Total positive curvature and the equality case in the relative isoperimetric inequality outside convex domains

Abstract

We settle the case of equality for the relative isoperimetric inequality outside any arbitrary convex set with not empty interior.

1 Introduction

In [5] Choe, Ghomi and Ritoré proved the following relative isoperimetric inequality outside convex sets, see also [14] for an alternative proof and [12] for a generalization to higher codimension.

Theorem 1.1

[5] Let ${\textbf{C}}\subset {\mathbb {R}}^N$ be a closed convex set with nonempty interior. For any set of finite perimeter and finite measure $\Omega \subset {\mathbb {R}}^N\setminus {\textbf{C}}$ we have

$$\begin{aligned} P(\Omega ;{\mathbb {R}}^N\setminus {\textbf{C}})\ge N\Big (\frac{\omega _N}{2}\Big )^{\frac{1}{N}}|\Omega |^{\frac{N-1}{N}}\,. \end{aligned}$$

(1.1)

Moreover, if ${\textbf{C}}$ has a $C^2$ boundary and $\Omega$ is a bounded set for which the equality in (1.1) holds, then $\Omega$ is a half ball.

Here and in what follows $P(\Omega ;{\mathbb {R}}^N\setminus {\textbf{C}})$ denotes the perimeter of a set $\Omega$ in ${\mathbb {R}}^N\setminus {\textbf{C}}$ in the sense of De Giorgi. As observed by the authors in [5] the equality case for general, possibly nonsmooth, convex sets does not follow from their methods as it cannot be handled by a simple approximation argument. However there are many situations in which nonsmooth convex sets naturally appear. For instance, in models of vapor-liquid-solid-grown nanowires the nanotube is often described as a semi-infinite convex cylinder with sharp edges and possibly nonsmooth cross sections. In these models super-saturated liquid droplets correspond to isoperimetric regions for the relative perimeter outside the cylinder or more in general for the capillarity energy, see [11, 17]. Experimentally it is observed that in some regimes preferred configurations are given by spherical caps lying on the top facet of the cylinder. Understanding these phenomena from a mathematical point of view was our first motivation to study the equality cases in (1.1) also for nonsmooth convex obstacles, beside the intrinsic geometric interest of the problem.

The main result of this paper reads as follows.

Theorem 1.2

(The equality case) Let ${\textbf{C}}\subset {\mathbb {R}}^N$ be a closed convex set with nonempty interior and let $\Omega \subset {\mathbb {R}}^N\setminus {\textbf{C}}$ be a set of finite perimeter such that equality holds in (1.1). Then $\Omega$ is a half ball supported on a facet of ${\textbf{C}}$.¹

Observe that, compared to the last part of Theorem 1.1, here we don’t have any restriction on the convex set ${\textbf{C}}$ and we allow for possibly unbounded competitors. As in [5] the starting point in order the get the characterization of the equality case in (1.1) is an estimate of the positive total curvature ${\mathcal {K}}^+(\Sigma )$ of a hypersurface $\Sigma \subset \overline{{\mathbb {R}}^N\setminus {\textbf{C}}}$ when the contact angle between $\partial {\textbf{C}}$ and $\Sigma$ is larger than or equal to a fixed $\theta \in (0,\pi )$. Here ${\mathcal {K}}^+(\Sigma )$ denotes, roughly speaking, the measure of the image of the Gauss map restricted to those points where there exists a support hyperplane, see Definition 1.3 below. To state more precisely our result we need to introduce some notation: Given $\theta \in (0,\pi )$ we denote by $S_\theta$ the spherical cap

$$\begin{aligned} S_\theta :=\left\{ y\in {\mathbb {S}}^{N-1}:\, y\cdot e_N\ge \cos \theta \right\} . \end{aligned}$$

Moreover, given $\Sigma \subset \overline{{\mathbb {R}}^N\setminus {\textbf{C}}}$ and a point $x\in \Sigma$ we denote by $N_x\Sigma$ the normal cone

$$\begin{aligned} N_x\Sigma =\left\{ \nu \in {\mathbb {S}}^{N-1}:\, (y-x)\cdot \nu \le 0\text { for all }y\in \Sigma \right\} , \end{aligned}$$

that is the set of (exterior) normals to support hyperplanes to $\Sigma$. We can now recall the definition of total positive curvature.

Definition 1.3

Let ${\textbf{C}}$ be a closed convex set with not empty interior, $\Omega \subset {\mathbb {R}}^N\setminus {\textbf{C}}$ a bounded open set and $\Sigma :=\overline{\partial \Omega \setminus {\textbf{C}}}$. The total positive curvature of $\Sigma$ is given by

$$\begin{aligned} \mathcal {K}^+(\Sigma ):={{{\mathcal {H}}}}^{N-1}\left( \bigcup _{x\in \Sigma \setminus {\textbf{C}}}N_x\Sigma \right) \,. \end{aligned}$$

The aforementioned estimate on the total positive curvature is provided by the following theorem, which will be proved in Sect. 3.

Theorem 1.4

Let ${\textbf{C}}\subset {\mathbb {R}}^N$ be a closed convex set of class $C^1$, $\Omega \subset {\mathbb {R}}^N\setminus {\textbf{C}}$ a bounded open set and $\Sigma :=\overline{\partial \Omega \setminus {\textbf{C}}}$. Let $\theta _0\in (0, \pi )$ such that

$$\begin{aligned} \nu \cdot \nu _{{\textbf{C}}}(x)\le \cos \theta _0 \quad \text {whenever}\ x\in \Sigma \cap {\textbf{C}},\,\,\nu \in N_x\Sigma , \end{aligned}$$

(1.2)

where $\nu _{{\textbf{C}}}(x)$ stands for the outer unit normal to ${\textbf{C}}$ at x. Then,

$$\begin{aligned} {\mathcal {K}}^+(\Sigma )\ge {{{\mathcal {H}}}}^{N-1}(S_{\theta _0})\,. \end{aligned}$$

(1.3)

Moreover, let $r>0$ be such that $\Sigma \cap {\textbf{C}}\subset B_r(0)$. For any $\varepsilon >0$ there exists $\delta$, depending on $\varepsilon ,\theta _0$ and r, but not on ${\textbf{C}}$ or $\Omega$, such that if

$$\begin{aligned} \nu \cdot \nu _{{\textbf{C}}}(x)\le \cos \theta _0+\delta \quad \text {whenever}\ x\in \Sigma \cap {\textbf{C}},\,\,\nu \in N_x\Sigma , \end{aligned}$$

(1.4)

and

$$\begin{aligned} {\mathcal {K}}^+(\Sigma )\le {{{\mathcal {H}}}}^{N-1}(S_{\theta _0})+\delta \,, \end{aligned}$$

(1.5)

then $\Sigma \cap {\textbf{C}}$ is not empty, $\textrm{width}(\Sigma \cap {\textbf{C}})\le \varepsilon$ and more precisely $\Sigma \cap {\textbf{C}}$ lies between two parallel $\varepsilon$-distant hyperplanes orthogonal to $\nu _{\textbf{C}}(x)$ for some $x\in \Sigma \cap {\textbf{C}}$. In particular, if (1.2) is satisfied and the equality in (1.3) holds, then $\Sigma \cap {\textbf{C}}$ is not empty and lies on a support hyperplane to ${\textbf{C}}$.

Note that in the previous statement $\textrm{width}(\Sigma \cap {\textbf{C}})$ denotes the distance between the closest pair of parallel hyperplanes which contains $\Sigma \cap {\textbf{C}}$ in between them, see (3.5). Even though the proof of this theorem follows the general strategy of [4] we are able to improve their result in three directions: (1) we consider a general contact angle $\theta _0\in (0,\pi )$, whereas in [4] only the case $\theta _0=\pi /2$ is considered; (2) we do not assume any regularity on $\Sigma$ and the contact angle condition can be replaced by the weaker condition (1.2); (3) we get a stability estimate on the ‘contact part’ $\Sigma \cap {\textbf{C}}$ which is independent of the shape of the convex set ${\textbf{C}}$. As we will explain below (2) and (3) are crucial in the proof of Theorem 1.2.

As a consequence of independent interest of the previous theorem we prove a sharp inequality for the Willmore energy, see Theorem 3.9.

Before outlining our strategy of the proof of Theorem 1.2 we briefly recall how in [5] it is proven that a bounded set $\Omega _0$ satisfying the equality in (1.1) is a half ball, when ${\textbf{C}}$ is sufficiently smooth. There the idea is to consider the isoperimetric profile

$$\begin{aligned} I(m)=\inf \left\{ P\left( E;{\mathbb {R}}^N\setminus {\textbf{C}}\right) :\, E\subset {\mathbb {R}}^N\setminus {\textbf{C}},\, |E|=m\right\} , \end{aligned}$$

defined for all $m\in (0,|\Omega _0|]$, and to show that $I(m)=N\big (\frac{\omega _N}{2}\big )^{\frac{1}{N}}m^{\frac{N-1}{N}}$, that is I(m) coincides with the isoperimetric profile $I_{\mathscr {H}}(m)$ of the half space. Moreover, since $I'(|\Omega _0|)=H_\Sigma$, where $H_\Sigma$ is the mean curvature of $\Sigma =\overline{\partial \Omega _0\setminus {\textbf{C}}}$,

$$\begin{aligned} \begin{aligned} I(|\Omega _0|)\big (I'(|\Omega _0|)\big )^{N-1}&=\int _{\Sigma \setminus {\textbf{C}}}H_\Sigma ^{N-1}\,d{{{\mathcal {H}}}}^{N-1}\ge (N-1)^{N-1}{{\mathcal {K}}}^+(\Sigma )\\&\ge (N-1)^{N-1}{{{\mathcal {H}}}}^{N-1}(S_{\pi /2})=I_{\mathscr {H}}(|\Omega _0|)\big (I_{{\mathscr {H}}}'(|\Omega _0|))^{N-1}\,, \end{aligned} \end{aligned}$$

(1.6)

where the first inequality follows from an application of coarea formula and the geometric-arithmetic mean inequality, see for instance the proof of Theorem 3.9, and the second one follows from the estimate of the total curvature proved in [5, Lemma 3.1]. Now, since $I(m)=I_{{\mathscr {H}}}(m)$ for all $m\in [0,|\Omega _0|]$, all the inequalities in (1.6) are equalities. In particular this implies that ${\mathcal K}^+(\Sigma )={{{\mathcal {H}}}}^{N-1}(S_{\pi /2})$ and that $\Sigma$ is umbilical. From this information, it is not difficult to see that $\Sigma$ must be a half ball.

Note that in the proof of [5, Lemma 3.1] it is crucial that the regular part of $\Sigma$ meets $\partial {\textbf{C}}$ orthogonally and in a $C^2$ fashion. This can be inferred from the boundary regularity theory for perimeter minimizers which can be applied only if ${\textbf{C}}$ is sufficiently smooth. Therefore the above argument fails for a general convex set.

In order to deal with this lack of regularity we implement a delicate argument based on the approximation of ${\textbf{C}}$ with more regular convex sets.

Let us describe the argument more in detail. Denote by $\Omega _0$ a set of finite perimeter satisfying the equality in (1.1). For $\eta >0$ sufficiently small we approximate ${\textbf{C}}$ with the closed $\eta$-neighborhood ${\textbf{C}}_\eta ={\textbf{C}}+\overline{B_\eta (0)}$, which is of class $C^{1,1}$. Now the idea is to consider the relative isoperimetric problem in ${\mathbb {R}}^N\setminus {\textbf{C}}_\eta$. In order to force the minimizers to converge to $\Omega _0$ when $\eta \rightarrow 0$ and the prescribed mass m converges to $|\Omega _0|$, we introduce the following constrained isoperimetric profiles with obstacle $\Omega _0$:

$$\begin{aligned} I_\eta (m)=\min \left\{ P\left( E;{\mathbb {R}}^N\setminus {\textbf{C}}_\eta \right) :\, E\subset \Omega _0\setminus {\textbf{C}}_\eta ,\, |E|=m\right\} \end{aligned}$$

(1.7)

for all $m\in (0,|\Omega _0\setminus {\textbf{C}}_\eta |]$. Denote by $\Omega _{\eta ,m}$ a minimizer of the above problem and set $\Sigma _{\eta ,m}:=\overline{\partial \Omega _{\eta ,m}\setminus {\textbf{C}}_\eta }$. Note that in the general N-dimensional case, both the obstacle $\Omega _0$ and the minimizers $\Omega _{\eta ,m}$ may have singularities. Thus, despite the fact that $\partial {\textbf{C}}_\eta$ is of class $C^{1,1}$, we cannot apply the known boundary regularity results at the points $x\in \partial \Omega _{\eta ,m}\cap \partial {\textbf{C}}_\eta \cap \partial \Omega _0$.

However, one useful observation is that $\Omega _{\eta ,m}$ is a restricted $\Lambda$-minimizer, i.e., a $\Lambda$-minimizer with respect to perturbations that do not increase the “wet part” $\partial \Omega _{\eta , m}\cap {\textbf{C}}_\eta$ (see Definition 4.1 below), with a $\Lambda >0$ which can be made uniform with respect to $\eta$ and locally uniform with respect to m (see Steps 1 and 2 of the proof of Theorem 1.2). Another important observation is that restricted $\Lambda$-minimizers satisfy uniform volume density estimates up to the boundary $\partial {\textbf{C}}_\eta$. All these facts are combined to show that the constrained isoperimetric profiles (1.7) are Lipschitz continuous and that their derivatives coincide a.e. with the constant mean curvature $H_{\Sigma ^*_{\eta , m}}$ of the regular part $\Sigma ^*_{\eta , m}$ of $\Sigma _{\eta , m}\setminus \partial \Omega _0$ (see Steps 3 and 4).

As in the argument of [5] another important ingredient is represented by the inequality

$$\begin{aligned} \mathcal K^+(\Sigma _{\eta ,m})\ge {{{\mathcal {H}}}}^{N-1}(S_{\pi /2})=\frac{1}{2} N\omega _N\,, \end{aligned}$$

(1.8)

which would hold by [5, Lemma 3.1] if we could show that $\Sigma _{\eta ,m}$ meets $\partial {\textbf{C}}_\eta$ orthogonally and in a sufficiently smooth fashion. However, as already observed, due the possible presence of boundary singularities at $\partial \Omega _{\eta ,m}\cap \partial {\textbf{C}}_\eta \cap \partial \Omega _0$ we cannot show that the aforementioned orthogonality condition is attained in a classical sense. An important step of our argument, which allows us to overcome this difficulty, consists in showing that restricted $\Lambda$-minimizers satisfy the $\pi /2$ contact angle condition with respect to $\partial {\textbf{C}}_\eta$ in a “viscosity” sense, namely that the following weak Young’s law holds:

$$\begin{aligned} \nu \cdot \nu _{{\textbf{C}}_\eta }(x)\le 0 \quad \text {whenever}\ x\in \Sigma _{\eta ,m}\cap {\textbf{C}}_\eta ,\,\,\nu \in N_x\Sigma _{\eta ,m}\,. \end{aligned}$$

(1.9)

This is achieved in Step 5 by combining a blow-up argument with a variant of the Strong Maximum Principle that we adapted from [9]. In turn, owing to (1.9) we may apply Theorem 1.4 to obtain (1.8). Having established the latter and with some extra work we can show that $I_\eta (m)\rightarrow I_{{\mathscr {H}}}(m)$ as $\eta \rightarrow 0$ for every $m\in (0,|\Omega _0|)$, where we recall $I_{\mathscr {H}}(m)=N\big (\frac{\omega _N}{2}\big )^{\frac{1}{N}}m^{\frac{N-1}{N}}$ is the isoperimetric profile of the half space (see Steps 6 and 7).

With the convergence of the isoperimetric profiles $I_\eta$ at hand and using again (1.8), we can then prove that for a.e. $m\in (0,|\Omega _0|)$

$$\begin{aligned} \mathcal K^+(\Sigma _{\eta ,m})\rightarrow \frac{1}{2} N\omega _N\,, \end{aligned}$$

(1.10)

and thus $\Sigma _{\eta ,m}$ almost satisfies the case of equality in (1.3) for $\eta$ sufficiently small. Thanks to the last part of Theorem 1.4 we may then infer that $\Sigma _{\eta ,m}\cap {\textbf{C}}_\eta$ is almost flat and with some extra work that the whole wet part $\partial \Omega _{\eta , m}\cap {\textbf{C}}_\eta$ has the same property. By showing that for suitable sequences $m_n\nearrow |\Omega _0|$ and $\eta _n\searrow 0$, $\partial \Omega _{\eta , m_n}\cap {\textbf{C}}_\eta \rightarrow \partial \Omega _0\cap {\textbf{C}}$ in the Hausdorff sense, we may finally conclude that $\partial \Omega _0\cap {\textbf{C}}$ is flat and lies on a facet of ${\textbf{C}}$ (see Step 8). We highlight here that in all the above argument it is crucial that the stability estimate on the width of $\Sigma _{\eta ,m}\cap {\textbf{C}}_\eta$ provided by our version Theorem 1.4 is independent of the shape of the convex set ${\textbf{C}}_\eta$.

