{n+1}$. Under parallel translations:

$\overline{V}$, ${\mathcal F}_\gamma$, ${\mathcal H}^n(D_j)$ are invariant. Hence, using the first variation formulas we gave above, we compute

By setting $u = (1, 0, \cdots, 0)$, $(0, 1, \cdots, 0)$, $\cdots$, $(0, \cdots, 0, 1)$, we have

On $\sigma_j$, since $\langle \tilde{\xi}, \tilde{N}_j\rangle = \omega_j$ and $\langle \tilde{\xi}, \rho_j\rangle = \langle \hat{\xi}_j, \rho_j\rangle = \hat{\gamma}_j$ hold, we can write

where $\tau$ is tangent to $C_j$. Then, we have

Note that, by the divergence theorem, it holds that

Hence, substituting Eq (4.4) into Eq (4.3), we obtain

Because $\tilde{N}_1$, $\cdots$, $\tilde{N}_k$ are linearly independent, Eq (4.5) implies Eq (4.2).

Now, consider the anisotropic parallel hypersurfaces $X_t: = X+t\tilde{\xi}$ ($t \in {\mathbb R}$, $|t| < < 1$) of $X$ (Figure 2, upper left). If $\omega_j > 0$ for some $j \in \{1, \cdots, k\}$, then $X_t$ does not satisfy the boundary condition, that is, the boundary $X_t(\partial M)$ of the hypersurface may not be included in the boundary $\partial \Omega \subset \Pi_1 \cup \cdots \cup \Pi_k$ of the wedge-shaped domain $\Omega$. We will prove that, by taking a suitable parallel translation $Z_t = X_t+t\vec{v}$ of $X_t$, $Z_t$ satisfies the boundary condition (Figure 2, upper right).

For any $a \in {\mathbb R}$, define the hyperplane $\Pi_j^a$ in ${\mathbb R}^{n+1}$ that is parallel to the hyperplane $\Pi_j$ as follows.

Then, we show:

Lemma 3. $X_t(\sigma_j)\subset \Pi_j^{t\omega_j}$ holds.

Proof. From Proposition 2 (ii), we have

Since $X(\sigma_j) \subset \Pi_j$, (4.7) gives the desired result.

By using Lemma 3, we will show the following.

Lemma 4. There exists some $\vec{v} \in {\mathbb R}^{n+1}$ such that

is satisfied, and the parallel translation $Z_t = X_t+t\vec{v}$ of $X_t$ satisfies the boundary condition, that is,

holds.

Proof. From Lemma 3,

holds. Hence, for any $\vec{v} \in {\mathbb R}^{n+1}$,

holds. Therefore, $X_t(\sigma_j) + t\vec{v}\subset \Pi_j$ if and only if $\omega_j\tilde{N}_j+\vec{v} \in \Pi_j$, which is equivalent to

that is,

Now set $\vec{v} = (v_1, \cdots, v_{n+1})$. Then,

is a system of linear equations in the $(n+1)$ variables $v_1, \cdots, v_{n+1}$ with $k$ equations satisfying $k \le n+1$. Hence, (4.12) has at least one solution $\vec{v}$, which proves the desired result.

We have proved that $Z_t = X_t+t\vec{v}$ satisfies the boundary condition. However, it is not a volume-preserving variation in general. We can take suitable homotheties

of $Z_t$ if necessary so that $Y_t:(M, \sigma_1, \cdots, \sigma_k)\to (\overline\Omega, \Pi_1, \cdots, \Pi_k)$ is a volume-preserving variation of $X$ (Figure 2).

Denote by $e(t)$ the total energy $E(Y_t)$ of $Y_t$. Then we obtain

Claim 1.

where $\hat{\Lambda}$ is the anisotropic curvature of $\chi_j$ for $\hat{\gamma}_j$.

