Weakly stable solutions of Serrin’s problem

Abstract.

We prove that weakly stable solutions of Serrin’s problem are compact and therefore round balls.

1. Introduction

Let $n\geq 2$ and $\Omega\subset\mathbb{R}^{n}$ a domain, i.e., open, nonempty, and such that $\bar{\Omega}$ is a properly embedded top-dimensional submanifold. We denote the mean curvature of $\partial\Omega$ with respect to the outward pointing normal $\mu(\Omega)$ by $H$. We call $\Omega$ a solution of Serrin’s problem (1) if there is $u\in C^{\infty}(\bar{\Omega})$ such that

(1) $$\begin{cases}\quad-\Delta u=n\qquad&\text{in $\Omega$},\\ \quad u>0&\text{in $\Omega$},\\ \quad u=0&\text{on $\partial\Omega$, and}\\ \quad|Du|=1&\text{on $\partial\Omega$}.\end{cases}$$

Note that $\mu(\Omega)=-Du$ on $\partial\Omega$ in this setting.

Using the method of moving planes, J. Serrin [19] has shown that if $\Omega$ is compact and a solution of (1), then $\Omega$ is a ball of radius 1. Subsequently, H. Weinberger [25] has found a short alternative proof of this fact. Such a result fails if $\Omega$ is allowed to be noncompact. Indeed, for all integers $m,\,n$ with $1\leq m\leq n-1$, the cylinder

(2) $$\{y\in\mathbb{R}^{m}:|y|<m/n\}\times\mathbb{R}^{n-m}$$

is a solution of (1). In view of this observation, H. Berestycki, L. Caffarelli, and L. Nirenberg [2, p. 1110] have conjectured that if $\Omega$ is noncompact and a solution of (1), then $\Omega$ is either a ball of radius $1$ or congruent to one of the cylinders described in (2). Indeed, they have made a similar conjecture for solutions $\Omega$ of the semilinear over-determined problem

(3) $$\begin{cases}\quad-\Delta u=f(u)\qquad&\text{in $\Omega$},\\ \quad u>0&\text{in $\Omega$},\\ \quad u=0&\text{on $\partial\Omega$, and}\\ \quad|Du|=1&\text{on $\partial\Omega$},\end{cases}$$

where $f\in C^{\infty}(\mathbb{R})$. We note that, by the work of J. Serrin [19], if $\Omega$ is compact and a solution of (3), then $\Omega$ is a ball. Counterexamples to this conjecture have been found in, e.g., [20, 9] for certain choices of $f$ and, notably, by M. Fall, I. Minlend, and T. Weth [13] for the choice $f=n$; see also [23] for a related positive result in this direction. In this paper, we show that, in the case where $f=n$, the conjecture of H. Berestycki, L. Caffarelli, and L. Nirenberg holds under a natural additional assumption on $\Omega$.

To explain this, recall that, by the pioneering work of H. Alt and L. Caffarelli [1], solutions of (3) are given by $\Omega=\{y\in\mathbb{R}^{n}:u(y)>0\}$ where $u\in C^{\infty}(\mathbb{R}^{n})$ is a critical point of the functional

$$J(u)=\frac{1}{2}\,\int_{\{y\in\mathbb{R}^{n}:u(y)>0\}}|Du|^{2}-\int_{\{y\in\mathbb{R}^{n}:u(y)>0\}}F(u).$$

Here, $F\in C^{\infty}(\mathbb{R})$ is such that $F^{\prime}=f$; see also, e.g., [23]. In the case where $f=n$, $J$ is not bounded from below. It is therefore natural to study $u\in C^{\infty}(\mathbb{R}^{n})$ that are critical for $J$ given the relative volume of $\{y\in\mathbb{R}^{n}:u(y)>0\}$. In this context, we call $\Omega$ weakly stable if

(4) $$\int_{\partial\Omega}(f(0)-H)\,\phi^{2}\leq\int_{\Omega}(|D\phi|^{2}-f^{\prime}(u)\,\phi^{2})$$

for all $\phi\in C_{c}^{\infty}(\bar{\Omega})$ such that

$$\int_{\partial\Omega}\phi=0.$$

We call $\Omega$ stable if (4) holds for all $\phi\in C_{c}^{\infty}(\bar{\Omega})$. As we explain in Appendix A, $\Omega$ is weakly stable if and only if $u$ passes the second derivative test for $J$ among compact perturbations that preserve the relative volume of $\{y\in\mathbb{R}^{n}:u(y)>0\}$.

Notably, domains $\Omega$ that are solutions of (1) and are weakly stable arise in the context of minimal surface theory. To explain this, let $S\subset\mathbb{R}^{n+1}$ be a closed hypersurface with positive mean curvature bounding an open and bounded set $M\subset\mathbb{R}^{n+1}$. Let $\Sigma\subset\bar{M}$ be a relative boundary. In particular, $\partial\Sigma\subset S$ bounds a closed subset $S(\Sigma)\subset S$ called the wetting surface of $\Sigma$. $\Sigma$ is called a minimal capillary surface supported on $S$ if the mean curvature of $\Sigma$ vanishes and if $\Sigma$ and $S$ intersect along $\partial\Sigma$ at a constant angle $\theta\in(0,\pi)$. Minimal capillary surfaces are critical for area given the area of their wetting surface. They are called weakly stable if they pass the second derivative test for area among perturbations that fix the area of their wetting surface. In the case where $n=2$, the authors [12] have shown weakly stable solutions $\Omega$ of (1) arise as certain blowup limits of weakly stable minimal capillary surfaces supported on $S$ with $\theta$ close to either $0$ or $\pi$. This has led to a characterization of all weakly stable minimal capillary surfaces supported on $S$ with angle close to $0$ or $\pi$; see [12, Theorem 6]. A fundamental ingredient in the proof is the following rigidity result for weakly stable solutions of (1).