Having established that the wet part $\partial \Omega _0\cap {\textbf{C}}$ is flat, more work is still needed in the final step of the proof to deduce again from (1.10) that $\Omega _0$ is umbilical and in turn a half ball supported on a facet of ${\textbf{C}}$.

The paper is organized as follows: in Sect. 2 we collect a few known results of the regularity theory of perimeter quasi minimizers needed in the paper. In Sect. 3 we prove Theorem 1.4, while the proof of Theorem 1.2 occupies the whole Sect. 4 with some of the most technical steps outsourced to Sect.  6. Section 5 contains further regularity properties of restricted $\Lambda$-minimizers that are needed in the proof of the main result and the proof of the version of the Strong Maximum Principle needed here.

2 Preliminaries

Throughout the paper we denote by $B_r(x)$ the ball in ${\mathbb {R}}^N$ of center x and radius $r>0$. In the following we shall often deal with sets of finite perimeter. For the definition and the basic properties of sets of (locally) finite perimeter we refer to the books [3, 15]. Here we fix some notation for later use. Given $E\subset {\mathbb {R}}^N$ of locally finite perimeter and a Borel set G we denote by P(E; G) the perimeter of E in G. The reduced boundary of E will be denoted by $\partial ^*E$, while $\partial ^eE$ will stand for the essential boundary defined as

$$\begin{aligned} \partial ^eE:={\mathbb {R}}^N\setminus \big (E^{(0)}\cup E^{(1)}\big ), \end{aligned}$$

where $E^{(0)}$ and $E^{(1)}$ are the sets of points where the density of E is 0 and 1, respectively. Moreover, we denote by $\nu _E$ the generalized exterior normal to E, which is well defined at each point of $\partial ^*E$, and by $\mu _E$ the Gauss-Green measure associated to E

(2.1)

In the following, when dealing with a set of locally finite perimeter E, we shall always tacitly assume that E coincides with a precise representative that satisfies the property $\partial E=\overline{\partial ^*E}$, see [15, Remark 16.11]. A possible choice is given by $E^{(1)}$ for which one may easily check that

$$\begin{aligned} \partial E^{(1)}=\overline{\partial ^*E}\,. \end{aligned}$$

(2.2)

We recall the well known notion of perimeter $(\Lambda ,r_0)$-minimizer and the main properties which will be used here.

Definition 2.1

Let $\Omega \subset {\mathbb {R}}^N$ be an open set. We say that a set of locally finite perimeter $E\subset {\mathbb {R}}^N$ is a perimeter $(\Lambda ,r_0)$-minimizer in $\Omega$, $\Lambda \ge 0$ and $r_0>0$, if for any ball $B_r(x_0)\subset \Omega$, with $0<r\le r_0$ and any $F\subset {\mathbb {R}}^N$ such that $E\Delta F\subset \!\subset B_r(x_0)$ we have

$$\begin{aligned} P(E;B_r(x_0))\le P(F;B_r(x_0))+\Lambda |E\Delta F|\,. \end{aligned}$$

In order to state a useful compactness theorem for $\Lambda$-minimizers we recall that a sequence $\{{\mathcal {C}}_n\}$ of closed sets converge in the Kuratoswki sense to a closed set ${\mathcal {C}}$ if the following conditions are satisfied:

• (i) if $x_n\in {\mathcal {C}}_n$ for every n, then any limit point of $\{x_n\}$ belongs to ${\mathcal {C}}$;

• (ii) any $x\in {\mathcal {C}}$ is the limit of a sequence $\{x_n\}$ with $x_n\in {\mathcal {C}}_n$.

One can easily see that ${\mathcal {C}}_n\rightarrow {\mathcal {C}}$ in the sense of Kuratowski if and only if dist$(\cdot ,{\mathcal {C}}_n)\rightarrow$ dist$(\cdot ,{\mathcal {C}})$ locally uniformly in ${\mathbb {R}}^N$. In particular, by the Arzelà-Ascoli Theorem any sequence of closed sets admits a subsequence which converge in the sense of Kuratowski.

Throughout the paper, with a common abuse of notation, we write $E_h\rightarrow E$ in $L^1$ ($L^1_{loc}$) instead of $\chi _{_{E_h}}\rightarrow \chi _{_{E}}$ in $L^1$ ($L^1_{loc}$). Moreover, given a sequence of Radon measures $\mu _h$ in an open set $\Omega$, we say that $\mu _h{\mathop {\rightharpoonup }\limits ^{*}}\mu$ weakly* in $\Omega$ in the sense of measures if

$$\begin{aligned} \int _\Omega \varphi \,d\mu _h\rightarrow \int _\Omega \varphi \,d\mu \qquad \text {for all}\ \varphi \in C^0_c(\Omega )\,. \end{aligned}$$

Next theorem is a well known result, see for instance [15, Ch. 21].

Theorem 2.2

Let $\Omega \subset {\mathbb {R}}^N$ be an open set and $\{E_n\}$ a sequence of locally finite perimeter sets contained in $\Omega$ satisfying the following property: there exists $r_0>0$ such that for every n, $E_n$ is a perimeter $(\Lambda _n,r_0)$-minimizer in $\Omega$, with $\Lambda _n\rightarrow \Lambda \in [0,+\infty )$. Then there exist $E\subset \Omega$ of locally finite perimeter and a subsequence $\{n_k\}$ such that

• (i) E is a $(\Lambda ,r_0)$-minimizer in $\Omega$;

• (ii) $E_{n_k}\rightarrow E$ in $L^1_{loc}(\Omega )$,

• (iii) $\partial E_{n_k}\rightarrow {\mathcal {C}}$ in the Kuratowski sense for some closed set ${\mathcal {C}}$ such that ${\mathcal {C}}\cap \Omega =\partial E\cap \Omega$;

• (iv) weakly* in $\Omega$ in the sense of measures.

Remark 2.3

From the definition of Kuratowski convergence it is not difficult to see that (ii) and (iii) of Theorem 2.2 imply that, up to extracting a further subsequence if needed, $\overline{E_{n_k}}\rightarrow K$ in the sense of Kuratowski, with $K\cap \Omega ={{\overline{E}}}\cap \Omega$.

Definition 2.4

Given a set of locally finite perimeter E, we say that a function $h\in L^1_{loc}(\partial ^*E)$ is the weak mean curvature of E if for any vector field $X\in C^1_c({\mathbb {R}}^N;{\mathbb {R}}^N)$ we have

$$\begin{aligned} \int _{\partial ^*E}\textrm{div}_\tau X\,d{{{\mathcal {H}}}}^{N-1}=\int _{\partial ^*E}h\, X\cdot \nu _E,d{{{\mathcal {H}}}}^{N-1}\,, \end{aligned}$$

where $\textrm{div}_\tau X:=\textrm{div}X-(\partial _{\nu _E}X)\cdot \nu _E$ stands for the tangential divergence of X along $\partial ^*E$. If such an h exists we will denote it by $H_{\partial E}$.

Note that if $\partial E$ is of class $C^2$ then $H_{\partial E}$ coincides with the classical mean curvature, or more precisely with the sum of all principal curvatures. In particular, if E coincides locally with the subgraph of a function u of class $C^2$ then locally

$$\begin{aligned} H_{\partial E}=-\textrm{div}\bigg (\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\bigg )\,. \end{aligned}$$

Concerning the above mean curvature operator, we recall the following useful Strong Maximum Principle, see for instance [18, Th. 2.3], which covers a more general class of quasilinear equations.

Theorem 2.5

Let $\Omega \subset {\mathbb {R}}^{N-1}$ be a connected open set and let $u,v\in C^2(\Omega )$ such that $u\le v$ and

$$\begin{aligned} \textrm{div}\bigg (\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\bigg )=\lambda =\textrm{div}\bigg (\frac{\nabla v}{\sqrt{1+|\nabla v|^2}}\bigg )\, \end{aligned}$$

for some constant $\lambda \in {\mathbb {R}}$. If $u(x_0)=v(x_0)$ for some $x_0\in \Omega$, then $u\equiv v$.

We recall the following classical regularity result for $\Lambda$-minimizers.

Theorem 2.6

Let E be a perimeter $(\Lambda ,r_0)$-minimizer in some open set $\Omega \subset {\mathbb {R}}^N$. Then

• (i) $\partial ^*E\cap \Omega$ is a hypersurface of class $C^{1,\alpha }$ for every $\alpha \in (0,1)$, relatively open in $\partial E\cap \Omega$. Moreover, $\text {dim}_{{{{\mathcal {H}}}}}((\partial E\setminus \partial ^* E)\cap \Omega )\le N-8$, where $\text {dim}_{{{{\mathcal {H}}}}}$ stands for the Hausdorff dimension;

• (ii) $H_{\partial E}\in L^\infty (\partial ^*E\cap \Omega )$, with $\Vert H_{\partial E}\Vert _{L^\infty }\le \Lambda$, and thus $\partial ^*E\cap \Omega$ is of class $W^{2,p}$ for all $p\ge 1$;

• (iii) if there exists a $C^1$ hypersurface $\Sigma$ touching $\partial E$ at $x\in \Omega$ and lying on one side with respect to $\partial E$ in a neighborhood of x, then $x\in \partial ^*E$.

Items (i) and (ii) are classical, see for instance Theorems 21.8 and 28.1 in [15] for (i) and Theorem 4.7.4 in [2] for (ii).

Concerning (iii) one can show that under the assumption on x the minimal cone obtained by blowing up E around x is contained in a half space. For the existence of such a minimal cone see Theorem 28.6 in [15]. Since any minimal cone contained in a half space is a half space, see for instance [8, Lemma 3], it follows that x is a regular point.

The so-called $\varepsilon$-regularity theory for $\Lambda$-minimizers underlying the proof of the above theorem yields that sequences of $\Lambda$-minimizers $E_h$ converging in $L^1$ to a smooth set E are regular for h large and in fact converge in a stronger sense. More precisely, we have the following result, which is well known to the experts.

Theorem 2.7

Let $E_n$, E be $(\Lambda ,r_0)$-minimizers in an open set $\Omega \subset {\mathbb {R}}^N$ such that $E_n\rightarrow E$ in $L^1_{loc}(\Omega )$. Let $x\in \partial ^*E\cap \Omega$. Then, up to rotations and translations, there exist a $(N-1)$-dimensional open ball $B'\subset {\mathbb {R}}^{N-1}$, functions $\varphi _n,\varphi \in W^{2,p}(B')$ for all $p\ge 1$, and $r>0$ such that $x\in B'\times (-r,r)$ and for n large

$$\begin{aligned} \begin{aligned}&\partial E_n\cap (B'\times (-r,r))=\{(x',\varphi _n(x')):\,x'\in B'\}, \\&\partial E\cap (B'\times (-r,r))=\{(x',\varphi (x')):\,x'\in B'\}, \\&\varphi _n\rightarrow \varphi \quad \text {in}\ C^{1,\alpha }(\overline{B'})\ \text {for some}\ \alpha \in (0,1)\,. \end{aligned} \end{aligned}$$

(2.3)

Moreover, $H_{\partial E_n}(x',\varphi _n(x')){\mathop {\rightharpoonup }\limits ^{*}}H_{\partial E}(x',\varphi (x'))$ in $L^\infty (B')$ and thus $\varphi _n\rightharpoonup \varphi$ in $W^{2,p}(B')$ for all $p\ge 1$.

Properties stated in (2.3) follow from the classical $\varepsilon$-regularity theory, see [22, Th. 1.9] (see also the arguments of Lemma 3.6 in [6]). The last part of the statement then easily follows from Theorem 2.6-(ii) combined with the classical Calderón-Zygmund estimates.

3 An estimate of the total positive curvature

This section is mainly devoted to the proof of Theorem 1.4 and to some applications.

We recall that a set $X\subset {\mathbb {S}}^{N-1}$ is called spherically convex (in short convex) if it is geodesically convex, that is, for any pair of points $x_1$, $x_2\in X$ there exists a distance minimizing geodesic connecting $x_1$ and $x_2$ contained in X.

If $x\in {\mathbb {S}}^{N-1}$ and $\theta \in (0,\pi )$ we denote by $S_{\theta , x}$ the spherical cap

$$\begin{aligned} S_{\theta , x}:=\{y\in {\mathbb {S}}^{N-1}:\, x\cdot y\ge \cos \theta \}\,. \end{aligned}$$

If $x=e_N$ we shall simply write $S_\theta$ instead of $S_{\theta ,e_N}$. Note that $S_{\pi -\theta , -x}$ coincides with $({\mathbb {S}}^{N-1}\setminus S_{\theta , x})\cup \partial S_{\theta ,x}$, where $\partial S_{\theta ,x}$ denotes the relative boundary of $S_{\theta , x}$ in ${\mathbb {S}}^{N-1}$. We recall that

$$\begin{aligned} {{{\mathcal {H}}}}^{N-1}(S_{\theta })=(N-1)\omega _{N-1}\int _0^\theta \sin ^{N-2}\sigma \,d\sigma \,, \end{aligned}$$

and ${{{\mathcal {H}}}}^{N-1}({\mathbb {S}}^{N-1})=N\omega _N$, where $\omega _N$ is the measure of the unit ball.

The following lemma extends [4, Proposition 3.1] to general angles.

Lemma 3.1

Let $X\subset {\mathbb {S}}^{N-1}$ be spherically convex and closed, with ${{{\mathcal {H}}}}^{N-1}(X)>0$, let $\theta \in (0,\pi )$ and fix $x\in X$. Then we have

$$\begin{aligned} {{{\mathcal {H}}}}^{N-1}(X\cap S_{\theta , x})\ge \frac{{{{\mathcal {H}}}}^{N-1}(S_\theta )}{N\omega _N} {{{\mathcal {H}}}}^{N-1} (X)\,. \end{aligned}$$

(3.1)

Moreover, the equality holds if and only if $-x\in X$. Finally, given $\theta _0\in (0,\pi )$, for every $\varepsilon >0$ there exists $\delta >0$, independent of X, such that if $\theta \in [\theta _0/2,\theta _0]$, then

$$\begin{aligned} {{{\mathcal {H}}}}^{N-1}(X\cap S_{\theta , x})\le \left( \frac{{{{\mathcal {H}}}}^{N-1}(S_\theta )}{N\omega _N}+\delta \right) {{{\mathcal {H}}}}^{N-1} (X)\quad \text {implies }\quad \textrm{dist}(-x, X)\le \varepsilon \,. \end{aligned}$$

Proof

We denote by A the subset of $S_{\theta , x}$ obtained by taking the union of all the minimal geodesics connecting x with the points of $X\cap \partial S_{\theta , x}$. Let $B:=S_{\theta , x}\setminus A$. Similarly denote by $A^-$ the subset of $S_{\pi -\theta , -x}$ obtained by taking the union of all the minimal geodesics connecting $-x$ with the points of $X\cap \partial S_{\theta , x}= X\cap \partial S_{\pi -\theta , -x}$, and set $B^-:=S_{\pi -\theta , -x}\setminus A^-$.

Assume first that ${{{\mathcal {H}}}}^{N-1}(A)>0$. We note that

$$\begin{aligned} \frac{{{{\mathcal {H}}}}^{N-1}(A^-)}{{{{\mathcal {H}}}}^{N-1}(A)}=\frac{{{{\mathcal {H}}}}^{N-1}(S_{\pi -\theta })}{{{{\mathcal {H}}}}^{N-1}(S_{\theta })}\,. \end{aligned}$$

Thus, we have

$$\begin{aligned} {{{\mathcal {H}}}}^{N-1}(X\cap A)={{{\mathcal {H}}}}^{N-1}(A)= \frac{{{{\mathcal {H}}}}^{N-1}(S_{\theta })}{{{{\mathcal {H}}}}^{N-1}(S_{\pi -\theta })}{{{\mathcal {H}}}}^{N-1}(A^-)\ge \frac{{{{\mathcal {H}}}}^{N-1}(S_{\theta })}{{{{\mathcal {H}}}}^{N-1}(S_{\pi -\theta })}{{{\mathcal {H}}}}^{N-1}(X\cap A^-)\,. \end{aligned}$$

Note now that $X\cap B^-=\emptyset$. Indeed, if $y\in X\cap B^-$, then the geodesic connecting y to x is contained in X and intersects $\partial S_{\theta ,x}$ at a point $z\in X\cap \partial S_{\theta , x}$. It follows in turn that y belongs to the geodesic connecting z with $-x$, and thus $y\in A^-$, which is a contradiction. Therefore,

$$\begin{aligned} \begin{aligned} {{{\mathcal {H}}}}^{N-1}(X\cap S_{\theta ,x})&={{{\mathcal {H}}}}^{N-1}(A)+{{{\mathcal {H}}}}^{N-1}(X\cap B)\ge {{{\mathcal {H}}}}^{N-1}(A)\\&= \frac{{{{\mathcal {H}}}}^{N-1}(S_{\theta })}{{{{\mathcal {H}}}}^{N-1}(S_{\pi -\theta })}{{{\mathcal {H}}}}^{N-1}(A^-)\\&\ge \frac{{{{\mathcal {H}}}}^{N-1}(S_{\theta })}{{{{\mathcal {H}}}}^{N-1}(S_{\pi -\theta })}{{{\mathcal {H}}}}^{N-1}(X\cap S_{\pi -\theta , -x})\,. \end{aligned} \end{aligned}$$

(3.2)

From this inequality (3.1) follows, recalling that ${{{\mathcal {H}}}}^{N-1}({\mathbb {S}}^{N-1})=N\omega _N$.