Claim 1 will be proved in §6. Note that, from Remark 2(i), each $k_j^\gamma$ is real. Since $X$ has constant anisotropic mean curvature $\Lambda$, the first term of the right hand side of Eq (4.13) is nonnegative if and only if $k_1^\gamma = \cdots = k_n^\gamma = \Lambda/n \ne 0$. Hence, by Corollary 1 in ^[15], $X(M) \subset (1/|\Lambda|)W_\gamma$ holds. Here we used $\Lambda\ne 0$ which is true because of Lemma 2.

Let us study the second term of the right hand side of the equation of $e"(0)$ in Claim 1. Set

Then we can prove the following statement (see §7 for its proof).

Claim 2. $B_j \ge 0$ holds and that the equality holds if and only if $\chi_j(\sigma_j) = r\hat{W}_j$ for some $r > 0$.

Now we are in the final position to complete the proof of the first half of Theorem 1. If the capillary hypersurface $X$ is stable, then $e"(0)\ge 0$ must hold. Hence, by the above observations, $X(M) \subset (1/|\Lambda|)W_\gamma$ holds.

5.   Proof of the second half of Theorem 1: the existence of the stable solution

Let $X:(M, \sigma_1, \cdots, \sigma_k)\to (\overline\Omega, \Pi_1, \cdots, \Pi_k)$ be an anisotropic capillary hypersurface, and we assume that $X$ is an embedding onto a part of $W_\gamma$ (up to translation and homothety). We will prove that $X$ is stable. It is sufficient to prove the stability for the case where $X$ is an embedding onto a part of $W_\gamma$. Then, there exists a point $Q \in {\mathbb R}^{n+1}$ such that

holds. Set

Then, $\Sigma$ is a closed convex piecewise-smooth hypersurface in ${\mathbb R}^{n+1}$. We will derive the support function of $\Sigma$ (^[16]). For any $x \in S^n$, we define a hyperplane $P(x)$ that is orthogonal to $x$ as follows.

Define a continuous function $\varphi: S^n \to {\mathbb R}_{ > 0}$ as follows.

Then, the homogeneous extension $\overline{\varphi}:{\mathbb R}^{n+1} \to {\mathbb R}$ of $\varphi$ is the support function of $\Sigma$. Then, $\Sigma$ is the Wulff shape for $\varphi$ (^[17]), that is, $\Sigma = W_\varphi$ holds. Note that

holds. Since $\gamma(x)\ge \varphi(x)$ holds, we have $\tilde{W}[\gamma] \supset \tilde{W}[\varphi]$. Hence we have the followings:

(i) If $x \in \nu(M)$, then $\varphi(x) = \gamma(x)$.

(ii) If $x = \tilde{N}_j$, then $\varphi(x) = \omega_j < \gamma(x)$, ($j = 1, \cdots, k$).

(iii) If $x \in S^n\setminus (\nu(M)\cup \{\tilde{N}_1, \cdots, \tilde{N}_k \})$, then $\varphi(x) < \gamma(x)$.

Now let $X_t:(M, \sigma_1, \cdots, \sigma_k)\to (\overline\Omega, \Pi_1, \cdots, \Pi_k)$ be a volume-preserving variation of $X$. Denote by $D_j(X_t)$ the domain bounded by $X_t(\sigma_j)$ in $\Pi_j$. Set

Then, because $\gamma \ge \varphi$ and $\Sigma$ is the minimizer of ${\mathcal F}_\varphi$ among all closed piecewise-smooth hypersurfaces enclosing the same $(n+1)$-dimensional volume, we obtain

holds. Therefore, $X$ is stable.

6.   Proof of Claim 1

Recall

$Y_t:(M, \sigma_1, \cdots, \sigma_k)\to (\overline\Omega, \Pi_1, \cdots, \Pi_k)$ is a volume-preserving variation of $X$ that satisfies the boundary condition. Note that the unit normal vector field along $X_t$, $Z_t$, and $Y_t$ coincide with $\nu:M\to S^n$ that is the unit normal vector field along $X$. Hence,

holds, where $dA_{Y_t}$ is the $n$-dimensional volume form of $Y_t$.