Theorem 1.

[12, Proposition 31] Let $n=2$. Assume that $\Omega$ is a weakly stable solution of (1) and that $\partial\Omega$ has bounded curvature. Then $\Omega$ is a ball of radius $1$.

In this paper, we show that Theorem 1 holds in all dimensions, even without the assumption that the curvature of $\partial\Omega$ is bounded. In the case where $f=n$, this gives an affirmative answer to the conjecture of H. Berestycki, L. Caffarelli, and L. Nirenberg under the, from a variational point of view, most natural additional assumption.

Theorem 2.

Let $n\geq 2$. Assume that $\Omega$ is a weakly stable solution of (1). Then $\Omega$ is a ball of radius $1$.

It is remarkable that Theorem 2 holds in all dimensions. Indeed, J. Serrin’s overdetermined problem (1) is closely related to the one-phase Bernoulli problem; see [1] and also [5] for a connection between the one-phase Bernoulli problem and minimal capillary surfaces. The one-phase Bernoulli problem is (3) with $f=0$, i.e.,

(5) $$\begin{cases}\quad-\Delta u=0\qquad&\text{in $\Omega$},\\ \quad u>0&\text{in $\Omega$},\\ \quad u=0&\text{on $\partial\Omega$, and}\\ \quad|Du|=1&\text{on $\partial\Omega$}.\end{cases}$$

In the case where $n\leq 3$, stable solutions $\Omega$ of (5) are known to be half-spaces; see [4] and, e.g., [15]. By contrast, in the case where $n\geq 7$, there are infinitely many geometrically distinct stable solutions of (5); see [8]. This shows that the assumption that $f=n$ in Theorem 2 cannot be dropped altogether. We note that, in a similar context, M. do Carmo [11] has conjectured that constant mean curvature hypersurfaces in $\mathbb{R}^{n+1}$ that pass the second derivative test for area among all hypersurfaces enclosing the same relative volume are round spheres for all $n\geq 2$. By contrast, stable minimal hypersurfaces in $\mathbb{R}^{n+1}$ are known to be hyperplanes in the case where $n\leq 5$ while there are infinitely many geometrically distinct stable minimal hypersurfaces in $\mathbb{R}^{n+1}$ in the case where $n\geq 7$; see, e.g., [10, 18, 14, 7, 6, 16] and [21].

Outline of our arguments

Our approach toward Theorem 2 differs substantially from our proof of Theorem 1 in [12], which is based on the Gauss-Bonnet theorem and therefore specific to the case where $n=2$. It also differs substantially from the arguments employed in the recent work of H. Chan, X. Fernández-Real, A. Figalli, and J. Serra [4] on (5), which are specific to the case where $n=3$ and, in view of the counterexamples found in [8], cannot possibly be adapted to the case where $n\geq 7$.

First, we suppose, for a contradiction, that $\Omega$ has infinite volume. Given $\lambda>0$, let $A(\lambda)=|B_{\lambda}(0)\cap\partial\Omega|$ and $V(\lambda)=|B_{\lambda}(0)\cap\Omega|$. Using (1), we show that

(6) $$A(\lambda)\approx n\,V(\lambda)$$

for suitable numbers $\lambda\gg 1$; see Proposition 14. In conjunction with a balancing argument, (6) implies that $\Omega$ is stable; see Lemma 20. The proof relies on certain gradient estimates for $u$ that we establish in Section 2. We then choose a sequence $\{\eta_{k}\}_{k=1}^{\infty}$ of suitable cutoff functions $\eta_{k}\in C^{\infty}_{c}(\mathbb{R}^{n})$ with $\eta_{k}\to 1$ locally smoothly. Applying the stability of $\Omega$ with $\phi=\eta_{k}$ and using (6), we obtain that

$$\int_{\partial\Omega}\eta_{k}^{2}\,(n-1-H)\leq o(1)\,A(\lambda_{k})-A(\lambda_{k}).$$

Applying the stability of $\Omega$ with $\phi=\eta_{k}\,\varphi$ where $\varphi=u+|Du|^{2}/2$ and using (6) again, we conclude that

$$\int_{\partial\Omega}\eta_{k}^{2}\,(n-H)\leq-(2+o(1))\,A(\lambda_{k});$$

see (20). A crucial ingredient here is that $\varphi$ is subharmonic in $\Omega$ and satisfies $\mu(\Omega)\cdot D\varphi=n-1-H$ on $\partial\Omega$; see Lemma 15. By contrast, the gradient estimates in Section 2 show that $n\geq H$; see Lemma 12. This leads to a contradiction.

It follows that $\Omega$ has finite volume. In view of the gradient estimates in Section 2, we show how to adapt the argument of H. Weinberger [25] to this setting; see Proposition 18. In particular, every component of $\Omega$ is a ball of radius 1. Using the weak stability of $\Omega$, we see that $\Omega$ is connected.

On the use of artificial intelligence (A.I.)

None of the arguments employed here have been suggested by or discussed with A.I. tools. The paper does not contain text written by A.I. tools. The authors have used Claude Opus 5 to check the typesetting of this paper.

Acknowledgments

This research was funded in whole or in part by the Austrian Science Fund (FWF) [10.55776/PAT1307525, 10.55776/PAT2423724].

2. Gradient estimates

Throughout, we assume that $n\geq 2$. We use $O(1)$ to denote constants that only depends on $n$.

The arguments in this section are well-known to experts in the field.

Lemma 3.