If instead ${{{\mathcal {H}}}}^{N-1}(A)=0$, then ${{{\mathcal {H}}}}^{N-1}(X\setminus S_{\theta ,x})=0$ and thus (3.1) holds trivially.

If (3.1) holds with the equality, then ${{{\mathcal {H}}}}^{N-1}(A)>0$ and all the inequalities in (3.2) are equalities. In particular, ${{{\mathcal {H}}}}^{N-1}(A^-)={{{\mathcal {H}}}}^{N-1}(X\cap S_{\pi -\theta , -x})>0$. In turn, recalling that $X\cap S_{\pi -\theta , -x} \subset A^-$ and by closedness we deduce that $-x\in X$. Conversely, if $-x\in X$ then by spherical convexity we have $A^-= X\cap S_{\pi -\theta , -x}$ and also $X\cap B=\emptyset$ since otherwise any geodesic connecting a point $y\in X\cap B$ to $-x$ would intersect $\partial S_{\theta , x}\cap X$, thus implying that y belongs to A, a contradiction. Therefore all the inequalities in (3.2) are equalities and the conclusion follows.

To establish the last part, we argue by contradiction assuming that there exist $\varepsilon >0$, a sequence of closed spherically convex sets $X_n\ni x$ such that ${{{\mathcal {H}}}}^{N-1}(X_n)>0$ and a sequence $\theta _n\in [\theta _0/2,\theta _0]$ converging to $\theta '$ such that

$$\begin{aligned} {{{\mathcal {H}}}}^{N-1}(X_n\cap S_{\theta _n, x})\le \left( \frac{{{{\mathcal {H}}}}^{N-1}(S_{\theta _n})}{N\omega _N}+\frac{1}{n}\right) {{{\mathcal {H}}}}^{N-1} (X_n)\quad \text {but }\quad \textrm{dist}(-x, X_n)\ge \varepsilon \,. \end{aligned}$$

(3.3)

We denote by $A_n$ and by $A^-_n$ the sets corresponding to $X_n$ and $S_{\theta _n,x}$ defined as above. Note that $X_n=(X_n\cap S_{\theta _n, x})\cup (X_n\cap A^-_n)$. From (3.3) it follows that ${{{\mathcal {H}}}}^{N-1}(A_n),\,{{{\mathcal {H}}}}^{N-1}(A_n^-)>0$ for n large and

$$\begin{aligned} \frac{{{{\mathcal {H}}}}^{N-1}(X_n\cap S_{\theta _n, x})}{{{{\mathcal {H}}}}^{N-1}(X_n\cap A^-_n)}\le \frac{{{{\mathcal {H}}}}^{N-1}(S_{\theta _n})}{{{{\mathcal {H}}}}^{N-1}(S_{\pi -\theta _n})}+O\Big (\tfrac{1}{n}\Big )\,. \end{aligned}$$

Since ${{{\mathcal {H}}}}^{N-1}(X_n\cap S_{\theta _n, x})\ge {{{\mathcal {H}}}}^{N-1}(A_n)\ge \frac{{{{\mathcal {H}}}}^{N-1}(S_{\theta _n})}{{{{\mathcal {H}}}}^{N-1}(S_{\pi -\theta _n})}{{{\mathcal {H}}}}^{N-1}(X_n\cap A^-_n)$, it follows that

$$\begin{aligned} \lim _{n\rightarrow \infty } \frac{{{{\mathcal {H}}}}^{N-1}(X_n\cap S_{\theta _n, x})}{{{{\mathcal {H}}}}^{N-1}(X_n\cap S_{\pi -\theta _n, -x})}=\lim _{n\rightarrow \infty }\frac{{{{\mathcal {H}}}}^{N-1}(A_n)}{{{{\mathcal {H}}}}^{N-1}(X_n\cap A^-_n)}=\frac{{{{\mathcal {H}}}}^{N-1}(S_{\theta '})}{{{{\mathcal {H}}}}^{N-1}(S_{\pi -\theta '})}\,. \end{aligned}$$

(3.4)

Note that we have

$$\begin{aligned} \frac{{{{\mathcal {H}}}}^{N-1}(S_{\theta _n})}{{{{\mathcal {H}}}}^{N-1}(S_{\pi -\theta _n})}= \frac{{{{\mathcal {H}}}}^{N-1}(A_n)}{{{{\mathcal {H}}}}^{N-1}( A^-_n)}\rightarrow \frac{{{{\mathcal {H}}}}^{N-1}(S_{\theta '})}{{{{\mathcal {H}}}}^{N-1}(S_{\pi -\theta '})} \end{aligned}$$

and thus, from (3.4) we get

$$\begin{aligned} \lim _{n\rightarrow \infty }\frac{{{{\mathcal {H}}}}^{N-1}(A^-_n)}{{{{\mathcal {H}}}}^{N-1}(X_n\cap A^-_n)}=1\,, \end{aligned}$$

which clearly contradicts the fact that by the second inequality in (3.3) we easily infer that ${{{\mathcal {H}}}}^{N-1}(A_n^-\setminus X_n)\ge C(\varepsilon ){{{\mathcal {H}}}}^{N-1}(A_n^-)$, for a positive constant $C(\varepsilon )$ depending only on $\varepsilon$. $\square$

Next we adapt to our case [4, Proposition 4.2]. To this aim we recall some preliminary definitions.

Definition 3.2

Given a set $X\subset {\mathbb {R}}^N$ and $x\in {\mathbb {R}}^N$ the unit normal cone of X at x is the (possibly empty) set defined as

$$\begin{aligned} N_xX:=\{\nu \in {\mathbb {S}}^{N-1}:\, (y-x)\cdot \nu \le 0\text { for all }y\in X\}\,. \end{aligned}$$

Any hyperplane passing through x and orthogonal to a direction $\nu \in N_xX$ is called a support hyperplane for X with outward normal $\nu$. In turn, we define the corresponding normal bundle of X as

$$\begin{aligned} NX:=\bigcup _{x\in X}N_xX\,. \end{aligned}$$

Given a map $\sigma : X\rightarrow {\mathbb {S}}^{N-1}$ and $\theta \in (0, \pi )$ we introduce the following restricted normal cone and restricted normal bundle respectively as

$$\begin{aligned} N_x^{\sigma , \theta } X:=N_xX\cap S_{\theta , \sigma (x)}\qquad \text {and}\qquad N^{\sigma , \theta } X:=\bigcup _{x\in X} N_x^{\sigma , \theta } X\,. \end{aligned}$$

Moreover, we say that a point $x\in X$ is exposed if there exists a support hyperplane $\Pi$ passing through x such that $X\cap \Pi =\{x\}$. Finally, we denote by $\textrm{width}(X)$ the distance between the closest pair of parallel hyperplanes which contains X in between them, i.e.,

$$\begin{aligned} \textrm{width}(X)=\inf _{\nu \in {\mathbb {S}}^{N-1}}\big (\sup \{x\cdot \nu :\,x\in X\}-\inf \{x\cdot \nu :\,x\in X\}\big )\,. \end{aligned}$$

(3.5)

Lemma 3.3

Let $r>0$ and let $X=\{x_1,\dots , x_k\}\subset B_r(0)$. Let $\sigma : X\rightarrow {\mathbb {S}}^{N-1}$ be such that $\sigma (x_i)\in N_{x_i}X$ whenever $N_{x_i}X$ is nonempty. Then

$$\begin{aligned} {{{\mathcal {H}}}}^{N-1}(N^{\sigma , \theta } X)\ge {{{\mathcal {H}}}}^{N-1}(S_{\theta })\,. \end{aligned}$$

(3.6)

Moreover, equality holds in (3.6) if and only if X lies in a hyperplane $\Pi$ such that $\sigma (x_i)\perp \Pi$ whenever $x_i$ is exposed. Finally, given $\theta _0\in (0,\pi )$, for every $\varepsilon >0$ there exists $\delta >0$ (depending also on $r>0$ and $\theta _0$, but not on $\sigma$ and not on X) such that if $\theta \in [\theta _0/2,\theta _0]$, then

$$\begin{aligned} {{{\mathcal {H}}}}^{N-1}(N^{\sigma , \theta }X)\le {{{\mathcal {H}}}}^{N-1}(S_{\theta })+\delta \quad \text {implies}\quad \textrm{width}(X)\le \varepsilon \, \end{aligned}$$

(3.7)

and more precisely there exist an exposed point $x\in X$ and two parallel hyperplanes orthogonal to $\sigma (x)$ with mutual distance equal to $\varepsilon$ such that X lies between them.

Proof

The proof is essentially the same as for [4, Proposition 4.2], using Lemma 3.1 in place of [4, Proposition 3.1]. We give the argument for the sake of completeness. Owing to the compactness of X, for every $\nu \in {\mathbb {S}}^{N-1}$ there exists a support hyperplane to X with outward normal equal to $\nu$. Thus, $NX={\mathbb {S}}^{N-1}$. Observe also that $\nu \in \textrm{int}_{{\mathbb {S}}^{N-1}} (N_{x_i}X)$ if and only if the hyperplane orthogonal to $\nu$ and passing through $x_i$ is a support hyperplane intersecting X only at $x_i$ (and thus $x_i$ is exposed). In turn, if $i\ne j$ we have

$$\begin{aligned} \textrm{int}_{{\mathbb {S}}^{N-1}}(N_{x_i}X)\cap \textrm{int}_{{\mathbb {S}}^{N-1}}(N_{x_j}X)=\emptyset \,. \end{aligned}$$

Since by [4, Lemma 4.1] every $N_{x_i}X$ with nonvanishing ${{{\mathcal {H}}}}^{N-1}$-measure is spherically convex, we may invoke Lemma 3.1 to conclude that

$$\begin{aligned} \begin{aligned} {{{\mathcal {H}}}}^{N-1}(N^{\sigma , \theta }X)=\sum _{i=1}^k{{{\mathcal {H}}}}^{N-1}(N^{\sigma , \theta }_{x_i}X)\ge \frac{{{{\mathcal {H}}}}^{N-1}(S_{\theta })}{N\omega _N} \sum _{i=1}^k{{{\mathcal {H}}}}^{N-1}(N_{x_i}X)={{{\mathcal {H}}}}^{N-1}(S_{\theta })\,, \end{aligned}\nonumber \\ \end{aligned}$$

(3.8)

thus establishing (3.6).

If equality holds in (3.6), then the above inequality is an equality and in particular

$$\begin{aligned} {{{\mathcal {H}}}}^{N-1}(N^{\sigma , \theta }_{x_i}X)=\frac{{{{\mathcal {H}}}}^{N-1}(S_{\theta })}{N\omega _N} {{{\mathcal {H}}}}^{N-1}(N_{x_i}X) \end{aligned}$$

(3.9)

whenever ${{{\mathcal {H}}}}^{N-1}(N_{x_i}X)>0$, that is whenever $x_i$ is exposed. Therefore, by Lemma 3.1$N^{\sigma , \theta }_{x_i}X$ contains both $\sigma (x_i)$ and $-\sigma (x_i)$ and thus X lies in the hyperplane orthogonal to $\sigma (x_i)$ and passing through $x_i$. Conversely, if X lies in a hyperplane orthogonal to $\sigma (x_i)$, for every $x_i$ exposed, then also $-\sigma (x_i)\in N_{x_i}X$ and thus by Lemma 3.1 (3.9) holds for all $x_i$ exposed. And thus equality holds also in (3.8).

To prove (3.7) and the last part of the lemma, let $X_n=\{x^n_1,\dots , x^n_{k_n}\}\subset B_r(0)$ and let $\sigma _n:X_n\rightarrow {\mathbb {S}}^{N-1}$, with $\sigma _n(x^n_i)\in N_{x^n_i}X_n$ whenever $x_i^n$ is exposed, $\theta _n\in [\theta _0/2,\theta _0]$ be such that

$$\begin{aligned} {{{\mathcal {H}}}}^{N-1}(N^{\sigma _n, \theta _n}X_n)- {{{\mathcal {H}}}}^{N-1}(S_{\theta _n})\rightarrow 0\,. \end{aligned}$$

Arguing as for (3.8) we then have, in particular, that for every $n\in {\mathbb {N}}$ there exists $i_n\in \{1, \dots , k_n\}$ such that

$$\begin{aligned} \frac{{{{\mathcal {H}}}}^{N-1}\left( N^{\sigma _n, \theta _n}_{x_{i_n}^n}X_n\right) }{{{{\mathcal {H}}}}^{N-1}\left( N_{x^n_{i_n}}X_n\right) }- \frac{{{{\mathcal {H}}}}^{N-1}(S_{\theta _n})}{N\omega _N}\rightarrow 0\,. \end{aligned}$$

By Lemma 3.1 this implies that $\textrm{dist}(-\sigma _n(x^n_{i_n}), N_{x_{i_n}^n}X_n)\rightarrow 0$. From this, owing to the equiboundedness of the $X_n$’s it follows that for every $k\in {\mathbb {N}}$ and for n large enough $X_n$ lies between the two parallel hyperplanes orthogonal to $\sigma _n(x^n_{i_n})$ and passing through the points $x^n_{i_n}$ and $x^n_{i_n}-\frac{1}{k} \sigma _n(x^n_{i_n})$. In particular, $\textrm{width}(X_n)\rightarrow 0$. $\square$

Next proposition extends the previous lemma to the case of a general compact set X and a continuous map $\sigma$.

Proposition 3.4

Let $X\subset B_r(0)$ be a compact set.

Let $\sigma : X\rightarrow {\mathbb {S}}^{N-1}$ be a continuous map such that $\sigma (x)\in N_x X$ for all $x\in X$ such that $N_x X\not =\emptyset$. Then,

$$\begin{aligned} {{{\mathcal {H}}}}^{N-1}(N^{\sigma , \theta }X)\ge {{{\mathcal {H}}}}^{N-1}(S_{\theta }) \end{aligned}$$

(3.10)

and if equality holds, then X lies in a hyperplane $\Pi$ which is orthogonal to $\sigma (x)$ for some $x\in X$. Moreover, given $\theta _0\in (0,\pi )$ and $\varepsilon >0$ there exists $\delta _0>0$ (depending also on r and $\theta _0$, but not on $\sigma$ and not on X) such that if $\theta \in [\theta _0/2,\theta _0]$, then

$$\begin{aligned} {{{\mathcal {H}}}}^{N-1}(N^{\sigma , \theta }X)\le {{{\mathcal {H}}}}^{N-1}(S_{\theta })+\delta _0\quad \text {implies}\quad \textrm{width}(X)\le \varepsilon \end{aligned}$$

(3.11)

and more precisely there exist $x\in X$ and two parallel hyperplanes orthogonal to $\sigma (x)$, with mutual distance equal to $\varepsilon$ such that X lies between them.

Proof

Let $\{X_i\}_{i\in {\mathbb {N}}}$ be an increasing sequence of discrete subsets of X such that $X_i\rightarrow X$ in the Hausdorff sense. We claim that

$$\begin{aligned} \chi _{_{N^{\sigma , \theta }X}}\ge \limsup _{i} \chi _{_{N^{\sigma , \theta }X_i}} \quad \text {pointwise in }{\mathbb {S}}^{N-1}\,. \end{aligned}$$

(3.12)

To this aim let $\nu \not \in N^{\sigma , \theta }X$ and assume by contradiction that (3.12) does not hold at $\nu$ and thus that there exist a subsequence $\{i_n\}$ and points $x_n\in X_{i_n}$ such that $\nu \in N^{\sigma , \theta }_{x_n}X_{i_n}$. Passing to a further (not relabelled) subsequence if needed, we may assume that $x_n\rightarrow {{\bar{x}}}\in X$. Observe that by the continuity of $\sigma (\cdot )$, $\nu \in S_{\theta , \sigma ({{\bar{x}}})}$. Fix now any $x\in X$ and due to the Hausdorff convergence find $y_n\in X_{i_n}$ such that $y_n\rightarrow x$. Since for every n, $(y_n-x_n)\cdot \nu \le 0$ passing to the limit we get $(x-{{\bar{x}}})\cdot \nu \le 0$. Due to the arbitrariness of x, we have shown that $\nu \in N_{{{\bar{x}}}} X$ and thus $\nu \in N^{\sigma , \theta }_{{{\bar{x}}}}X$, a contradiction.