Set

and

Then

In order to compute $e"(0)$, we need to compute $\mu'(0), \mu"(0), f'(0)+w'(0), f"(0)$, and $w"(0)$. First, using the Steiner-type formula (2.4), we obtain the followings.

Before computing the other derivatives, we prepare two useful lemmas.

Lemma 5. We have the following equalities.

(i)

(ii) In the special case where $k = 0$, that is $\partial M = \emptyset$, we have

Proof. Since

by using the first variation formula (3.2) for the volume, we have

We compute the last term of (6.16). Denote by $dA(\Pi_j^{t\omega_j})$ the standard volume form on the $n$-plane $\Pi_j^{t\omega_j}$. Using the divergence formula and the equality $\langle \vec{v}, \tilde{N}_j\rangle = -\omega_j$ (see Lemma 4), we have

(6.16) with (6.17) gives the desired equality (6.14). The proof of (6.15) is similar.

Lemma 6. We have the following equality.

Proof. Consider the variation ${\hat X}_\epsilon: = (1+\epsilon)X$ of $X$, and set $F(\epsilon): = {\mathcal F}_\gamma({\hat X}_\epsilon)$. Then,

Hence,

On the other hand, the first variation formula (3.1) of ${\mathcal F}_\gamma$ gives

We compute the integrand of the second term of (6.20) on $\sigma_j$. Proposition 2 (ii) gives $\langle \tilde{\xi}, \tilde{N}_j\rangle = \omega_j$ on $\sigma_j$. Hence,

Using (6.21) and the equality $\langle X, \tilde{N}_j\rangle = 0$, we have

Inserting (6.22) to (6.20), we obtain

(6.19) with (6.23) gives the desired equality (6.18).

Let us continue the proof of Claim 1. Using the equalities (6.1), (6.10), and (6.14), we have

Hence we have

Next we compute $f'(0)+w'(0)$. Note that $e'(0) = 0$ because $X$ is a capillary hypersurface. Using (6.8) and (6.25), we obtain

which implies that

holds.

Now we compute $w"(0)$. Using the first variation formula (3.2) for volume, we obtain

where $\rho_t$ is the outward-pointing unit normal vector field along $X_t|_{\sigma_j}:\sigma_j \to \Pi_j^{t\omega_j}$, and $ds_t$ is the $(n-1)$-dimensional volume form of $X_t|_{\sigma_j}$. Using (6.28) and the first variation formula (3.1) of the anisotropic energy, we obtain

Finally we compute $\mu"(0)$ by using $v"(0) = 0$ and (6.11). From (6.14), we have

here in the last equality we used (6.8). Inserting (6.25), (6.30), (6.31) to (6.11), we compute to obtain

Inserting (6.13), (6.25), (6.8), (6.29), and (6.32) to (6.9) at $t = 0$, we obtain

From Lemma 6, we have

Inserting this to (6.33), we obtain

Now we are in the final stage to prove Claim 1. The balancing formula (4.2) implies

Hence

holds. This with (6.34) proves Claim 1.

7.   Proof of Claim 2

In this section, we examine the sign of

where

First, we recall two known useful propositions, which we prove for completeness.

Proposition 3. Let $\gamma:S^m \to {\mathbb R}_{ > 0}$ be a positive strictly convex function of class at least $C^2$, and $W \subset {\mathbb R}^{m+1}$ be its Wulff shape. Denote by ${\mathcal F}_\gamma(W)$ the anisotropic energy of $W$ for $\gamma$, and by ${\mathcal H}^{m+1}(W)$ the $(m+1)$-dimensional Hausdorff measure of the domain bounded by $W$ in ${\mathbb R}^{m+1}$. Then,

holds.