There holds $\Omega\neq\mathbb{R}^{n}$.

Proof.

Suppose, for a contradiction, that $\Omega=\mathbb{R}^{n}$. Then $v\in C^{\infty}(\mathbb{R}^{n})$ given by

$$v(x)=u(x)+\frac{1}{2}\,|x|^{2}$$

is positive and harmonic. By the Liouville theorem, $v$ is a positive constant. Since $u>0$, this leads to a contradiction. ∎

Given $x\in\mathbb{R}^{n}$ and $r>0$, let

$$B_{r}(x)=\{y\in\mathbb{R}^{n}:|y-x|<r\}.$$

Given $x\in\Omega$, let

$$r(x)=\operatorname{dist}(x,\partial\Omega).$$

By Lemma 3, $r(x)<\infty$. Note that $B_{r(x)}(x)\subset\Omega$ and $\partial\Omega\cap\partial B_{r(x)}(x)\neq\emptyset$. Moreover,

$$\mu(\Omega)(z)=\frac{z-x}{|z-x|}$$

for all $z\in\partial\Omega\cap\partial B_{r(x)}(x)$.

Lemma 4 (cp. [12, Lemma 25]).

Let $x\in\Omega$. There holds

$$r(x)\leq 1.$$

Proof.

Let $v\in C^{\infty}(\mathbb{R}^{n})$ be given by

$$v(y)=\frac{1}{2}\,r(x)^{2}-\frac{1}{2}\,|y-x|^{2}.$$

Note that $u-v$ is harmonic in $B_{r(x)}(x)$ and nonnegative on $\partial B_{r(x)}(x)$. By the maximum principle, $u\geq v$ in $B_{r(x)}(x)$. Also note that $u(z)=v(z)$ for all $z\in\partial\Omega\cap\partial B_{r(x)}(x)$. At such $z$,

$$0\geq\mu(\Omega)\cdot Du-\mu(\Omega)\cdot Dv=-1+r(x).$$

This completes the proof of the lemma. ∎

Lemma 5.

There holds

$$\frac{1}{2}\,\Delta|Du|^{2}\geq n$$

in $\Omega$.

Proof.

By the Bochner formula,

(7) $$\frac{1}{2}\,\Delta|Du|^{2}=Du\cdot D\Delta u+|D^{2}u|^{2}=|D^{2}u|^{2}.$$

By the inequality of arithmetic and geometric means,

(8) $$|D^{2}u|^{2}\geq\frac{(\Delta u)^{2}}{n}=n.$$

This completes the proof of the lemma.

∎

Corollary 6.

Assume that $\Omega$ is connected and that

$$\frac{1}{2}\,\Delta|Du|^{2}=n.$$

There is $x\in\mathbb{R}^{n}$ such that $\Omega=B_{1}(x)$.

Proof.

By assumption, equality holds in (8). It follows that $D^{2}u=-\operatorname{Id}$. Equivalently, there is $x\in\mathbb{R}^{n}$ such that

$$u(y)+\frac{1}{2}\,|y-x|^{2}$$

is constant in $\Omega$. Since $u=0$ and $|Du|=1$ on $\partial\Omega$, $\Omega=B_{1}(x),$ as asserted. ∎

Lemma 7 (cp. [24, Lemma 2.3]).

There holds

$$\sup\{|Du(x)|:x\in\Omega\}<\infty.$$

Proof.

Fix $x\in\Omega$. Let $v:\mathbb{R}^{n}\to\mathbb{R}$ be given by

$$v(y)=\frac{1}{2}\,r(x)^{2}-\frac{1}{2}\,|y-x|^{2}.$$

Note that $u-v$ is harmonic in $B_{r(x)}(x)$ and that $u-v\geq 0$ on $\partial B_{r(x)}(x)$ with equality for all $z\in\partial\Omega\cap\partial B_{r(x)}(x)$. By the strong maximum principle, either $u(y)=v(y)$ for all $y\in B_{r(x)}(x)$ or $u(y)>v(y)$ for all $y\in B_{r(x)}(x)$. Since $Dv(x)=0$, we may assume that $u(y)>v(y)$ for all $y\in B_{r(x)}(x)$. Let

$$\kappa=\inf\left\{u(y)-v(y):y\in\partial B_{r(x)/2}(x)\right\}.$$

By the Harnack inequality,

(9) $$u(x)-v(x)=O(1)\,\kappa.$$

Let $h:\mathbb{R}^{n}\setminus\{x\}\to\mathbb{R}$ be the harmonic function given by

$$h(y)=\frac{1}{\log 2}\,\kappa\,(\log r(x)-\log|y-x|)$$

if $n=2$ and

$$h(y)=\frac{r(x)^{n-2}}{2^{n-2}-1}\,\kappa\,(|y-x|^{2-n}-r(x)^{2-n})$$

if $n\geq 3$. Note that

$$h(y)=\begin{cases}\kappa\qquad&\text{if $|y-x|={r(x)}/2$ and}\\ 0&\text{if $|y-x|=r(x)$.}\end{cases}$$

By the maximum principle, $u-v\geq h$ in $\bar{B}_{r(x)}(x)\setminus B_{r(x)/2}(x)$ with equality for all $z\in\partial\Omega\cap\partial B_{r(x)}(x)$. At such $z$, there holds

$$0\geq\mu(\Omega)\cdot D(u-v)-\mu(\Omega)\cdot Dh=-1+r(x)+\frac{1}{\log 2}\,\frac{\kappa}{r(x)}\quad\,\,\,\,\,$$

if $n=2$ and

$$0\geq\mu(\Omega)\cdot D(u-v)-\mu(\Omega)\cdot Dh=-1+r(x)+\frac{n-2}{2^{n-2}-1}\,\frac{\kappa}{r(x)}$$

if $n\geq 3$. It follows that $\kappa=O(1)\,r(x)$. In conjunction with (9) and the gradient estimate for positive harmonic functions,

$$|D(u-v)(x)|=O(1)\,\frac{u(x)-v(x)}{r(x)}=O(1).$$

Since $Dv(x)=0$, the assertion follows. ∎

Corollary 8 (cf. [24, Proposition 2.1]).