Using the first part of Lemma 3.3 (with X replaced by $X_i$), (3.12) and Fatou’s Lemma we get

$$\begin{aligned} {{{\mathcal {H}}}}^{N-1} (N^{\sigma , \theta }X)\ge \limsup _{i} {{{\mathcal {H}}}}^{N-1}({N^{\sigma , \theta }X_i})\ge \liminf _i {{{\mathcal {H}}}}^{N-1}({N^{\sigma , \theta }X_i})\ge {{{\mathcal {H}}}}^{N-1}(S_{\theta })\,. \end{aligned}$$

Assume now that the first inequality (3.11) holds for some $\theta \in [\theta _0/2,\theta _0]$, with $\delta _0=\frac{\delta }{2}$, where $\delta$ is the constant provided by Lemma 3.3. Then the previous inequality yields for i sufficiently large, depending on $\theta$,

$$\begin{aligned} {{{\mathcal {H}}}}^{N-1}({N^{\sigma , \theta }X_i}) \le {{{\mathcal {H}}}}^{N-1}(S_{\theta })+\delta \end{aligned}$$

and thus, thanks to second part of Lemma 3.3 we infer that there exists $x_i\in X_i$ and two parallel hyperplanes orthogonal to $\sigma (x_i)$ with mutual distance equal to $\varepsilon$ such that $X_i$ lies between them. By a compactness argument and the continuity of $\sigma$, letting $i\rightarrow \infty$ we get that there exist $x\in X$ and two parallel hyperplanes orthogonal to $\sigma (x)$ with mutual distance equal to $\varepsilon$ such that X lies between them. Thus, in particular $\textrm{width}(X)\le \varepsilon$. This establishes (3.11), which in turn, again by a compactness argument and the continuity of $\sigma$, yields the conclusion in the equality case. $\square$

Next we prove a result in the spirit of [4, Theorem 1.1]. In the following ${\textbf{C}}$, $\Omega$ and $\Sigma$ will be as in Definition 1.3. Moreover if $x\in \Sigma$ is a point where the tangent hyperplane to $\Sigma$ exists we denote by $\nu _\Sigma (x)$ the normal to this hyperplane pointing outward with respect to $\Omega$. We give the following definition.

Definition 3.5

We denote by $\Sigma ^+$ the set of points in $\Sigma \setminus {\textbf{C}}$ such that there exists a support hyperplane $\Pi _x$ with the property that $\Pi _x\cap \Sigma =\{x\}$.

We recall the following result, see [20, Theorem 2.2.9]:

Theorem 3.6

Let $K\subset {\mathbb {R}}^N$ be a compact convex set. Then for ${{{\mathcal {H}}}}^{N-1}$-almost every $\nu \in {\mathbb {S}}^{N-1}$ the support hyperplane for K orthogonal to $\nu$ intersects K at a single point.

Corollary 3.7

Let ${\textbf{C}}$ and $\Sigma \subset {\mathbb {R}}^N$ be as in Definition 1.3. With the notation above, we have that

$$\begin{aligned} \mathcal {K}^+(\Sigma )={{{\mathcal {H}}}}^{N-1}\left( \bigcup _{x\in \Sigma ^+}N_x\Sigma \right) \,, \end{aligned}$$

where ${\mathcal {K}}^+(\Sigma )$ is the total positive curvature defined in Definition 1.3.

Proof

Let K denote the closed convex hull of $\Sigma$. By Theorem 3.6 we have that for ${{{\mathcal {H}}}}^{N-1}$-a.e. direction $\nu \in \bigcup _{x\in \Sigma \setminus {\textbf{C}}} N_x\Sigma$ the corresponding support plane for K intersects K at a single point that necessarily belongs to $\Sigma \setminus {\textbf{C}}$ and thus to $\Sigma ^+$. $\square$

Proof of Theorem 1.4

Observe that if $\Sigma \cap {\textbf{C}}=\emptyset$ then

$$\begin{aligned} \bigcup _{x\in \Sigma \setminus {\textbf{C}}}N_x\Sigma ={\mathbb {S}}^{N-1}\,, \end{aligned}$$

hence (1.3) trivially holds.

Hence in the following we may assume that $\Sigma \cap {\textbf{C}}\not =\emptyset$.

We denote by $\nu _{\textbf{C}}$ the outward normal to ${\textbf{C}}$. We start by proving (1.3). Let us define $\sigma : \Sigma \cap {\textbf{C}}\rightarrow {\mathbb {S}}^{N-1}$ as $\sigma (x):=\nu _{\textbf{C}}(x)$. Note that since ${\textbf{C}}$ is convex the direction $\sigma (x)$ belongs to $N_x{\textbf{C}}$ and thus to $N_x(\Sigma \cap {\textbf{C}})$ for every $x\in \Sigma \cap {\textbf{C}}$.

Given $\nu \in {\mathbb {S}}^{N-1}$, we denote by $\nu ^\perp$ the hyperplane orthogonal to $\nu$ and passing through the origin and we set

$$\begin{aligned} {{\bar{t}}}:=\max \{t\in {\mathbb {R}}:\, (t\nu +\nu ^\perp )\cap \Sigma \ne \emptyset \}\,. \end{aligned}$$

Clearly, by definition for every $\nu \in {\mathbb {S}}^{N-1}$ the hyperplane ${{\bar{t}}}\nu +\nu ^\perp$ is a support hyperplane for $\Sigma$. Fix $\theta \in (0,\theta _0)$. We claim that for every $x \in \Sigma \cap {\textbf{C}}$

$$\begin{aligned} \nu \in N_x(\Sigma \cap {\textbf{C}})\cap S_{\theta , \sigma (x)}\quad \text {implies}\quad {{\bar{t}}}\nu +\nu ^\perp \cap \Sigma \subset \Sigma \setminus {\textbf{C}}\,. \end{aligned}$$

(3.13)

Let $t_0\in {\mathbb {R}}$ be such that $x+\nu ^\perp = t_0\nu +\nu ^\perp$ and observe that since $\nu \cdot \sigma (x)\ge \cos \theta >\cos \theta _0$ then by assumption (1.2) $\nu \not \in N_x\Sigma$, hence the hyperplane $t_0\nu +\nu ^\perp$ enters $\Omega$. Thus it easily follows that ${{\bar{t}}}>t_0$. Let $y\in {{\bar{t}}}\nu +\nu ^\perp \cap \Sigma$. Then $y\not \in \Sigma \cap {\textbf{C}}$, since otherwise this would contradict the fact that $t_0\nu +\nu ^\perp$ is a support hyperplane for $\Sigma \cap {\textbf{C}}$. This establishes (3.13). From (3.13) it follows that

$$\begin{aligned} N^{\sigma , \theta }(\Sigma \cap {\textbf{C}})\subset \bigcup _{x\in \Sigma \setminus {\textbf{C}}}N_x\Sigma \,. \end{aligned}$$

(3.14)

Recall that by Definition 1.3

$$\begin{aligned} {{{\mathcal {H}}}}^{N-1}\left( \bigcup _{x\in \Sigma \setminus {\textbf{C}}}N_x\Sigma \right) =\mathcal K^+(\Sigma )\,. \end{aligned}$$

Combining the equality above with (3.14), the inequality (1.3) follows from (3.10) with $X=\Sigma \cap {\textbf{C}}$, letting $\theta \rightarrow \theta _0^-$.

Given $\varepsilon >0$, let $\delta _0$ be the constant provided by Proposition 3.4 and let $\theta \in [\theta _0/2,\theta _0)$ such that

$$\begin{aligned} {{{\mathcal {H}}}}^{N-1}(S_{\theta _0})\le {{{\mathcal {H}}}}^{N-1}(S_{\theta })+\frac{\delta _0}{2}\,. \end{aligned}$$

(3.15)

Assume that (1.4) and (1.5) hold for some $\delta \in (0,\delta _0/2)$ such that $\cos \theta _0+\delta <\cos \theta$. Then, using the assumption (1.4), the same argument as before yields (3.13), hence (3.14). Thus, from (1.5) and (3.15) we have in particular

$$\begin{aligned} {{{\mathcal {H}}}}^{N-1}(N^{\sigma , \theta }(\Sigma \cap {\textbf{C}}))\le \mathcal K^+(\Sigma )\le {{{\mathcal {H}}}}^{N-1}(S_{\theta _0})+\delta \le {{{\mathcal {H}}}}^{N-1}(S_{\theta })+\delta _0\,. \end{aligned}$$

The conclusion follows from Proposition 3.4. $\square$

Remark 3.8

Observe that the equality case in (1.3) does not imply $\partial \Omega \cap {\textbf{C}}$ lies on a facet of ${\textbf{C}}$. In fact it may happen that $\partial \Omega \cap {\textbf{C}}$ is contained in a convex set of Hausdorff dimension strictly less than $N-1$, see Fig. 1.

Fig. 1

Both $\Sigma _1=\overline{\partial \Omega _1\setminus {\textbf{C}}}$ and $\Sigma _2=\overline{\partial \Omega _2\setminus {\textbf{C}}}$ meet ${\textbf{C}}$ with contact angle $\pi /4$ and satisfy the equality in (1.3) with $\theta _0=3\pi /4$. Note that $\partial \Omega _1\cap {\textbf{C}}$ is a point and $\partial \Omega _2\cap {\textbf{C}}$ is a segment

It is well known that for surfaces $\Sigma \subset {\mathbb {R}}^3$ without boundary the following inequality holds

$$\begin{aligned} \int _{\Sigma }|H_\Sigma |^{2}\,d{{{\mathcal {H}}}}^{2}\ge 16\pi \,. \end{aligned}$$

with equality achieved if and only if $\Sigma$ is a sphere. We now apply Theorem 1.4 to extend this inequality to the following extension of the Willmore energy in N-dimensions

$$\begin{aligned} \int _{\Sigma \setminus {\textbf{C}}}|H_\Sigma |^{N-1}\,d{{{\mathcal {H}}}}^{N-1}\,, \end{aligned}$$

for $C^{1,1}$ hypersurfaces with boundary supported on convex sets and with contact angle larger than a given $\theta _0\in (0,\pi )$. Note that in the next theorem we do not assume any regularity on the convex set ${\textbf{C}}$.

Theorem 3.9

(A Willmore type inequality.) Let ${\textbf{C}}$, $\Omega$ and $\Sigma$ be as in Definition 1.3 and let $\theta _0\in (0, \pi )$. Assume that $\Sigma \setminus {\textbf{C}}$ is of class $C^{1,1}$. Set $H_\Sigma :=\textrm{div}_\Sigma {\nu _\Sigma }$ (where $\nu _\Sigma$ is the unit normal to $\Sigma$ pointing outward with respect to $\Omega$). Assume also

$$\begin{aligned} \nu \cdot \nu '\le \cos \theta _0 \quad \text {whenever}\ x\in \Sigma \cap {\textbf{C}},\,\,\nu \in N_x\Sigma \,\,\ \text {and}\ \,\,\nu '\in N_x{\textbf{C}}\,. \end{aligned}$$

(3.16)

Then,

$$\begin{aligned} \int _{\Sigma \setminus {\textbf{C}}}|H_\Sigma |^{N-1}\,d{{{\mathcal {H}}}}^{N-1}\ge (N-1)^{N-1}{{{\mathcal {H}}}}^{N-1}(S_{\theta _0})\,. \end{aligned}$$

(3.17)

Moreover, if equality holds in (3.17) and $H_\Sigma \not =0$ a.e., then $\Sigma \setminus {\textbf{C}}$ coincides, up to a rigid motion, with an omothetic of $S_{\theta _0}$ sitting on a facet of ${\textbf{C}}$.

Proof

Without loss of generality we may assume that

$$\begin{aligned} \int _{\Sigma \setminus {\textbf{C}}}|H_\Sigma |^{N-1}\,d{{{\mathcal {H}}}}^{N-1}<\infty \,. \end{aligned}$$

Set for any $\eta >0$ sufficiently small ${\textbf{C}}_\eta :={\textbf{C}}+\overline{B_\eta (0)}$ and $\Sigma _\eta :=\overline{\partial \Omega \setminus {\textbf{C}}_\eta }$. Observe that ${\textbf{C}}_\eta$ satisfies both a outer and inner uniform ball condition and thus is of class $C^{1,1}$, see [7, 16]. Note also that there exists $\theta _\eta \in (0, \theta _0)$ such that

$$\begin{aligned} \nu \cdot \nu _{{\textbf{C}}_\eta }(x)\le \cos \theta _\eta \quad \text {whenever}\ x\in \Sigma _\eta \cap {\textbf{C}}_\eta ,\,\,\nu \in N_x\Sigma _\eta \,, \end{aligned}$$

(3.18)

with $\theta _\eta \rightarrow \theta _0$ as $\eta \rightarrow 0^+$. Indeed, if not, there would exist a sequence $\eta _h\rightarrow 0$, a sequence of points $x_h\in \Sigma _{\eta _h}\cap {\textbf{C}}_{\eta _h}$ and a sequence $\nu _h\in N_{x_h}\Sigma _{\eta _h}$, such that $\nu _h\cdot \nu _{{\textbf{C}}_{\eta _h}}(x_h)\ge \cos \theta '$ for some $\theta '\in (0,\theta _0)$. We may assume that $x_h\rightarrow x\in \Sigma \cap {\textbf{C}}$, $\nu _h\rightarrow \nu$ and $\nu _{{\textbf{C}}_{\eta _h}}(x_h)\rightarrow \nu '$. Clearly $\nu \in N_x\Sigma$, $\nu '\in N_x{\textbf{C}}$ and $\nu \cdot \nu _{\textbf{C}}(x)\ge \cos \theta '$, a contradiction to (1.2).

We set

$$\begin{aligned} {{\widetilde{\Sigma }}}=\{x\in \Sigma \setminus {\textbf{C}}:\, N_x\Sigma \not =\emptyset \},\qquad \widetilde{\Sigma }_\eta =\{x\in \Sigma \setminus {\textbf{C}}_\eta :\, N_x\Sigma _\eta \not =\emptyset \}\,. \end{aligned}$$

We claim that

$$\begin{aligned} \chi _{_{{{\widetilde{\Sigma }}}_\eta }}\rightarrow \chi _{_{{{\widetilde{\Sigma }}}}} \quad \hbox {pointwise in}\ \Sigma \setminus {\textbf{C}}\ \hbox {as}\ \eta \rightarrow 0\,. \end{aligned}$$

(3.19)

First of all note that ${{\widetilde{\Sigma }}}\setminus {\textbf{C}}_\eta \subset {{\widetilde{\Sigma }}}_\eta$ for all $\eta$, whence

$$\begin{aligned} \chi _{_{{{\widetilde{\Sigma }}}}}=\lim _{\eta \rightarrow 0+}\chi _{_{{{\widetilde{\Sigma }}}\setminus {\textbf{C}}_\eta }} \le \liminf _{\eta \rightarrow 0^+} \chi _{_{{{\widetilde{\Sigma }}}_\eta }}\quad \text {pointwise in}\ \Sigma \setminus {\textbf{C}}\end{aligned}$$

If otherwise $x\not \in {{\widetilde{\Sigma }}}$, we show that $x\not \in {{\widetilde{\Sigma }}}_\eta$ for $\eta$ small. Indeed, assume by contradiction that there exist $\nu _h\in N_x(\Sigma _{\eta _h})$, for a sequence $\eta _h\rightarrow 0$. Then, passing to a subsequence, if needed, $\nu _h\rightarrow \nu \in N_x\Sigma$, a contradiction. This proves that

$$\begin{aligned} \chi _{_{{{\widetilde{\Sigma }}}}}\ge \limsup _{\eta \rightarrow 0^+} \chi _{_{{{\widetilde{\Sigma }}}_\eta }}\quad \text {pointwise in}\; \Sigma \setminus {\textbf{C}}\end{aligned}$$

and thus (3.19) holds.