Proof. Let $\xi:S^m \to{\mathbb R}^{m+1}$ be the Cahn-Hoffman map of $\gamma$ defined by $\xi(\nu) = D\gamma|_{\nu} +\gamma(\nu)\nu$. Then $\xi$ is an embedding onto $W$. Hence, denoting by $dA_{\xi}$ the volume form of $\xi$, we have

which proves (7.2).

Proposition 4 (Anisotropic isoperimetric inequality). Let $\gamma$, $W$ be the same as in Proposition 3. We also use the same notation as in Proposition 3. Then, for any closed embedded piecewise-smooth hypersurface $M$ in ${\mathbb R}^{m+1}$, it holds that

where the equality holds if and only if $M$ coincides with $W$ up to homothety and translation in ${\mathbb R}^{m+1}$.

Proof. Recall that the Wulff shape $W$ is the minimizer of ${\mathcal F}_\gamma$ among closed hypersurfaces enclosing the same $(m+1)$-dimensional Hausdorff measure. Set

Then, the hypersurface

similar to $M$ encloses the same $(m+1)$-dimensional Hausdorff measure as $W$. Hence,

holds, where the equality holds if and only if $M$ coincides with $W$ up to homothety and translation. On the other hand,

The inequality (7.4) combined with the equalities (7.2) and (7.5) gives (7.3).

Now we examine $B_j$. From now on, for simplicity, we identify an embedded closed hypersurface in an euclidean space with the domain bounded by this hypersurface. We also identify an embedded closed hypersurface with its representation mapping.

Using Proposition 4, we have

First we study the special case where $n = 2$. In this case, using (7.6) with Proposition 3, we have

We use the representation (2.2) of the anisotropic mean curvature in Remark 1. Note that $\sigma_j$ is topologically $S^1$. Denote by $G:{\hat{W}}_j\to S^1$ the outward-pointing unit normal vector field on ${\hat{W}}_j$ and by $\hat{s}$ the arc-length parameter of ${\hat{W}}_j$. Then $G = \hat{\xi}_j^{-1}$, where $\hat{\xi}_j:S^1 \to {\hat{W}}_j$ is the Cahn-Hoffman map of ${\hat{W}}_j$. Hence, if we take $\rho \in S^1$ as the parameter of ${\hat{W}}_j$ through $\hat{\xi}_j$, $G$ is the identity mapping on $S^1$. We also denote by $\kappa_j$, $\hat{\kappa}_j$ the curvatures of $\chi_j$, ${\hat{W}}_j$ with respect to the outward-pointing unit normal, respectively. Then, using (2.2), we have

Inserting (7.8) into (7.7), we have $B_j\ge 0$.

Next we assume $n\ge 3$. We assume that $D_j$ are convex. Below, for simplicity, we omit the subscript $j$, that is, we write $D$ instead of $D_j$, for instance. On the Minkowski sum $D+t\hat{W}$, there holds

where $v(K_1, \cdots, K_n)$ is the so-called the mixed volume of convex bodies $K_1, \cdots, K_n$ in ${\mathbb R}^n$ (^[16, Theorem 5.1.7]). On the other hand, from Lemma 5 and the Steiner-type formula (2.4), we have

Comparing (7.9) with (7.10), we obtain

The Minkowski's second inequality (^[16,Theorem 7.2.1]) gives

where the equality holds if and only if $D$ is homothetic to $\hat{W}$. Combining (7.11)–(7.14), we obtain

here the equality holds if and only if $D$ is homothetic to $\hat{W}$. This completes the proof of Claim 2.

Acknowledgments

This work was partially supported by JSPS KAKENHI Grant Number JP20H01801, JSPS Grant-in-Aid for Scientific Research on Innovative Areas "Discrete Geometric Analysis for Materials Design": Grant Numbers JP18H04487 and JP20H04642, JST CREST GrantNumber JPMJCR1911, and JSPS(Japan)-FWF(Austria) Joint Research Project (2018-2019, Project leader (Japan): Miyuki Koiso).

Conflict of interest

The authors declare no conflict of interest.