There holds $|Du|<1$ in $\Omega$.

Proof.

Let $0<\varepsilon<1$. By Lemma 5, $\psi_{\varepsilon}\in C^{\infty}(\bar{\Omega})$ given by

$$\psi_{\varepsilon}(y)=|Du(y)|^{2}-\varepsilon\,|y|^{2}$$

is strictly subharmonic. By the maximum principle and Lemma 7, $\psi_{\varepsilon}$ attains its maximum on $\partial\Omega$. In particular, $|Du(y)|^{2}-\varepsilon\,|y|^{2}<1$ for all $y\in\Omega$. Letting $\varepsilon\to 0$, we obtain that $|Du(y)|\leq 1$ for all $y\in\Omega$. By the strong maximum principle, either $|Du|=1$ in $\Omega$ or $|Du|<1$ in $\Omega$. In view of Lemma 5, the former alternative is impossible. This completes the proof of the corollary. ∎

Remark 9.

The estimate

$$\frac{1}{2}\,|Du|^{2}\leq F(u)$$

for solutions of (3) where $F\in C^{\infty}(\mathbb{R})$ is such that $F(0)=1/2$ and $F^{\prime}=f$ proven in [24, Proposition 6.3] gives

$$|Du|^{2}\leq 1+\frac{n}{2}\,u$$

in the case where $f=n$; see also [17]. Note that, unlike Corollary 8, this estimate is not optimal.

Corollary 10.

There holds $0<u<1$ in $\Omega$.

Proof.

This follows from Lemma 4 and Corollary 8. ∎

Lemma 11.

There holds

$$D^{2}u(Du,Du)=H-n$$

on $\partial\Omega$.

Proof.

There holds

$$\Delta v=\Delta^{\partial\Omega}v+D^{2}v(\mu(\Omega),\mu(\Omega))+H\,\mu(\Omega)\cdot Dv$$

for every $v\in C^{\infty}(\bar{\Omega})$. The assertion follows from this, taking $v=u$. ∎

Lemma 12 (cf. [24, Lemma 6.3] and [3]).

There holds $H\leq n$ on $\partial\Omega$.

Proof.

By Corollary 8,

$$\mu(\Omega)\cdot D|Du|^{2}\geq 0$$

and $\mu(\Omega)\cdot D|Du|^{2}=-2\,D^{2}u(Du,Du)$ on $\partial\Omega$. The assertion follows from this and Lemma 11. ∎

3. Asymptotics

Given $\lambda>0$, let

$$A(\lambda)=|B_{\lambda}(0)\cap\partial\Omega|\qquad\text{and}\qquad V(\lambda)=|B_{\lambda}(0)\cap\Omega|.$$

Note that $V(\lambda)\leq|B_{\lambda}(0)|=\lambda^{n}\,|B_{1}(0)|$ for all $\lambda>0$ and that $V$ is nondecreasing and Lipschitz. For almost every $\lambda>0$, $\partial B_{\lambda}(0)$ and $\partial\Omega$ intersect transversely. $V$ is differentiable at such $\lambda>0$ and there holds

$$V^{\prime}(\lambda)=|\partial B_{\lambda}(0)\cap\Omega|.$$

Moreover, by the divergence theorem,

$$A(\lambda)=-\int_{B_{\lambda}(0)\cap\partial\Omega}\mu(\Omega)\cdot Du=n\,V(\lambda)+\int_{\Omega\cap\partial B_{\lambda}(0)}\frac{x}{|x|}\cdot Du.$$

Let $E:(0,\infty)\to\mathbb{R}$ be given by

$$E(\lambda)=\int_{\Omega\cap\partial B_{\lambda}(0)}\frac{x}{|x|}\cdot Du.$$

By Corollary 8,

$$|E(\lambda)|\leq V^{\prime}(\lambda)$$

for all $\lambda>0$ as above and

(10) $$A(\lambda)=n\,V(\lambda)+E(\lambda).$$

Lemma 13.

There is a sequence $\{m_{k}\}_{k=1}^{\infty}$ of positive integers with $m_{k}\to\infty$ such that

$$\frac{V(m_{k}+2)-V(m_{k})}{V(m_{k})}=o(1).$$

Proof.

We may assume that $V(1)>0$. For every integer $M\geq 1$, there holds

$$\sum_{m=1}^{M}\frac{V(m+2)-V(m)}{V(m+2)} \leq\sum_{m=1}^{M}\int_{V(m)}^{V(m+2)}\frac{1}{t}\,\mathrm{d}t$$

$$\leq 2\,\int_{V(1)}^{V(M+2)}\frac{1}{t}\,\mathrm{d}t$$

$$=2\,\log\frac{V(M+2)}{V(1)}$$

$$=O(1)\,\log M.$$

It follows that there is a sequence $\{m_{k}\}_{k=1}^{\infty}$ of positive integers with $m_{k}\to\infty$ such that

$$\frac{V(m_{k}+2)-V(m_{k})}{V(m_{k}+2)}=o(1).$$

In particular,

$$\frac{V(m_{k}+2)}{V(m_{k})}=1+o(1).$$

The assertion follows from these estimates. ∎

Proposition 14.