Let $(\Sigma _\eta )^+$ the subset of ${{\widetilde{\Sigma }}}_\eta$ defined as in Definition 3.5 with $\Sigma$ replaced by $\Sigma _\eta$. Denote by $K_\Sigma$ the Gaussian curvature of $\Sigma \setminus {\textbf{C}}$ and observe that on ${{\widetilde{\Sigma }}}_\eta$ all principal curvatures are nonnegative. By the arithmetic-geometric mean inequality $(N-1)^{N-1}K_\Sigma \le H_\Sigma ^{N-1}$ on ${{\widetilde{\Sigma }}}_\eta$. Then by Theorem 1.4 we get

$$\begin{aligned} \int _{\Sigma \setminus {\textbf{C}}_\eta }|H_\Sigma |^{N-1}\,d{{{\mathcal {H}}}}^{N-1}&\ge \int _{{{\widetilde{\Sigma }}}_\eta }H_\Sigma ^{N-1}\,d{{{\mathcal {H}}}}^{N-1}\ge (N-1)^{N-1}\int _{{{\widetilde{\Sigma }}}_\eta }K_\Sigma \,d{{{\mathcal {H}}}}^{N-1}\nonumber \\&\ge (N-1)^{N-1}\int _{(\Sigma _\eta )^+}K_\Sigma \,d{{{\mathcal {H}}}}^{N-1}\nonumber \\&=(N-1)^{N-1}\int _{(\Sigma _\eta )^+}\textrm{det}(D\nu _\Sigma )\,d{{{\mathcal {H}}}}^{N-1}\nonumber \\&=(N-1)^{N-1}{\mathcal K}^+(\Sigma _\eta )\ge (N-1)^{N-1}{{{\mathcal {H}}}}^{N-1}(S_{\theta _\eta })\,, \end{aligned}$$

(3.20)

where in the second equality we have used Corollary 3.7 and the area formula, since $\nu _{\Sigma }$ is a Lipschitz map in a neighborhood of $\Sigma ^+$. Then, letting $\eta \rightarrow 0$ and recalling (3.19) and the fact that $\theta _\eta \rightarrow \theta _0$, we get

$$\begin{aligned} \begin{aligned} \int _{\Sigma \setminus {\textbf{C}}}|H_\Sigma |^{N-1}\,d{{{\mathcal {H}}}}^{N-1}&\ge \int _{{{\widetilde{\Sigma }}}}H_\Sigma ^{N-1}\,d{{{\mathcal {H}}}}^{N-1}\\&\ge (N-1)^{N-1}\int _{{{\widetilde{\Sigma }}}}K_\Sigma \,d{{{\mathcal {H}}}}^{N-1}\ge (N-1)^{N-1}{{{\mathcal {H}}}}^{N-1}(S_{\theta _0})\,. \end{aligned}\nonumber \\ \end{aligned}$$

(3.21)

In particular (3.17) follows.

If equality holds in (3.17) holds, from (3.20) we have that ${\mathcal {K}}^+(\Sigma _\eta )-{{{\mathcal {H}}}}^{N-1}(S_{\theta _\eta })\rightarrow 0$. In turn from the second part of Theorem 1.4 we get that $\textrm{width}(\Sigma _\eta \cap {\textbf{C}}_\eta )\rightarrow 0$ and more precisely that $\Sigma _\eta \cap {\textbf{C}}_\eta$ lies between two parallel hyperplanes orthogonal to $\nu _{{\textbf{C}}_\eta }(x_\eta )$ for some $x_\eta \in \Sigma _\eta \cap {\textbf{C}}_\eta$ with mutual distance going to zero. Passing to the limit by a simple compactness argument we infer that $\Sigma \cap {\textbf{C}}$ lies on a support hyperplane to ${\textbf{C}}$.

Note also that in the equality case, if $H_\Sigma \not =0$ ${{{\mathcal {H}}}}^{N-1}$-a.e., then (3.21) implies that $\Sigma \setminus {\textbf{C}}={{\widetilde{\Sigma }}}$. In turn this yields that every $x\in \Sigma \cap {\textbf{C}}$ has a support hyperplane to $\Sigma$. Moreover, (3.21) yields also that $H_\Sigma ^{N-1}=(N-1)^{N-1}K_\Sigma$. In turn this implies that $\Sigma$ is umbilical and thus, by a classical result, see for instance [19, Th. 3.1], each connected component $\Sigma _i$ of $\Sigma$ is contained in a sphere. Since $\Sigma \cap {\textbf{C}}$ is contained in a hyperplane $\Pi$ tangent to ${\textbf{C}}$, each $\Sigma _i$ is either a spherical cap supported on $\Pi$ and satisfying (3.16) with $\Sigma$ replaced by $\Sigma _i$, or a sphere not intersecting ${\textbf{C}}$.

In either case, since (3.16) is satisfied at every point in $\Sigma \cap {\textbf{C}}$ (recall that $\Sigma \setminus {\textbf{C}}={{\widetilde{\Sigma }}}$), we may apply (3.17) to infer that for every connected component $\Sigma _i$ we have

$$\begin{aligned} \int _{\Sigma _i\setminus {\textbf{C}}}H_{\Sigma _i}^{N-1}\,d{{{\mathcal {H}}}}^{N-1}\ge (N-1)^{N-1}{{{\mathcal {H}}}}^{N-1}(S_{\theta _0})\,. \end{aligned}$$

In particular, since we are in the equality case, there must be only one connected component. Thus $\Sigma$ is a spherical cap homothetic to $S_{\theta _0}$ up to a rigid motion. Finally $\Sigma \cap {\textbf{C}}$ by convexity must lie on a facet of ${\textbf{C}}$. $\square$

4 The equality case in the relative isoperimetric inequality outside a convex set

In this section we give the proof of Theorem 1.2. Throughout this proof we will denote by ${\mathscr {H}}$ the half space

$$\begin{aligned} {\mathscr {H}}:=\{(x', x_N)\in {\mathbb {R}}^N:\, x_N>0\}\,. \end{aligned}$$

(4.1)

We will also need the following notions of $(\Lambda ,r_0)$-minimizer and restricted $(\Lambda ,r_0)$-minimizer for the relative perimeter, which extend the standard notion of perimeter $(\Lambda ,r_0)$-minimizer recalled in Definition 2.1.

Definition 4.1

Let ${\textbf{C}}\subset {\mathbb {R}}^N$ be a closed convex set with nonempty interior and let $\Lambda ,r_0>0$. We say that a set of finite perimeter $E\subset {\mathbb {R}}^N\setminus {\textbf{C}}$ is a $(\Lambda ,r_0)$-minimizer of the relative perimeter $P(\cdot ;{\mathbb {R}}^N\setminus {\textbf{C}})$ if for any $F\subset {\mathbb {R}}^N\setminus {\textbf{C}}$ such that diam$(E\Delta F)\le r_0$ we have

$$\begin{aligned} P(E;{\mathbb {R}}^N\setminus {\textbf{C}})\le P(F;{\mathbb {R}}^N\setminus {\textbf{C}})+\Lambda |E\Delta F|\,. \end{aligned}$$

Moreover, we say that $E\subset {\mathbb {R}}^N\setminus {\textbf{C}}$ is a restricted $(\Lambda ,r_0)$-minimizer if the above inequality holds for every set $F\subset {\mathbb {R}}^N\setminus {\textbf{C}}$ such that diam$(E\Delta F)\le r_0$ and $\partial ^*F\cap {\textbf{C}}\subset \partial ^*E\cap {\textbf{C}}$ up to a ${{{\mathcal {H}}}}^{N-1}$-negligible set.

Proof of Theorem 1.2

Let $m_0>0$ be a given mass and let $\Omega _0$ be a minimizer of the perimeter outside ${\textbf{C}}$ such that $|\Omega _0|=m_0$ and

$$\begin{aligned} P(\Omega _0;{\mathbb {R}}^N\setminus {\textbf{C}})=N\Big (\frac{\omega _N}{2}\Big )^{\frac{1}{N}}m_0^{\frac{N-1}{N}}\,. \end{aligned}$$

(4.2)

Since $\Omega _0$ solves the isoperimetric problem we have that $\Omega _0$ is a $(\Lambda _0,r_0)$-minimizer of the relative perimeter in ${\mathbb {R}}^N\setminus {\textbf{C}}$, see Definition 4.1, for some $\Lambda _0, r_0>0$, depending on $\Omega _0$, see for instance the argument of [15, Example 21.3].² In turn by Proposition 5.2$\Omega _0$ satisfies uniform volume density estimates and thus it easily follows that $\Omega _0$ is bounded.

We fix a sufficiently large ball $B_R(0)$ containing $\overline{\Omega _0}$. Note that by standard argument, see also the argument of Step 1 below, $\Omega _0$ solves the following penalized minimum problem

$$\begin{aligned} \min \{P(E;{\mathbb {R}}^N\setminus {\textbf{C}})+\Lambda _0||E|-m_0|:\,\,E\subset B_R\setminus {\textbf{C}}\}\,, \end{aligned}$$

for a possibly larger $\Lambda _0$. In particular we have

$$\begin{aligned} P(\Omega _0;{\mathbb {R}}^N\setminus {\textbf{C}})\le P(E;{\mathbb {R}}^N\setminus {\textbf{C}})+\Lambda _0|\Omega _0\Delta E|\qquad \text {for all}\; E\subset B_R\setminus {\textbf{C}}. \end{aligned}$$

(4.3)

Since in the remaining part of the proof we will always work inside $B_R$, up to replacing ${\textbf{C}}$ with ${\textbf{C}}\cap \overline{B_R}$, we may assume without loss of generality that ${\textbf{C}}$ is bounded.

Observe that by Theorem 2.6 we may assume that $\Omega _0$ is an open set and that $\partial \Omega _0\setminus {\textbf{C}}$ coincides with the reduced boundary $\partial ^*\Omega _0\setminus {\textbf{C}}$ up to an ${{{\mathcal {H}}}}^{N-1}$-negligible set. Let us show that $\Omega _0$ is connected. Indeed, if otherwise $\Omega _0=\Omega _1\cup \Omega _2$, with $\Omega _1$ and $\Omega _2$ open, $\Omega _1$ a connected component of $\Omega _0$ with $0<|\Omega _1|<m_0$, we have by Theorem 1.1

$$\begin{aligned} P(\Omega _0;{\mathbb {R}}^N\setminus {\textbf{C}})&=P(\Omega _1;{\mathbb {R}}^N\setminus {\textbf{C}})+P(\Omega _2;{\mathbb {R}}^N\setminus {\textbf{C}})\\&\ge N\Big (\frac{\omega _N}{2}\Big )^{\frac{1}{N}}|\Omega _1|^{\frac{N-1}{N}}+N\Big (\frac{\omega _N}{2}\Big )^{\frac{1}{N}}|\Omega _2|^{\frac{N-1}{N}}>N\Big (\frac{\omega _N}{2}\Big )^{\frac{1}{N}}m_0^{\frac{N-1}{N}}\,, \end{aligned}$$

which is a contradiction to (4.2).

For every $\eta \ge 0$ we set ${\textbf{C}}_\eta ={\textbf{C}}+\overline{B_\eta (0)}$ and, for $\eta \in [0,{{\bar{\eta }}}]$ we set $m_\eta :=|\Omega _0\setminus {\textbf{C}}_\eta |$, where ${{\bar{\eta }}}>0$ is such that $|\Omega _0\setminus {\textbf{C}}_{{{\bar{\eta }}}}|>0$. Correspondingly, we set for $m\in (0,m_\eta ]$

$$\begin{aligned} I_\eta (m)=\min \{P(E;{\mathbb {R}}^N\setminus {\textbf{C}}_\eta ):\,\,E\subset \Omega _0\setminus {\textbf{C}}_\eta ,\,\,|E|=m\} \end{aligned}$$

(4.4)

and denote by $\Omega _{\eta ,m}$ any minimizer of the above problem. Note that $\Omega _{0,m_0}=\Omega _0$. Observe also that

$$\begin{aligned} \sup _{\eta \in [0,{{\bar{\eta }}}]}\sup _{m\in (0,m_\eta )}P(\Omega _{\eta ,m})<\infty \,. \end{aligned}$$

(4.5)

Indeed, given $\eta \in [0,{{\bar{\eta }}}]$ and $0<m\le m_\eta$ there exists $\eta '\ge \eta$ such that $|\Omega _0\setminus {\textbf{C}}_{\eta '}|=m$. Thus

$$\begin{aligned} P(\Omega _{\eta ,m})&\le P(\Omega _{\eta ,m};{\mathbb {R}}^N\setminus {\textbf{C}}_\eta )+P({\textbf{C}}_\eta ;B_R)\le P(\Omega _0\setminus {\textbf{C}}_{\eta '})+P({\textbf{C}}_\eta ;B_R)\\&\le P(\Omega _0;{\mathbb {R}}^N\setminus {\textbf{C}})+2\sup _{s\ge 0}P({\textbf{C}}_s;B_R)\le P(\Omega _0;{\mathbb {R}}^N\setminus {\textbf{C}})+2N\omega _NR^{N-1}\,. \end{aligned}$$

Let us fix $m', m''\in (0,m_0)$, with $m'<m''$. We claim that there exists ${{\tilde{\eta }}}\in (0,{{\bar{\eta }}}]$ such that

$$\begin{aligned} \text {if}\ \eta \in [0,{{\tilde{\eta }}}]\ \text {and}\ U\ \text {is a connected component of}\ \Omega _0\setminus {\textbf{C}}_\eta ,\ \text {then} |U|\not \in [m',m'']. \end{aligned}$$

(4.6)

To prove (4.6) we fix $x_0\in \Omega _0$ and for every $\eta$ we denote by $U_\eta$ the connected component of $\Omega _0\setminus {\textbf{C}}_\eta$ containing $x_0$. Note that $U_\eta$ increases as $\eta$ becomes smaller. Given any other point $x\in \Omega _0$ there exists a path connecting $x_0$ and x contained in $\Omega _0$, thus $x\in U_\eta$ for $\eta$ small enough. Hence $|U_\eta |\rightarrow m_0$, and the claim follows. Note that this argument implies also that for $\eta$ sufficiently small

$$\begin{aligned} \partial \Omega _{\eta ,m}\cap (\Omega _0\setminus {\textbf{C}}_\eta )\not =\emptyset \qquad \text {for all}\ \eta \in [0,{{\tilde{\eta }}}]\ \text {and}\ m\in [m',m'']. \end{aligned}$$

(4.7)

We split the remaining part of the proof in several steps. Some of the long technical claims contained in these steps will be proved in “Appendix B” so as not to break the line of reasoning.

Step 1 (Equivalence with a volume penalized problem). Fix $0<m'<m''<m_0$ and let $0<{{\tilde{\eta }}}\le {{\bar{\eta }}}$ be as in (4.6). We claim that there exists $\Lambda '>0$ with the following property: for every $\eta \in [0,{{\tilde{\eta }}}]$ and $m\in [m',m'']$ we have that $\Omega _{\eta ,m}$ is a minimizer of the following problem

$$\begin{aligned} \min \{P(E;{\mathbb {R}}^N\setminus {\textbf{C}}_\eta )+\Lambda '||E|-m|:\,\,E\subset \Omega _0\setminus {\textbf{C}}_\eta \}\,. \end{aligned}$$

(4.8)

The proof of this claim will be given in the “Appendix B”.

Step 2 ($\Omega _{\eta ,m}$ is a restricted $\Lambda$-minimizer). Fix $0<m'<m''<m_0$, let $0<{{\tilde{\eta }}}\le {{\bar{\eta }}}$ be as in (4.6) and set $\Lambda =\max \{\Lambda ',\Lambda _0\}$, where $\Lambda '$ is as in Step 1. We claim that for every $\eta \in [0,{{\tilde{\eta }}}]$ and $m\in [m',m'']$, $\Omega _{\eta ,m}$ is a restricted $\Lambda$-minimizer under the constraint that $\partial ^*E\cap {\textbf{C}}_\eta \subset \partial ^*(\Omega _0\setminus {\textbf{C}}_\eta )\cap {\textbf{C}}_\eta$. More precisely, for every set of finite perimeter $E\subset B_R(0)\setminus {\textbf{C}}_\eta$ such that $\partial ^*E\cap {\textbf{C}}_\eta \subset \partial ^*(\Omega _0\setminus {\textbf{C}}_\eta )\cap {\textbf{C}}_\eta$ up to a ${{{\mathcal {H}}}}^{N-1}$-negligible set

$$\begin{aligned} P(\Omega _{\eta ,m};{\mathbb {R}}^N\setminus {\textbf{C}}_\eta )\le P(E;{\mathbb {R}}^N\setminus {\textbf{C}}_\eta )+\Lambda |\Omega _{\eta ,m}\Delta E|\,. \end{aligned}$$

(4.9)

In particular $\Omega _{\eta ,m}$ is a restricted $(\Lambda ,r_0)$-minimizer according to Definition 4.1, choosing for instance $r_0:=$dist$(\Omega _0,\partial B_R(0))$.

Given E as above, from Step 1 we get

$$\begin{aligned} P\left( \Omega _{\eta ,m};{\mathbb {R}}^N\setminus {\textbf{C}}_\eta \right)&\le P(E\cap \Omega _0;{\mathbb {R}}^N\setminus {\textbf{C}}_\eta )+\Lambda '||E\cap \Omega _0|-|\Omega _{\eta ,m}|| \nonumber \\&={{{\mathcal {H}}}}^{N-1}(\partial ^*E\cap (\Omega _0\setminus {\textbf{C}}_\eta ))+{{{\mathcal {H}}}}^{N-1}(\partial ^*E\cap \partial ^*\Omega _0\cap \{\nu _E=\nu _{\Omega _0}\}\setminus {\textbf{C}}_\eta )\nonumber \\&\quad +{{{\mathcal {H}}}}^{N-1}(\partial ^*\Omega _0\cap E^{(1)})+\Lambda '||E\cap \Omega _0|-|\Omega _{\eta ,m}||\,. \end{aligned}$$

(4.10)

Then, using (4.3) and the condition $\partial ^*E\cap {\textbf{C}}_\eta \subset \partial ^*(\Omega _0\setminus {\textbf{C}}_\eta )\cap {\textbf{C}}_\eta$, we have

$$\begin{aligned}&{{{\mathcal {H}}}}^{N-1}((\partial ^*\Omega _0\cap {\textbf{C}}_\eta )\setminus {\textbf{C}})+{{{\mathcal {H}}}}^{N-1}(\partial ^*\Omega _0\setminus {\textbf{C}}_\eta )=P(\Omega _0;{\mathbb {R}}^N\setminus {\textbf{C}})\\&\quad \le P(E\cup \Omega _0;{\mathbb {R}}^N\setminus {\textbf{C}})+\Lambda _0|E\setminus \Omega _0|\\&\quad ={{{\mathcal {H}}}}^{N-1}((\partial ^*\Omega _0\cap {\textbf{C}}_\eta )\setminus {\textbf{C}})+{{{\mathcal {H}}}}^{N-1}((\partial ^*\Omega _0\cap E^{(0)})\setminus {\textbf{C}}_\eta )\\&\qquad +{{{\mathcal {H}}}}^{N-1}(\partial ^*E\cap \partial ^*\Omega _0\cap \{\nu _E=\nu _{\Omega _0}\}\setminus {\textbf{C}}_\eta )+{{{\mathcal {H}}}}^{N-1}(\partial ^*E\setminus {{{\overline{\Omega }}}_0})+\Lambda _0|E\setminus \Omega _0|\,. \end{aligned}$$

Simplifying the above inequality, we get

$$\begin{aligned} {{{\mathcal {H}}}}^{N-1}(\partial ^*\Omega _0\cap E^{(1)})+{{{\mathcal {H}}}}^{N-1}(\partial ^*E\cap \partial ^*\Omega _0\cap \{\nu _E=-\nu _{\Omega _0}\})\le {{{\mathcal {H}}}}^{N-1}(\partial ^*E\setminus {{{\overline{\Omega }}}_0})+\Lambda _0|E\setminus \Omega _0|\,. \end{aligned}$$

Combining this inequality with (4.10) we conclude that

$$\begin{aligned} P(\Omega _{\eta ,m};{\mathbb {R}}^N\setminus {\textbf{C}}_\eta )&\le {{{\mathcal {H}}}}^{N-1}(\partial ^*E\cap (\Omega _0\setminus {\textbf{C}}_\eta ))+{{{\mathcal {H}}}}^{N-1}(\partial ^*E\cap \partial ^*\Omega _0\cap \{\nu _E=\nu _{\Omega _0}\}\setminus {\textbf{C}}_\eta )\\&\quad +{{{\mathcal {H}}}}^{N-1}(\partial ^*E\setminus {{{\overline{\Omega }}}_0})+\Lambda '||E\cap \Omega _0|-|\Omega _{\eta ,m}||+\Lambda _0|E\setminus \Omega _0|\\&\le P(E;{\mathbb {R}}^N\setminus {\textbf{C}}_\eta )+\max \{\Lambda ',\Lambda _0\}|E\Delta \Omega _{\eta ,m}| \end{aligned}$$

so that the claim is proven.

Step 3 (Monotonicity and Lipschitz equicontinuity of the isoperimetric profiles). We claim that $I_\eta$ [see (4.4)] is strictly increasing in $[0,m_\eta ]$ for all $\eta \in [0,{{\bar{\eta }}}]$. Moreover, for any fixed $0<m'<m''<m_0$ and for $0<{{\tilde{\eta }}}\le {{\bar{\eta }}}$ be as in (4.6), we claim that for $\eta \in [0,{{\tilde{\eta }}}]$, $I_\eta$ is $\Lambda '$-Lipschitz in $[m',m'']$, where $\Lambda '$ is as in Step 2.

We postpone the proof to “Appendix B”.

Step 4 (A formula for $I'_\eta$). Fix $0<m'<m''<m_0$ and let $0<{{\tilde{\eta }}}\le {{\bar{\eta }}}$ be as in (4.6). For $m\in [m',m'']$ and $\eta \in [0,{{\tilde{\eta }}}]$ we set $\Sigma _{\eta ,m}:=\overline{\partial \Omega _{\eta ,m}\setminus {\textbf{C}}_\eta }$ and denote by $\Sigma ^*_{\eta ,m}$ the regular free part of $\Sigma _{\eta ,m}$, that is $\Sigma ^*_{\eta ,m}:=\partial ^*\Omega _{\eta ,m}\setminus (\partial \Omega _0\cup {\textbf{C}}_\eta )$. Observe that by (4.7) $\Sigma ^*_{\eta ,m}$ is nonempty. We recall that by a standard first variation argument $\Sigma ^*_{\eta ,m}$ is a constant mean curvature manifold. We denote by $H_{\Sigma ^*_{\eta ,m}}$ such a mean curvature.

We claim that at any point $m\in (m',m'')$ of differentiability for $I_\eta$, $\eta \in [0,{{\tilde{\eta }}}]$, we have

$$\begin{aligned} I'_\eta (m)=H_{\Sigma ^*_{\eta ,m}}\,. \end{aligned}$$

(4.11)

To this end we fix $x\in \Sigma ^*_{\eta ,m}$ and a ball $B_r(x)\subset \!\subset \Omega _0\setminus {\textbf{C}}_\eta$ such that $\Sigma ^*_{\eta ,m}\cap B_r(x)=\partial \Omega _{\eta ,m}\cap B_r(x)$. Let X be a smooth vector field compactly supported in $B_r(x)$ such that

$$\begin{aligned} \int _{\Sigma ^*_{\eta ,m}}X\cdot \nu _{\Omega _{\eta ,m}}\,d{{{\mathcal {H}}}}^{N-1}\not =0\,. \end{aligned}$$

Consider now the flow associated with X, that is the solution in ${\mathbb {R}}^N\times {\mathbb {R}}$ of

$$\begin{aligned} {\left\{ \begin{array}{ll} \frac{\partial \Phi }{\partial t}(x,t)=X(\Phi (x,t)) \\ \Phi (x,0)=x \end{array}\right. } \end{aligned}$$

and set $\Omega _{\eta ,m}(t):=\Phi (\Omega _{\eta ,m},t)$. Clearly, $P(\Omega _{\eta ,m}(t));{\mathbb {R}}^N\setminus {\textbf{C}}_\eta )\ge I_\eta (|\Omega _{\eta ,m}(t)|)$, with the equality at $t=0$. Therefore

$$\begin{aligned} \frac{d}{dt}\Big (P(\Omega _{\eta ,m}(t));{\mathbb {R}}^N\setminus {\textbf{C}}_\eta )\Big )_{\big |_{t=0}}=\frac{d}{dt}\Big (I_\eta (|\Omega _{\eta ,m}(t)|)\Big )_{\big |_{t=0}}\,. \end{aligned}$$

Note that

$$\begin{aligned}&\frac{d}{dt}\Big (P(\Omega _{\eta ,m}(t));{\mathbb {R}}^N\setminus {\textbf{C}}_\eta \Big )_{\big |_{t=0}} =H_{\Sigma ^*_{\eta ,m}}\int _{\Sigma ^*_{\eta ,m}}X\cdot \nu _{\Omega _{\eta ,m}}\,d{{{\mathcal {H}}}}^{N-1}\,, \\&\frac{d}{dt}\Big (I_\eta (|\Omega _{\eta ,m}(t)|)\Big )_{\big |_{t=0}}=I'_\eta (m)\frac{d}{dt}\big (|\Omega _{\eta ,m}(t)|\big )_{\big |_{t=0}}= I'_\eta (m)\int _{\Sigma ^*_{\eta ,m}}X\cdot \nu _{\Omega _{\eta ,m}}\,d{{{\mathcal {H}}}}^{N-1}\,, \end{aligned}$$

where we have used the well known formulas for the first variation of the perimeter and the volume, see for instance [15, Chap. 17]. Thus (4.11) follows.

Step 5 (A weak Young’s law). Fix $0<m'<m''<m_0$ and let $0<{{\tilde{\eta }}}\le {{\bar{\eta }}}$ be as in (4.6). We claim that if $\eta \in [0,{{\tilde{\eta }}}]$ and $m\in [m',m'']$, the following weak Young’s law holds:

$$\begin{aligned} \nu \cdot \nu _{{\textbf{C}}_\eta }(x)\le 0\quad \text {whenever}\ x\in \Sigma _{\eta ,m}\cap {\textbf{C}}_{\eta }\ \text {and}\ \nu \in N_x\Sigma _{\eta ,m}. \end{aligned}$$

(4.12)

Let $x\in \Sigma _{\eta ,m}\cap {\textbf{C}}_{\eta }$ and $\nu \in N_x\Sigma _{\eta ,m}$. Without loss of generality, by rotating the coordinate system if needed, we may assume that $x=0$, $\nu _{{\textbf{C}}_\eta }(0)=e_N$ and $\nu =(\nu _1,0,\dots ,0,\nu _N)$ with $\nu _1\le 0$. Note that (4.12) will be proven if we show that

$$\begin{aligned} \hbox {} \nu _N\ge \;0\; \hbox {implies that} \;\nu _N=0\,. \end{aligned}$$

(4.13)

Set $E_h=h\Omega _{\eta ,m}$, $h\in {\mathbb {N}}$ and ${\textbf{C}}_{\eta ,h}=h{\textbf{C}}_\eta$ and observe that, since $\nu _1\le 0$ and $\nu _N\ge 0$, $E_h\subset \{x_1\ge 0\}$. Note also that by (4.9) we have that

$$\begin{aligned} P(E_h;{\mathbb {R}}^N\setminus {\textbf{C}}_{\eta ,h})\le P( G;{\mathbb {R}}^N\setminus {\textbf{C}}_{\eta ,h})+\frac{1}{h}\Lambda |E_h\Delta G| \end{aligned}$$

(4.14)

for all sets $G\subset B_{hR}(0)\setminus {\textbf{C}}_{\eta ,h}$ such that $\partial ^*G\cap {\textbf{C}}_{\eta ,h}\subset \partial ^*E_h\cap {\textbf{C}}_{\eta ,h}$ up to a ${{{\mathcal {H}}}}^{N-1}$-negligible set. Using the density estimate proved in Proposition 5.2 and passing possibly to a not relabelled subsequence we may assume that $E_h$ converge in $L^1_{loc}({\mathbb {R}}^N)$ to some set $E\subset {\mathscr {H}}\cap \{x_1>0\}$ [see (4.1)] of locally finite perimeter and that $\mu _{E_h}{\mathop {\rightharpoonup }\limits ^{*}}\mu _E$ as Radon measures in ${\mathbb {R}}^N$, see (2.1) for the definition of $\mu _E$. Finally, given $r>0$, from the volume density estimate in Proposition 5.2 we get that for h large enough $|E_h\cap B_r(0)|\ge cr^N$ and thus, passing to the limit, we have $|E\cap B_r(0)|\ge cr^N$ for all $r>0$. This in turn implies that

$$\begin{aligned} 0\in \partial ^eE\subset \partial E\,. \end{aligned}$$

(4.15)

Since each $E_h$ is a $\frac{\Lambda }{h}$-minimizer, by Theorem 2.2 we have that E is a 0-minimizer that is

$$\begin{aligned} P(E;B_r(x_0))\le P(F; B_r(x_0))\text { for any}\ F, B_r(x_0)\ \text {s.t.}\ E\Delta F\subset \!\subset B_r(x_0)\subset \subset {\mathscr {H}}\,. \end{aligned}$$

(4.16)

We claim that also the minimality with respect to inner perturbations passes to the limit. More precisely we want to show that E satisfies the following minimality property: for any cube $Q_r(0)=(-r,r)^N$ and any open set with Lipschitz boundary $V\subset \subset Q_r(0)$³

$$\begin{aligned} {{{\mathcal {H}}}}^{N-1}(\partial E\cap \partial V\cap {\mathscr {H}})=0\quad \text {implies}\quad P(E;{\mathscr {H}}\cap Q_r(0))\le P(E\setminus V;{\mathscr {H}}\cap Q_r(0))\,.\nonumber \\ \end{aligned}$$

(4.17)

We postpone the proof of this claim to “Appendix B”.

We now denote by ${{\widehat{E}}}=E\cup R(E)$ where R denotes the reflection map $R(x',x_N)=(x',-x_N)$. From (4.17) one can easily check that given an open set with Lipschitz boundary $V\subset \!\subset Q_r(0)$ such that ${{{\mathcal {H}}}}^{N-1}(\partial {{\widehat{E}}}\cap \partial V)=0$ we have

$$\begin{aligned} P({{\widehat{E}}}; Q_r(0))\le P({{\widehat{E}}}\setminus V; Q_r(0))\,. \end{aligned}$$

We claim that $\partial {{\widehat{E}}}$ contains $\{x_1=0\}$. In turn this implies (4.13).

To see this assume first that $\partial {{\widehat{E}}}$ intersects $\{x_1=0\}\setminus \{x_N=0\}$ at some point $x_0$. Then, by Theorem 2.6-(iii) $\partial {{\widehat{E}}}$ is a smooth minimal surface in a neighborhood of $x_0$. In turn, by the Strong Maximum Principle Theorem 2.5 it coincides with the hyperplane $\{x_1=0\}$ in a neighborhood of $x_0$. The same argument shows that $\partial {{\widehat{E}}}\cap \{x_1=0\}$ is both relatively closed and open in $\{x_1=0\}$ and therefore the connected component of $\partial {{\widehat{E}}}$ containing $x_0$ coincides with $\{x_1=0\}$. Otherwise, $\partial {{\widehat{E}}}\cap \{x_1=0\}\subset \{x_1=0\}\cap \{x_N=0\}$ and thus in particular ${{{\mathcal {H}}}}^{N-1}(\partial {{\widehat{E}}}\cap \{x_1=0\})=0$. We may then apply Lemma 5.3 to conclude that $0\not \in \partial {{\widehat{E}}}$, thus getting a contradiction to (4.15).

Step 6 (Convergence of the isoperimetric profiles). We claim that

$$\begin{aligned} \lim _{\eta \rightarrow 0}I_\eta (m)=I_0(m)\,\,\text {for all}\ m\in [0,m_0)\quad \ \text {and}\quad \lim _{\eta \rightarrow 0}I_\eta (m_\eta )=I_0(m_0)\,. \end{aligned}$$

(4.18)

Let $\eta _n$ be a sequence converging to zero such that $I_{\eta _n}(m)\rightarrow \liminf _{\eta \rightarrow 0}I_\eta (m)$. Since the perimeters of $\Omega _{\eta _n,m}$ are equibounded, see (4.5), up to a subsequence we may assume that $\Omega _{\eta _n,m}$ converge in $L^1$ to a set of finite perimeter $E\subset \Omega _0$ with $|E|=m$. Thus, by lower semicontinuity,

$$\begin{aligned} I_0(m)\le P(E;{\mathbb {R}}^N\setminus {\textbf{C}})\le \liminf _{n}P(\Omega _{\eta _n,m},{\mathbb {R}}^N\setminus {\textbf{C}}_{\eta _n})=\liminf _{\eta \rightarrow 0}I_\eta (m)\,. \end{aligned}$$

(4.19)

Recall that $\Omega _{0,m}$ denotes a minimizer for the problem defining $I_0(m)$. Since

$$\begin{aligned} I_\eta (m-|\Omega _{0,m}\cap {\textbf{C}}_\eta |)\le P(\Omega _{0,m};{\mathbb {R}}^N\setminus {\textbf{C}}_\eta )\le I_0(m)\,, \end{aligned}$$

using the equilipschitz continuity of $I_\eta$ proved in Step 3, by letting $\eta$ tend to 0 in the previous inequality and recalling (4.19) we obtain the first equality in (4.18). The second one follows simply from the fact that $\Omega _{\eta ,m_\eta }=\Omega _0\setminus {\textbf{C}}_\eta$.

Note that the above argument shows in particular that if $m\in (0,m_0)$, $\eta _n\rightarrow 0$ and $\Omega _{\eta _n,m}$ is a sequence converging in $L^1$ to a set E, then E is a minimizer for the minimum problem in (4.4) with $\eta =0$. Recall that any such minimizer is denoted by $\Omega _{0,m}$.