There is a sequence $\{\lambda_{k}\}_{k=1}^{\infty}$ of positive numbers with $\lambda_{k}\to\infty$ such that

(11) $$\frac{V(\lambda_{k}+1)-V(\lambda_{k})}{A(\lambda_{k})}=o(1)$$

and

(12) $$\frac{n\,V(\lambda_{k})}{A(\lambda_{k})}=1+o(1).$$

Proof.

Let $\{m_{k}\}_{k=1}^{\infty}$ be as in Lemma 13. Since $V$ is Lipschitz, there is $\lambda_{k}\in(m_{k},m_{k}+1)$ such that $\partial B_{\lambda_{k}}(0)$ and $\partial\Omega$ intersect transversely and

$$V^{\prime}(\lambda_{k})\leq V(m_{k}+1)-V(m_{k}).$$

Using also that $V$ is nondecreasing,

$$V^{\prime}(\lambda_{k})\leq V(m_{k}+2)-V(m_{k})=o(1)\,V(m_{k})\leq o(1)\,V(\lambda_{k}).$$

In conjunction with (10),

$$A(\lambda_{k})=(1+o(1))\,n\,V(\lambda_{k}).$$

Note that

$$V(\lambda_{k}+1)-V(\lambda_{k})\leq V(m_{k}+2)-V(m_{k})=o(1)\,V(m_{k}).$$

Using that

$$n\,V(m_{k})\leq n\,V(\lambda_{k})=(1+o(1))\,A(\lambda_{k}),$$

the assertion follows. ∎

4. Finite volume

Let $\varphi\in C^{\infty}(\bar{\Omega})$ be given by

(13) $$\varphi(x)=u(x)+\frac{1}{2}\,|Du(x)|^{2}.$$

Lemma 15.

There holds

$$-\Delta\varphi\leq 0$$

in $\Omega$ and

$$\varphi=\frac{1}{2}\qquad\text{and}\qquad\mu(\Omega)\cdot D\varphi=n-1-H$$

on $\partial\Omega$.

Proof.

This follows from Lemma 5 and Lemma 11. ∎

Corollary 16.

Assume that $\Omega$ is connected and that $\varphi=1/2$ in $\Omega$. There is $x\in\mathbb{R}^{n}$ such that $\Omega=B_{1}(x)$.

Proof.

In view of Lemma 5, the proof of Lemma 15 shows that

$$\frac{1}{2}\,\Delta|Du|^{2}=n.$$

The assertion now follows from Corollary 6. ∎

Lemma 17.

There holds $\varphi<3/2$ in $\Omega$.

Proof.

The assertion follows from Corollary 8 and Corollary 10. ∎

The proof of the following proposition adapts that of [25, Theorem 1] to the case where $\Omega$ is not a-priori bounded.

Proposition 18.

Assume that $\Omega$ is connected and has finite volume. There is $x\in\mathbb{R}^{n}$ such that $\Omega=B_{1}(x)$.

Proof.

Since $\Omega$ has finite volume, there is a sequence $\{\lambda_{k}\}_{k=1}^{\infty}$ of positive numbers such that $\lambda_{k}\to\infty$ and

(14) $$|(B_{\lambda_{k}+1}(0)\setminus B_{\lambda_{k}}(0))\cap\Omega|=o(1)\,\frac{1}{\lambda_{k}}.$$

Let $\eta_{k}\in C_{c}^{\infty}(\mathbb{R}^{n}),$ $k\geq 1,$ be such that

• $\eta_{k}(x)=1$ for all $x\in B_{\lambda_{k}}(0)$,

• $\eta_{k}(x)=0$ for all $x\in\mathbb{R}^{n}\setminus B_{\lambda_{k}+1}(0)$,

• $|D\eta_{k}|=O(1)$, and

• $|D^{2}\eta_{k}|=O(1)$.

Note that

$$-\Delta(x\cdot Du)=2\,n.$$

By the divergence theorem,

$$2\,n\,\int_{\Omega}\eta_{k}\,u-n\,\int_{\Omega}\eta_{k}\,x\cdot Du$$

$$\qquad=-\int_{\Omega}\eta_{k}\,u\,\Delta(x\cdot Du)+\int_{\Omega}\eta_{k}\,(x\cdot Du)\,\Delta u$$

$$\qquad=\int_{\partial\Omega}\eta_{k}\,(x\cdot Du)\,\mu(\Omega)\cdot Du+\int_{\Omega}u\,D(x\cdot Du)\cdot D\eta_{k}-\int_{\Omega}(x\cdot Du)\,Du\cdot D\eta_{k}$$

$$\qquad=-\int_{\partial\Omega}\eta_{k}\,(x\cdot Du)-\int_{\Omega}u\,(x\cdot Du)\,\Delta\eta_{k}-2\,\int_{\Omega}(x\cdot Du)\,Du\cdot D\eta_{k}.$$

Moreover, using that $\operatorname{div}x=n$,

$$-\int_{\partial\Omega}\eta_{k}\,(x\cdot Du)=\int_{\partial\Omega}\eta_{k}\,\mu(\Omega)\cdot x=n\,\int_{\Omega}\eta_{k}+\int_{\Omega}x\cdot D\eta_{k}.$$

By (14),

$$\int_{\Omega}x\cdot D\eta_{k}=o(1).$$

By Corollary 8, Corollary 10, and (14),

$$\int_{\Omega}u\,(x\cdot Du)\,\Delta\eta_{k}=o(1).$$

By Corollary 8 and (14),

$$\int_{\Omega}(x\cdot Du)\,Du\cdot D\eta_{k}=o(1).$$

By the divergence theorem,

$$\int_{\Omega}\eta_{k}\,x\cdot Du=-n\,\int_{\Omega}\eta_{k}\,u-\int_{\Omega}u\,x\cdot D\eta_{k}.$$