Step 7 ($I_0=I_{{\mathscr {H}}}$ and any minimizer $\Omega _{0,m}$ is a connected open set). We set

$$\begin{aligned} I_{\mathscr {H}}(m)=N\Big (\frac{\omega _N}{2}\Big )^{\frac{1}{N}}m^{\frac{N-1}{N}}\,, \end{aligned}$$

(4.20)

that is the isoperimetric profile of half spaces. We claim that

$$\begin{aligned} I_0(m)=I_{{\mathscr {H}}}(m)\qquad \text {for all}\ m\in [0,m_0]\,. \end{aligned}$$

(4.21)

To this end we fix $0<m'<m''<m_0$ and let $0<{{\tilde{\eta }}}\le {{\bar{\eta }}}$ be as in (4.6). Recall that by Step 2 for all $\eta \in [0,{{\tilde{\eta }}}]$, $\Omega _{\eta ,m}$ is a restricted $(\Lambda ,r_0)$-minimizer for all $m\in [m',m'']$. We claim that for any such $\eta$ if $x_0\in \Sigma ^+_{\eta ,m}$ then $\Sigma _{\eta ,m}$ is of class $C^{1,1}$ in a neighborhood of $x_0$. Here $\Sigma ^+_{\eta ,m}$ is defined as in Definition 3.5 with $\Sigma$ and ${\textbf{C}}$ replaced by $\Sigma _{\eta ,m}$ and ${\textbf{C}}_\eta$. Indeed, observe first that if $x_0\in \Sigma ^+_{\eta ,m}$ then by Theorem 2.6$\Sigma _{\eta ,m}$ is of class $C^{1,\alpha }$ in a neighborhood of $x_0$. Moreover, if $x_0\in \Omega _0$ then, since $H_{\Sigma _{\eta ,m}}$ is constant in a neighborhood of $x_0$, we have that in fact $\Sigma _{\eta ,m}$ is analytic in such a neighborhood.

If instead $x_0\in \partial \Omega _0$, since $\Omega _0$ is a $(\Lambda ,r_0)$-minimizer and $\partial \Omega _0$ lies on one side with respect to $\Sigma _{\eta ,m}$ which is of class $C^{1,\alpha }$ in a neighborhood of $x_0$, again by Theorem 2.6 we infer that $\partial \Omega _0$ is of class $C^{1,\alpha }$, hence analytic in a neighborhood of $x_0$. The claim then follows from Proposition 5.5.

To prove (4.21) observe that the very same argument of (3.20) (with ${{\widetilde{\Sigma }}}_\eta$ replaced by $\Sigma _{\eta ,m}^+$ and $\theta _\eta$ replaced by $\pi /2$) yields that

$$\begin{aligned} \int _{\Sigma _{\eta ,m}^+\setminus {\textbf{C}}_\eta }H_{\Sigma _{\eta ,m}}^{N-1}\,d{{{\mathcal {H}}}}^{N-1}\ge (N-1)^{N-1}N\frac{\omega _N}{2}\,. \end{aligned}$$

(4.22)

Indeed this argument only requires that $\Sigma _{\eta ,m}$ is of class $C^{1,1}$ in a neighborhood of $\Sigma _{\eta ,m}^+$ and that (3.18) holds. Recall that the latter condition with $\theta _\eta =\pi /2$ is ensured by Step 5. Observe also that if $\Sigma _{\eta ,m}^+$ intersects $\partial \Omega _0$ in a set of positive ${{{\mathcal {H}}}}^{N-1}$ measure then for ${{{\mathcal {H}}}}^{N-1}$-a.e. x on such a set

$$\begin{aligned} H_{\Sigma _{\eta ,m}}(x)=H_{\partial \Omega _0}\le H_{\Sigma _{\eta ,m}^*} \end{aligned}$$

(4.23)

where $\Sigma _{\eta ,m}^*$ is the regular free part defined in Step 4 and the inequality follows from Proposition 5.5. Here, with a slight abuse of notation, we denote by $H_{\partial \Omega _0}$ the constant curvature of $\partial ^*\Omega _0\setminus {\textbf{C}}$. Therefore the previous inequality, (4.22) and (4.11) imply in particular that for a.e. $m\in (m',m'')$ and for all $\eta \in [0,{{\tilde{\eta }}}]$

$$\begin{aligned} \begin{aligned} I_\eta (m)(I'_\eta (m))^{N-1}&=P(\Omega _{\eta ,m};{\mathbb {R}}^N\setminus {\textbf{C}}_\eta ) H_{\Sigma _{\eta ,m}^*}^{N-1}\\&\ge (N-1)^{N-1}N\frac{\omega _N}{2}=I_{{\mathscr {H}}}(m)(I_{\mathscr {H}}'(m))^{N-1}\,, \end{aligned} \end{aligned}$$

(4.24)

where the last equality follows from (4.20). Recalling that $I_\eta$ is Lipschitz in $[m',m'']$ and thus absolutely continuous, raising the above inequality to the power $\frac{1}{N-1}$ and integrating in $[m,m'']$, for any $m\in (m',m'')$ we get

$$\begin{aligned} I_\eta (m'')^{\frac{N}{N-1}}-I_\eta (m)^{\frac{N}{N-1}} \ge I_{{\mathscr {H}}}(m'')^{\frac{N}{N-1}}-I_{\mathscr {H}}(m)^{\frac{N}{N-1}} \end{aligned}$$

for all $\eta \in [0,{{\tilde{\eta }}}]$. Passing to the limit as $\eta \rightarrow 0$ and using Step 6 we get

$$\begin{aligned} I_0(m'')^{\frac{N}{N-1}}-I_0(m)^{\frac{N}{N-1}} \ge I_{\mathscr {H}}(m'')^{\frac{N}{N-1}}-I_{{\mathscr {H}}}(m)^{\frac{N}{N-1}} \end{aligned}$$

(4.25)

for all $0<m<m''<m_0$. Observe now that $\lim _{m''\rightarrow m_0}I_0(m'')=I_0(m_0)$ (this follows by a simple semicontinuity argument and by the fact that $I_0$ is increasing). Thus, passing to the limit in (4.25) as $m''\rightarrow m_0$, recalling that by assumption $I_0(m_0)=I_{{\mathscr {H}}}(m_0)$ and that by Theorem 1.1$I_0(m)\ge I_{{\mathscr {H}}}(m)$, we get $I_0(m)= I_{{\mathscr {H}}}(m)$ for all $m\in (0,m_0)$, as claimed.

Finally, since $\Omega _{0,m}$ satisfies the equality case in (1.1), the same argument used for $\Omega _0$ shows that $\Omega _{0,m}$ is a connected open set.

Step 8 ($\partial \Omega _0\cap {\textbf{C}}$ is flat). In this step we prove that $\partial \Omega _0\cap {\textbf{C}}$ lies on a hyperplane $\Pi$.

To this aim we start by showing that

$$\begin{aligned} \left( I_{\eta }^{\frac{N}{N-1}}\right) '\rightarrow \left( I_{\mathscr {H}}^{\frac{N}{N-1}}\right) '\qquad \hbox { in}\ L^1_{loc}(0,m_0)\,. \end{aligned}$$

(4.26)

Indeed, given $0<m'<m''<m_0$ from (4.24) and the fact that $I_\eta \rightarrow I_{{\mathscr {H}}}$, we have that for a.e. $m\in (m',m'')$ and $\eta \in [0,{{\tilde{\eta }}}]$, $\left( I_{\eta }^{\frac{N}{N-1}}\right) '(m) \ge \left( I_{{\mathscr {H}}}^{\frac{N}{N-1}}\right) '(m)$ and

$$\begin{aligned} \int _{m'}^{m''}\left( I_{\eta }^{\frac{N}{N-1}}\right) '(t)\,dt \rightarrow \int _{m'}^{m''}\left( I_{\mathscr {H}}^{\frac{N}{N-1}}\right) '(t)\,dt\quad \text {as}\ \eta \rightarrow 0\,. \end{aligned}$$

Hence, (4.26) follows.

Returning to the proof of the flatness of $\partial \Omega _0\cap {\textbf{C}}$, observe that by a simple diagonal argument we can construct two sequences $m_n\rightarrow m_0$ and $\eta _n\rightarrow 0$ such that $\Omega _{\eta _n,m_n}$ is a $\Lambda _n$-minimizer for some $\Lambda _n>0$ (possibly going to $+\infty$) and

$$\begin{aligned} I_{\eta _n}(m_n)\rightarrow I_0(m_0),\quad \left( I_{\eta _n}^{\frac{N}{N-1}}\right) '(m_n)\rightarrow \left( I_{\mathscr {H}}^{\frac{N}{N-1}}\right) '(m_0)=N\left( N\frac{\omega _N}{2}\right) ^{\frac{1}{N-1}}\,. \end{aligned}$$

This is possible thanks to Step 2, Step 6 and (4.26). Given $\varepsilon >0$, let $\delta >0$ be as in Theorem 1.4 with $\theta _0=\pi /2$. Recall that $\delta$ depends only on $\varepsilon$ and on diam$(\Omega _0)$. Recall also that $\Sigma _{\eta _n,m_n}$ is of class $C^{1,1}$ in a neighborhood of $\Sigma ^+_{\eta _n,m_n}$, thanks to Step 7. Then from the above convergence, arguing as in the proof of (3.20) with ${{\widetilde{\Sigma }}}_\eta$ replaced by $\Sigma ^+_{\eta _n,m_n}$, and recalling that the weak Young’s inequality (4.12) holds for $\Sigma _{\eta _n,m_n}$, we have that for n large

$$\begin{aligned} \begin{aligned} \frac{N\omega _N}{2}\le {\mathcal {K}}^+(\Sigma _{\eta _n,m_n})&\le (N-1)^{1-N}\int _{\Sigma ^+_{\eta _n,m_n}}H^{N-1}_{\Sigma _{\eta _n,m_n}}\,d{{{\mathcal {H}}}}^{N-1}\\&\le (N-1)^{1-N}P(\Omega _{\eta _n,m_n};{\mathbb {R}}^N\setminus {\textbf{C}}_{\eta _n}) H_{\Sigma _{\eta _n,m_n}^*}^{N-1}\\&=\left[ \frac{1}{N} \left( I_{\eta _n}^{\frac{N}{N-1}}\right) '(m_n)\right] ^{N-1}<\frac{N\omega _N}{2}+\delta \,. \end{aligned} \end{aligned}$$

Note that in the third inequality above we have used (4.23). Thus from Theorem 1.4 we get that

$$\begin{aligned} \text {width}(\Sigma _{\eta _n,m_n}\cap {\textbf{C}}_{\eta _n}):=\varepsilon _n\rightarrow 0\,. \end{aligned}$$

More precisely, for n sufficiently large there exists $x_n\in \partial {\textbf{C}}_{\eta _n}$ such that

$$\begin{aligned} \Sigma _{\eta _n,m_n}\cap {\textbf{C}}_{\eta _n}\subset \{x: -\varepsilon _n\le (x-x_n)\cdot \nu _{{\textbf{C}}_{\eta _n}}(x_n)\le 0\}\,. \end{aligned}$$

(4.27)

Observe that, up to a not relabelled subsequence,

$$\begin{aligned} x_n\rightarrow {\overline{x}}\in {\textbf{C}},\qquad \nu _{{\textbf{C}}_{\eta _n}}(x_n)\rightarrow {{\overline{\nu }}}\in N_{{{\overline{x}}}}({\textbf{C}})\,. \end{aligned}$$

(4.28)

Denote by $\Pi$ the support hyperplane passing through ${{\overline{x}}}$ and orthogonal to ${{\overline{\nu }}}$ and by $\Pi ^{\pm }$ the half spaces $\{x:\, (x-{\overline{x}})\cdot {{\overline{\nu }}}\gtrless 0\}$. We claim that $\partial \Omega _0\cap {\textbf{C}}\subset \Pi$ up to a set of ${{{\mathcal {H}}}}^{N-1}$-measure zero.

To prove the claim we first show that, passing possibly to a further subsequence,

$$\begin{aligned} \partial \Omega _{\eta _n,m_n}\cap {\textbf{C}}_{\eta _n}\rightarrow K\quad \text { for some}\; K\subset \partial \Omega _0\cap {\textbf{C}}\ \text {s.t.}\ {{{\mathcal {H}}}}^{N-1}(\partial \Omega _0\cap {\textbf{C}}\setminus K)=0\,, \end{aligned}$$

(4.29)

where the convergence is meant in the Kuratowski sense. The existence of a subsequence converging to $K\subset \partial \Omega _0\cap {\textbf{C}}$ follows easily from the compactness properties of Kuratowski convergence, see Sect. 2. To show that ${{{\mathcal {H}}}}^{N-1}(\partial \Omega _0\cap {\textbf{C}}\setminus K)=0$ observe first that since ${\textbf{C}}_{\eta _n}\cap \overline{B_R(0)}$ is a sequence of convex sets converging to the convex set ${\textbf{C}}\cap \overline{B_R(0)}$ in the sense of Kuratowski then $P({\textbf{C}}_{\eta _n}\cap \overline{B_R(0)})\rightarrow P({\textbf{C}}\cap \overline{B_R(0)})$. This in turn yields that and in particular that

(4.30)

We claim that

$$\begin{aligned} \limsup _{n}{{{\mathcal {H}}}}^{N-1}(\partial \Omega _{\eta _n,m_n}\cap {\textbf{C}}_{\eta _n})\le {{{\mathcal {H}}}}^{N-1}(K)\,. \end{aligned}$$

(4.31)

To this aim set $K_\sigma =K+\overline{B_\sigma (0)}\subset B_R(0)$ for $\sigma >0$ sufficiently small. Then for n sufficiently large $\partial \Omega _{\eta _n,m_n}\cap {\textbf{C}}_{\eta _n}\subset K_\sigma \cap \partial {\textbf{C}}_{\eta _n}$, hence

$$\begin{aligned} {{{\mathcal {H}}}}^{N-1}(\partial \Omega _{\eta _n,m_n}\cap {\textbf{C}}_{\eta _n})\le {{{\mathcal {H}}}}^{N-1}(K_\sigma \cap \partial {\textbf{C}}_{\eta _n})\,. \end{aligned}$$

From this inequality we then have

$$\begin{aligned} \limsup _{n}{{{\mathcal {H}}}}^{N-1}(\partial \Omega _{\eta _n,m_n}\cap {\textbf{C}}_{\eta _n})\le \limsup _{n}{{{\mathcal {H}}}}^{N-1}(K_\sigma \cap \partial {\textbf{C}}_{\eta _n}) \le {{{\mathcal {H}}}}^{N-1}(K_\sigma \cap \partial {\textbf{C}})\,, \end{aligned}$$

where in the last inequality we have used (4.30). Then (4.31) follows letting $\sigma \rightarrow 0$. On the other hand, $\Omega _{\eta _n,m_n}\rightarrow \Omega _0$ in $L^1$ and by the lower semicontinuity of perimeter and (4.31)

$$\begin{aligned} P(\Omega _0)=I_0(m_0)+{{{\mathcal {H}}}}^{N-1}(\partial ^*\Omega _0\cap {\textbf{C}})&\le \liminf _{n}P(\Omega _{\eta _n,m_n})\\&=\liminf _{n}\left[ I_{\eta _n}(m_n)+{{{\mathcal {H}}}}^{N-1}(\partial \Omega _{\eta _n,m_n}\cap {\textbf{C}}_{\eta _n})\right] \\&\le I_0(m_0)+{{{\mathcal {H}}}}^{N-1}(K)\,. \end{aligned}$$

Recall that by the volume estimate Proposition 5.2-(ii) $\partial ^*\Omega _0\cap {\textbf{C}}$ coincides ${{{\mathcal {H}}}}^{N-1}$-a.e. with $\partial \Omega _0\cap {\textbf{C}}$. Thus the above inequality implies that K coincides ${{{\mathcal {H}}}}^{N-1}$-a.e. with $\partial \Omega _0\cap {\textbf{C}}$. Hence, (4.29) follows.

We finally claim that for n large

$$\begin{aligned} \partial \Omega _{\eta _n,m_n}\cap {\textbf{C}}_{\eta _n}\subset \{x: -{\varepsilon _n}\le (x-x_n)\cdot \nu _{{\textbf{C}}_{\eta _n}}(x_n)\le 0\}\,. \end{aligned}$$

(4.32)

To prove this we argue by contradiction assuming that for infinitely many n there exists $y_n\in \partial \Omega _{\eta _n,m_n}\cap {\textbf{C}}_{\eta _n}$ such that $(y_n-x_n)\cdot \nu _{{\textbf{C}}_{\eta _n}}(x_n)<-\varepsilon _n$. Observe that, if this is the case for all such n,

$$\begin{aligned} F_n:=\partial {\textbf{C}}_{\eta _n}\cap \{x: (x-x_n)\cdot \nu _{{\textbf{C}}_{\eta _n}}(x_n)<-\varepsilon _n\}\subset \partial \Omega _{\eta _n,m_n}\cap {\textbf{C}}_{\eta _n}\,. \end{aligned}$$

(4.33)

Indeed, otherwise there exists $z_n\in F_n\setminus \partial \Omega _{\eta _n,m_n}$ and in turn a continuous path $\gamma \subset F_n$ connecting $z_n$ to $y_n$ (recall that ${\textbf{C}}_{\eta _n}$ is bounded). But then this arc must contain a point in $\partial _{{\textbf{C}}_{\eta _n}}(\partial \Omega _{\eta _n,m_n}\cap {\textbf{C}}_{\eta _n})\subset \Sigma _{\eta _n,m_n}\cap {\textbf{C}}_{\eta _n}$, which contradicts (4.27). Therefore, from (4.33), (4.28) and (4.29) we have that

$$\begin{aligned} \partial {\textbf{C}}\cap \{x:\, (x-\overline{x})\cdot {{\overline{\nu }}}<0\}=\partial {\textbf{C}}\cap \Pi ^-\subset \partial \Omega _0\cap {\textbf{C}}\,. \end{aligned}$$

Then, let ${{\bar{t}}}:=\min \{t\le 0:\, \Pi +t{{\overline{\nu }}}\cap {\textbf{C}}\not =\emptyset \}$ and set for $t\in (\bar{t},0)$, ${\textbf{C}}^t:={\textbf{C}}\cap (\Pi ^++t{{\overline{\nu }}})$. Note that, from the above inclusion, $P(\Omega _0\cup ({\textbf{C}}\setminus {\textbf{C}}^t);{\mathbb {R}}^N\setminus {\textbf{C}}^t)=P(\Omega _0;{\mathbb {R}}^N\setminus {\textbf{C}})=I_{\mathscr {H}}(m_0)$, but this contradicts (1.1) since $|\Omega _0\cup ({\textbf{C}}\setminus {\textbf{C}}^t)|>m_0$. Hence (4.32) holds for n large enough.