By Corollary 10 and (14),

$$\int_{\Omega}u\,x\cdot D\eta_{k}=o(1).$$

By the dominated convergence theorem, using Corollary 10, we conclude that

(15) $$(n+2)\,\int_{\Omega}u=|\Omega|.$$

Suppose, for a contradiction, that $\Omega$ is unbounded. Let $\{x_{\ell}\}_{\ell=1}^{\infty}$ be a sequence of points in $\Omega$ such that $|x_{\ell}|\to\infty$. Since $\Omega$ has finite volume, there holds $r(x_{\ell})=o(1)$. By Corollary 8, $u(x_{\ell})=o(1)$. Using Corollary 8 again, we conclude that $\varphi(x_{\ell})\leq 1/2+o(1)$. By Lemma 15 and the maximum principle,

(16) $$\varphi\leq\frac{1}{2}$$

in $\Omega$. By the divergence theorem,

$$\int_{\Omega}\eta_{k}\,|Du|^{2}=n\,\int_{\Omega}\eta_{k}\,u-\int_{\Omega}u\,Du\cdot D\eta_{k}.$$

By the dominated convergence theorem, using Corollary 8, Corollary 10, and (14),

$$\int_{\Omega}|Du|^{2}=n\,\int_{\Omega}\,u.$$

Thus,

$$2\,\int_{\Omega}\varphi=2\,\int_{\Omega}u+\int_{\Omega}|Du|^{2}=(n+2)\,\int_{\Omega}u.$$

In view of (15) and (16), we conclude that $\varphi=1/2$. By Corollary 16, there is $x\in\mathbb{R}^{n}$ such that $\Omega=B_{1}(x)$, contradicting that $\Omega$ is unbounded.

It follows that $\Omega$ is bounded. As in the previous paragraph, we see that there is $x\in\mathbb{R}^{n}$ such that $\Omega=B_{1}(x)$. This completes the proof of the proposition. ∎

5. Stability and weak stability

The following lemma has essentially been observed in [22, Theorem 3.5].

Lemma 19.

Let $\eta\in C_{c}^{\infty}(\bar{\Omega})$. There holds

(17) $$\int_{\Omega}|D(|Du|\,\eta)|^{2}+\int_{\partial\Omega}(H-n)\,\,\eta^{2}\leq\int_{\Omega}|D\eta|^{2}.$$

Proof.

By the divergence theorem, using (7) and Lemma 11,

$$\frac{1}{2}\,\int_{\Omega}D|Du|^{2}\cdot D\eta^{2}=-\int_{\Omega}|D^{2}u|^{2}\,\eta^{2}+\int_{\partial\Omega}(n-H)\,\eta^{2}.$$

Thus,

$$\int_{\Omega}|D(|Du|\,\eta)|^{2}+\int_{\partial\Omega}(H-n)\,\eta^{2}$$

$$\qquad=-\int_{\Omega}|D^{2}u|^{2}\,\eta^{2}+\int_{\Omega}|D|Du||^{2}\,\eta^{2}+\int_{\Omega}|Du|^{2}\,|D\eta|^{2}.$$

Note that $|D^{2}u|\geq|D|Du||$ almost everywhere. Using also Corollary 8, we obtain (17). ∎

In the case where $n=2$ and $\partial\Omega$ has bounded curvature, the following lemma has been obtained in [12, Lemma 26].

Lemma 20.

Assume that $\Omega$ has infinite volume and is weakly stable. Then $\Omega$ is stable.

Proof.

Let $\phi\in C_{c}^{\infty}(\bar{\Omega})$ be such that

$$\int_{\partial\Omega}\phi=1.$$

Let $\lambda>2$ be such that $\operatorname{spt}(\phi)\subset B_{\lambda-2}(0)$. Let $\{\lambda_{k}\}_{k=1}^{\infty}$ be as in Proposition 14. We may assume that $\lambda_{k}>\lambda$ for all $k$. We choose $\eta_{k}\in C_{c}^{\infty}(\mathbb{R}^{n}),$ $k\geq 1$, such that

• $\eta_{k}(x)=1$ for all $x\in B_{\lambda_{k}}(0)\setminus B_{\lambda}(0)$,

• $\eta_{k}(x)=0$ for all $x\in B_{\lambda-1}(0)$ and $x\in\mathbb{R}^{n}\setminus B_{\lambda_{k}+1}(0)$,

• $0\leq\eta_{k}\leq 1$, and

• $|D\eta_{k}|=O(1)$.

By (12), using that $V(\lambda_{k})\to\infty$, there holds $A(\lambda_{k})\to\infty$ and

$$\int_{\partial\Omega}\eta_{k}\to\infty.$$

In conjunction with (11),

(18) $$\int_{\Omega}|D\eta_{k}|^{2}=o(1)\,\int_{\partial\Omega}\eta_{k}.$$

Note that $\operatorname{spt}\eta_{k}\cap\operatorname{spt}\phi=\emptyset$. Let $\varepsilon_{k}>0$, $k\geq 1$, be such that

$$\int_{\partial\Omega}\tilde{\phi}_{k}=0$$

where $\tilde{\phi}_{k}=\phi-\varepsilon_{k}\,|Du|\,\eta_{k}$. Note that

$$\varepsilon_{k}\,\int_{\partial\Omega}\eta_{k}=O(1).$$

In particular, $\varepsilon_{k}=o(1)$. By (18),

$$\varepsilon^{2}_{k}\,\int_{\Omega}|D\eta_{k}|^{2}=o(1).$$

By the weak stability of $\Omega$,

$$\int_{\partial\Omega}(n-H)\,\tilde{\phi}_{k}^{2}\leq\int_{\Omega}|D\tilde{\phi}_{k}|^{2}.$$

In conjunction with (17), we see that

$$\int_{\partial\Omega}(n-H)\,\phi^{2}\leq\int_{\Omega}|D\phi|^{2}+\varepsilon^{2}_{k}\,\int_{\Omega}|D\eta_{k}|^{2}=\int_{\Omega}|D\phi|^{2}+o(1).$$

This completes the proof of the lemma. ∎

6. Proof of Theorem 2

Lemma 21.

Suppose that $\Omega$ is stable. Let $\eta\in C_{c}^{\infty}(\bar{\Omega})$. There holds

(19) $$\int_{\partial\Omega}\eta^{2}\,(n-H)\leq 9\,\int_{\Omega}|D\eta|^{2}+2\,\int_{\partial\Omega}\eta^{2}\,(n-1-H).$$

Proof.

Recall the definition (13) of $\varphi\in C^{\infty}(\bar{\Omega})$. Since $\Omega$ is stable and $\varphi=1/2$ on $\partial\Omega$,

$$\frac{1}{4}\,\int_{\partial\Omega}(n-H)\,\eta^{2}\leq\int_{\Omega}|D(\eta\,\varphi)|^{2}.$$

Note that

$$|D(\eta\,\varphi)|^{2}=\varphi^{2}\,|D\eta|^{2}+2\,\eta\,\varphi\,D\eta\cdot D\varphi+\eta^{2}\,|D\varphi|^{2}.$$

Integrating by parts and using Lemma 15, we have

$$2\,\int_{\Omega}\eta\,\varphi\,D\eta\cdot D\varphi+\int_{\Omega}\eta^{2}\,|D\varphi|^{2} =-\int_{\Omega}\eta^{2}\,\varphi\Delta\varphi+\int_{\partial\Omega}\eta^{2}\,\varphi\,\mu(\Omega)\cdot D\varphi$$

$$\leq\frac{1}{2}\,\int_{\partial\Omega}\eta^{2}\,\mu(\Omega)\cdot D\varphi$$

$$=\frac{1}{2}\,\int_{\partial\Omega}\eta^{2}\,(n-1-H).$$

Thus,

$$\int_{\partial\Omega}\eta^{2}\,(n-H)\leq 4\,\int_{\Omega}\varphi^{2}\,|D\eta|^{2}+2\,\int_{\partial\Omega}\eta^{2}\,(n-1-H).$$

The assertion follows from Lemma 17. ∎

Proof of Theorem 2.

Suppose, for a contradiction, that $\Omega$ has infinite volume. By Lemma 20, $\Omega$ is stable. Let $\{\lambda_{k}\}_{k=1}^{\infty}$ be as in Proposition 14. We choose $\eta_{k}\in C_{c}^{\infty}(\mathbb{R}^{n}),$ $k\geq 1$, such that

• $\eta_{k}(x)=1$ for all $x\in B_{\lambda_{k}}(0)$,

• $\eta_{k}(x)=0$ for all $x\in\mathbb{R}^{n}\setminus B_{\lambda_{k}+1}(0)$, and

• $|D\eta_{k}|\leq 2$.

By the stability of $\Omega$ and (11),

$$\int_{\partial\Omega}\eta_{k}^{2}\,(n-H)\leq\int_{\Omega}|D\eta_{k}|^{2}=o(1)\,A(\lambda_{k}).$$

It follows that

$$\int_{\partial\Omega}\eta_{k}^{2}\,(n-1-H)\leq o(1)\,A(\lambda_{k})-\int_{\partial\Omega}\eta_{k}^{2}\leq-(1+o(1))\,A(\lambda_{k}).$$

In conjunction with (19), using (11) again,

(20) $$\int_{\partial\Omega}\eta_{k}^{2}\,(n-H)\leq-(2+o(1))\,A(\lambda_{k})+36\,(V(\lambda_{k}+1)-V(\lambda_{k}))=-(2+o(1))\,A(\lambda_{k}).$$

By Lemma 12, the left side is nonnegative. Since $A(\lambda_{k})\to\infty$ by (12), this leads to a contradiction.

It follows that $\Omega$ has finite volume. Applying Proposition 18 to each component of $\Omega$, we see that $\Omega$ is the union of finitely many balls of radius 1. If $\Omega$ has at least two components, say $\Omega_{1}$ and $\Omega_{2}$, then the function $f=\chi_{\Omega_{1}}-\chi_{\Omega_{2}}$ satisfies

$$\int_{\partial\Omega}f=0,\qquad\int_{\Omega}|Df|^{2}=0,\qquad\text{and}\qquad\int_{\partial\Omega}(n-H)f^{2}=2\,n\,|B_{1}(0)|.$$

This contradicts the weak stability of $\Omega$. Thus, $\Omega$ is connected. This completes the proof of the theorem. ∎

Appendix A First and second variation

Let $F\in C^{\infty}(\mathbb{R})$ and $U\subset\mathbb{R}^{n}$ be nonempty, open, and bounded. Given $u\in C^{\infty}(\mathbb{R}^{n})$, let

$$J_{U}(u)=\frac{1}{2}\,\int_{U\cap\{y\in\mathbb{R}^{n}:u(y)>0\}}|Du|^{2}-\int_{U\cap\{y\in\mathbb{R}^{n}:u(y)>0\}}F(u).$$

Let $\Omega=\{y\in\mathbb{R}^{n}:u(y)>0\}$. We assume that $\Omega$ is nonempty and that $Du(y)\neq 0$ for all $y\in\partial\Omega$. By the regular value theorem, $\Omega$ is a domain with outward normal

$$\mu(\Omega)=-\frac{Du}{|Du|}.$$

Given $\varepsilon>0$, let

(21) $$\{u_{t}:t\in(-\varepsilon,\varepsilon)\}$$

be a smooth family of functions $u_{t}\in C^{\infty}(\mathbb{R}^{n})$ such that $u_{0}=u$. Let $\Omega_{t}=\{y\in\mathbb{R}^{n}:u_{t}(y)>0\}$. We assume that there is $K\Subset U$ such that

$$\{y\in\mathbb{R}^{n}:u(y)\neq u_{t}(y)\}\subset K$$

for all $t\in(-\varepsilon,\varepsilon)$. Shrinking $\varepsilon>0$, if necessary, we may assume that $Du_{t}(y)\neq 0$ for all $t\in(-\varepsilon,\varepsilon)$ and $y\in\partial\Omega_{t}$. By the regular value theorem, $\Omega_{t}$ is a domain. Let $\phi\in C^{\infty}_{c}(\mathbb{R}^{n})$ be given by

$$\phi=\frac{d}{dt}\bigg|_{t=0}u_{t}.$$

Let $\gamma_{t}$ be the normal speed of the variation $\{\partial\Omega_{t}:t\in(-\varepsilon,\varepsilon)\}$ of hypersurfaces with respect to the normal pointing out of $\Omega_{t}$. Note that

(22) $$\gamma_{0}=\frac{\phi}{|Du|}.$$

We assume that

$$\frac{d}{dt}\bigg|_{t=0}|U\cap\Omega_{t}|=\frac{d^{2}}{dt^{2}}\bigg|_{t=0}|U\cap\Omega_{t}|=0.$$

It follows that

(23) $$\int_{\partial\Omega}\frac{\phi}{|Du|}=0\qquad\text{and}\qquad\frac{d}{dt}\bigg|_{t=0}\int_{\partial\Omega_{t}}\gamma_{t}=0.$$

Lemma 22.

There holds

(24) $$\frac{d}{dt}\bigg|_{t=0}J_{U}(u_{t})=-\int_{\Omega}(\Delta u+F^{\prime}(u))\,\phi-\frac{1}{2}\,\int_{\partial\Omega}|Du|\,\phi.$$

Proof.

Using (22) and (23), we have

$$\frac{d}{dt}\bigg|_{t=0}J_{U}(u_{t})=\int_{\Omega}(Du\cdot D\phi-F^{\prime}(u)\,\phi)+\frac{1}{2}\,\int_{\partial\Omega}|Du|\,\phi.$$

The assertion now follows from the divergence theorem. ∎

Assume that, for every choice of $U$, $u$ is critical for $J_{U}$ among all variations (21). In particular, the right side of (24) vanishes for all $\phi\in C^{\infty}_{c}(\mathbb{R}^{n})$ with

$$\int_{\partial\Omega}\frac{\phi}{|Du|}=0.$$

It follows that there is $\rho>0$ such that

(25) $$\begin{cases}\quad-\Delta u=F^{\prime}(u)\qquad&\text{in $\Omega$ and}\\ \quad|Du|=\rho&\text{on $\partial\Omega$}.\end{cases}$$

Lemma 23.

Assume that $u\in C^{\infty}(\mathbb{R}^{n})$ satisfies (25). There holds

$$\frac{d^{2}}{dt^{2}}\bigg|_{t=0}J_{U}(u_{t})=\int_{\Omega}(|D\phi|^{2}-F^{\prime\prime}(u)\,\phi^{2})+\int_{\partial\Omega}\left(H-\frac{F^{\prime}(0)}{\rho}\right)\,\phi^{2}.$$

Proof.

Given $t\in(-\varepsilon,\varepsilon)$, let $\Phi_{t}:\partial\Omega\to\partial\Omega_{t}$ be the normal parametrization of $\partial\Omega_{t}$. By (22),

$$\frac{d}{dt}\bigg|_{t=0}\Phi_{t}=-\frac{\phi}{|Du|}\,\frac{Du}{|Du|}=-\frac{\phi}{\rho^{2}}\,Du.$$

Note that

$$\frac{d}{dt}\bigg|_{t=0}|(Du_{t})(\Phi_{t})|^{2}=-\frac{2}{\rho^{2}}\,D^{2}u(Du,Du)\,\phi+2\,Du\cdot D\phi.$$

In conjunction with Lemma 22, we obtain

$$\frac{d^{2}}{dt^{2}}\bigg|_{t=0}J_{U}\left(u_{t}\right) =-\int_{\Omega}(\Delta\phi+F^{\prime\prime}(u)\,\phi)\,\phi-\frac{1}{\rho}\,\int_{\partial\Omega}\phi\,Du\cdot D\phi+\frac{1}{\rho^{3}}\,\int_{\partial\Omega}\phi^{2}\,D^{2}u(Du,Du).$$

By the divergence theorem,

$$\int_{\Omega}\phi\,\Delta\phi=-\int_{\Omega}|D\phi|^{2}-\frac{1}{\rho}\,\,\int_{\partial\Omega}D\phi\cdot Du\,\phi.$$

Moreover, on $\partial\Omega$, using (25),

$$-F^{\prime}(0)=\Delta u=\Delta^{\partial\Omega}u+D^{2}u(\mu(\Omega),\mu(\Omega))+H\,\mu(\Omega)\cdot Du=\frac{1}{\rho^{2}}\,D^{2}u(Du,Du)-H\,\rho.$$

The assertion follows from these identities. ∎