Finally, from (4.32) and (4.29) we have that $\partial \Omega _0\cap {\textbf{C}}\subset \Pi$ up to a set of vanishing ${{{\mathcal {H}}}}^{N-1}$ measure.

Step 9 (Conclusion). In this final step we show that $\Omega _0$ is a half ball.

To this aim we fix $m\in (0,m_0)$ and a sequence $\eta _n\rightarrow 0$ such that

$$\begin{aligned} I_{\eta _n}(m)\rightarrow I_0(m)=I_{\mathscr {H}}(m),\quad \left( I_{\eta _n}^{\frac{N}{N-1}}\right) '(m)\rightarrow \left( I_{\mathscr {H}}^{\frac{N}{N-1}}\right) '(m)=N\Big (N\frac{\omega _N}{2}\Big )^{\frac{1}{N-1}}\,. \end{aligned}$$

(4.34)

Owing to Steps 6-8 we can find such a sequence for a.e. $m\in (0,m_0)$. Thanks to Step 2, we may assume that there exists $\Lambda >0$ such that $\Omega _{\eta _n,m}$ is a $\Lambda$-minimizer for all n. By Theorem 2.6-(ii) this implies in particular that $|H_{\Sigma _{\eta _n,m}}|\le \Lambda$ ${{{\mathcal {H}}}}^{N-1}$-a.e. on $\partial ^*\Omega _{\eta _n,m}\setminus {\textbf{C}}_{\eta _n}$. Arguing as in the previous step, see also the proof of (3.20), we have then

$$\begin{aligned} \begin{aligned} \frac{N\omega _N}{2}\le {\mathcal {K}}^+(\Sigma _{\eta _n,m})&=\int _{\Sigma ^+_{\eta _n,m}}K_{\Sigma _{\eta _n,m}}\,d{{{\mathcal {H}}}}^{N-1} \le (N-1)^{1-N}\int _{\Sigma ^+_{\eta _n,m}}H^{N-1}_{\Sigma _{\eta _n,m}}\,d{{{\mathcal {H}}}}^{N-1}\\&\le (N-1)^{1-N}P(\Omega _{\eta _n,m};{\mathbb {R}}^N\setminus {\textbf{C}}_{\eta _n}) H_{\Sigma _{\eta _n,m}^*}^{N-1}\\&=\Big [\frac{1}{N} \Big (I_{\eta _n}^{\frac{N}{N-1}}\Big )'(m)\Big ]^{N-1}\rightarrow \frac{N\omega _N}{2}\,, \end{aligned} \end{aligned}$$

(4.35)

where we recall $K_{\Sigma _{\eta _n,m}}$ is the Gaussian curvature of $\Sigma _{\eta _n,m}$ and we used (4.23). We start by observing that, since $H_{\Sigma _{\eta _n,m}}(x)\le H_{\Sigma _{\eta _n,m}^*}$ for ${{{\mathcal {H}}}}^{N-1}$-a.e. $x\in \Sigma ^+_{\eta _n,m}$, from the third inequality in (4.35) we have in particular that

$$\begin{aligned} \lim _{n}{{{\mathcal {H}}}}^{N-1}(\Sigma ^+_{\eta _n,m})=\lim _{n}P(\Omega _{\eta _n,m};{\mathbb {R}}^N\setminus {\textbf{C}}_{\eta _n})={{{\mathcal {H}}}}^{N-1}(\partial ^*\Omega _{0,m}\setminus {\textbf{C}})\,. \end{aligned}$$

(4.36)

Note that $H_{\Sigma _{\eta _n,m}}$ may only take the constant values $H_{\partial \Omega _0}$ or $H_{\Sigma _{\eta _n,m}^*}$. Then, again from (4.35) and from (4.23), it follows that

$$\begin{aligned} \text {either}\ \quad H_{\Sigma _{\eta _n,m}^*}\rightarrow H_{\partial \Omega _0}\quad \text {or}\quad {{{\mathcal {H}}}}^{N-1}\big ((\partial \Omega _{\eta _n,m}\cap \partial \Omega _{0})\setminus {\textbf{C}}_{\eta _n}\big )\rightarrow 0\,. \end{aligned}$$

(4.37)

Fix now $x\in \partial ^*\Omega _{0,m}\setminus {\textbf{C}}$. Since $\Omega _{\eta _n,m}\rightarrow \Omega _{0,m}$ in $L^1$ and $P(\Omega _{\eta _n,m};{\mathbb {R}}^N\setminus {\textbf{C}}_{\eta _n})\rightarrow P(\Omega _{0,m};{\mathbb {R}}^N\setminus {\textbf{C}})$ thanks to the first condition in (4.34), we have that in ${\mathbb {R}}^N\setminus {\textbf{C}}$. In turn, by Theorem 2.7 it follows that, up to rotations and translations, there exist a $(N-1)$-dimensional ball $B'\subset {\mathbb {R}}^{N-1}$, functions $\varphi _n,\varphi \in W^{2,p}(B')$, and $r>0$ such that $x\in B'\times (-r,r)$ and

$$\begin{aligned}&\partial \Omega _{\eta _n,m}\cap (B'\times (-r,r))=\{(x',\varphi _n(x')):\,x'\in B'\},\\&\partial \Omega _{0,m}\cap (B'\times (-r,r))=\{(x',\varphi (x')):\,x'\in B'\},\\&\varphi _n\rightharpoonup \varphi \quad \hbox {in}\,\, W^{2,p}(B')\,\, \hbox {for all} p\ge 1,\\&H_{\Sigma _{\eta _n,m}}(x',\varphi _n(x'))\rightharpoonup H_{\Sigma _{0,m}}(x',\varphi (x'))\quad \text {in}\ L^p(B')\ \text {for all}\ p\ge 1\,, \end{aligned}$$

Recalling (4.37) the fourth condition above implies that

$$\begin{aligned} H_{\Sigma _{\eta _n,m}}(x',\varphi _n(x'))\rightarrow H_{\Sigma _{0,m}}(x',\varphi (x'))\equiv H_{\Sigma ^*_{0,m}} \end{aligned}$$

strongly in $L^p(B')$ for all $p\ge 1$. In turn, see for instance [1, Lemma 7.2], this implies

$$\begin{aligned} \varphi _n\rightarrow \varphi \quad \text {strongly in}\ W^{2,p}(B')\ \text {for all}\ p\ge 1\,. \end{aligned}$$

(4.38)

Note also that, since from (4.36) ${{{\mathcal {H}}}}^{N-1}(\Sigma _{\eta _n,m}\setminus \Sigma ^+_{\eta _n,m})\rightarrow 0$, we have that for every $y\in (B'\times (-r,r))\cap \Sigma _{0,m}$ there exists a sequence $y_n\in (B'\times (-r,r))\cap \Sigma ^+_{\eta _n,m}$ such that $y_n\rightarrow y$. Therefore, using the $L^1$ convergence of $\Omega _{\eta _n,m}$ to $\Omega _{0,m}$ we conclude that the tangent hyperplane to $\partial \Omega _{0,m}$ at y is also a support hyperplane. Thus we have shown that all principal curvatures at any point in $(B'\times (-r,r))\cap \Sigma _{0,m}$ are nonnegative. Thus, from the second inequality in (4.35), recalling (4.36) and (4.38) we may conclude that

$$\begin{aligned} K_{\Sigma _{0,m}}=(N-1)^{1-N}H^{N-1}_{\Sigma _{0,m}}=(N-1)^{1-N}H^{N-1}_{\Sigma ^*_{0,m}}\quad \text {on}\ (B'\times (-r,r))\cap \Sigma _{0,m}\,. \end{aligned}$$

The equality above implies that $\Sigma _{0,m}\cap (B'\times (-r,r)))$ is umbilical. Hence $\partial ^*\Omega _{0,m}\setminus {\textbf{C}}$ is umbilical, thus each connected component of $\partial ^*\Omega _{0,m}\setminus {\textbf{C}}$ lies on a sphere of radius $R_m=(N-1)/H_{\Sigma ^*_{0,m}}$. Consider the unique unbounded connected component of $U:={\mathbb {R}}^N\setminus \overline{\Omega _{0,m}}$. Then, recalling Step 8 and that $\Omega _{0,m}$ is connected (see Step 7), $\partial U\setminus {\textbf{C}}$ is contained in a sphere of radius $R_m$ intersecting ${\textbf{C}}$ on $\Pi$. Thus $\partial U\setminus {\textbf{C}}$ is a spherical cap and $\Omega _{0,m}$ is contained in the region enclosed by $\partial U\setminus {\textbf{C}}$ and $\Pi$. In particular $\Omega _{0,m}$ is contained in the half space $\Pi ^+$ determined by $\Pi$ not containing ${\textbf{C}}$. Since $P(\Omega _{0,m};\Pi ^+)=P(\Omega _{0,m};{\mathbb {R}}^N\setminus {\textbf{C}})=N\big (\frac{\omega _N}{2}\big )^{\frac{1}{N}}m^{\frac{N-1}{N}}$, by Theorem 19.21 in [15] for a.e. m we conclude that for such m $\Omega _{0,m}$ is a half ball. Since the argument above can be carried out for a.e. $m\in (0,m_0)$, in particular there exists a sequence $m_n\rightarrow m_0$ such that $\Omega _{0,m_n}$ is a half ball. Hence $\Omega _0$ is a half ball. $\square$

Appendices

Appendix A: some auxiliary results

In this section we collect some auxiliary results needed in the proof of Theorem 1.2.

1.1 Density estimates

Density estimates for $(\Lambda ,r_0)$-minimizers are well known. However for the sake of completeness we give the proof of the proposition below showing that such density estimates are independent of the convex obstacle.

Lemma 5.1

Let ${\textbf{C}}$ be a closed convex set with nonempty interior and $F\subset {\mathbb {R}}^N\setminus {\textbf{C}}$ a bounded set of finite perimeter. Then

$$\begin{aligned} P(F;\partial {\textbf{C}})\le P(F;{\mathbb {R}}^N\setminus {\textbf{C}}) \end{aligned}$$

Proof

Assume that ${\textbf{C}}$ is bounded and let $H_i$ be a sequence of closed half spaces such that ${\textbf{C}}=\displaystyle \bigcap _{i=1}^{\infty }H_i$. Since ${\textbf{C}}=({\textbf{C}}\cup F)\cap \displaystyle \bigcap _{i=1}^{\infty }H_i$ we have

$$\begin{aligned} P({\textbf{C}})\le \liminf _nP\left( ({\textbf{C}}\cup F)\cap \bigcap _{i=1}^{n}H_i\right) \le P({\textbf{C}}\cup F)\,, \end{aligned}$$

where the last inequality follows by applying repeatedly the inequality $P(G\cap H_i)\le P(G)$ where G is a set of finite perimeter. Since $P({\textbf{C}}\cup F)={{{\mathcal {H}}}}^{N-1}(\partial {\textbf{C}}\cap F^{(0)})+{{{\mathcal {H}}}}^{N-1}(\partial ^*F\setminus {\textbf{C}})$, the conclusion follows observing that $P({\textbf{C}})={{{\mathcal {H}}}}^{N-1}(\partial {\textbf{C}}\cap F^{(0)})+{{{\mathcal {H}}}}^{N-1}(\partial {\textbf{C}}\cap \partial ^*F)$.

If ${\textbf{C}}$ is not bounded, since $F\subset \!\subset B$ for a suitable closed ball B, the conclusion follows by the same argument as before, replacing ${\textbf{C}}$ with ${\textbf{C}}\cap B$. $\square$

Proposition 5.2

Let ${\textbf{C}}$ be a closed convex set with nonempty interior and let $E\subset {\mathbb {R}}^N\setminus {\textbf{C}}$ be a restricted $(\Lambda ,r_0)$-minimizer of the relative perimeter $P(\cdot ;{\mathbb {R}}^N\setminus {\textbf{C}})$ according to Definition 4.1. Then there are positive constants $c_1=c_1(N)$ and $C_1=C_1(N)$ independent of ${\textbf{C}}$ such that for all $r\in (0,\min \{r_0,N/(4\Lambda )\})$ we have:

• (i) for all $x\in {\mathbb {R}}^N\setminus \text {int}({\textbf{C}})$

  $$\begin{aligned} P(E; B_r(x))\le C_1 r^{N-1}\;, \end{aligned}$$

• (ii) for all $x\in \partial ^*E$

  $$\begin{aligned} |E\cap B_r(x)|\ge c_1 r^N\,. \end{aligned}$$

Moreover E is equivalent to an open set $\Omega$ such that $\partial \Omega =\partial ^e\Omega$, hence ${{{\mathcal {H}}}}^{N-1}(\partial \Omega \setminus \partial ^*\Omega )=0$, and (ii) holds at any point $x\in \partial \Omega$.

Proof

Given $x\in {\mathbb {R}}^N\setminus \text {int}({\textbf{C}})$ and $r<\min \{r_0,N/(4\Lambda )\}$, we set $m(r):=|E\cap B_r(x)|$. Recall that for a.e. such r we have $m'(r)={{{\mathcal {H}}}}^{N-1}(E^{(1)}\cap \partial B_r(x))$ and ${{{\mathcal {H}}}}^{N-1}(\partial ^*E\cap \partial B_r(x))=0$. For any such r we set $F:=E\setminus B_r(x)$. Then, using Definition 4.1, we have

$$\begin{aligned} P(E; B_r(x)\setminus {\textbf{C}})\le {{{\mathcal {H}}}}^{N-1}(\partial B_r(x)\cap E^{(1)})+\Lambda |E\cap B_r(x)| \le C_1 r^{N-1} \end{aligned}$$

(5.1)

for a suitable constant $C_1$. In turn

$$\begin{aligned} P(E; B_r(x))\le P(E; B_r(x)\setminus {\textbf{C}})+{{{\mathcal {H}}}}^{N-1}(\partial ({\textbf{C}}\cap B_r(x)))\le C_1 r^{N-1}+{{{\mathcal {H}}}}^{N-1}(\partial B_r(x))\,, \end{aligned}$$

where in the last inequality we estimated the perimeter of ${\textbf{C}}\cap B_r(x)$ with the perimeter of the larger convex set $B_r(x)$. Thus (i) follows by taking $C_1$ larger.

Observe now that by Lemma 5.1

$$\begin{aligned} P(E\cap B_r(x); \partial {\textbf{C}})\le P(E\cap B_r(x); {\mathbb {R}}^N\setminus {\textbf{C}})\,. \end{aligned}$$

Thus, using also (5.1), we have

$$\begin{aligned} P(E\cap B_r(x))&=P(E\cap B_r(x);{\mathbb {R}}^N\setminus {\textbf{C}})+P(E\cap B_r(x); \partial {\textbf{C}})\\&\le 2 P(E\cap B_r(x); {\mathbb {R}}^N\setminus {\textbf{C}})= 2 P(E; B_r(x)\setminus {\textbf{C}})+2m'(r)\\&\le 4 m'(r)+2\Lambda m(r)\,. \end{aligned}$$

In turn, using the isoperimetric inequality and the fact that $2\Lambda r<N/2$ we get

$$\begin{aligned} N\omega _N^{\frac{1}{N}}m(r)^{\frac{N-1}{N}}&\le P(E\cap B_r(x))\le 4m'(r)+2\Lambda m(r)\\&\le 4m'(r)+2\Lambda r\omega _N^{\frac{1}{N}}m(r)^{\frac{N-1}{N}}\le 4m'(r)+\frac{N}{2}\omega _N^{\frac{1}{N}}m(r)^{\frac{N-1}{N}}\,. \end{aligned}$$

Then from the previous inequality we get

$$\begin{aligned} \frac{N}{2}\omega _N^{\frac{1}{N}}m(r)^{\frac{N-1}{N}}\le 4m'(r)\,. \end{aligned}$$

Observe now that if in addition $x\in \partial ^* E$, then $m(r)>0$ for all r as above. Thus, we may divide the previous inequality by $m(r)^{\frac{N-1}{N}}$, and integrate the resulting differential inequality thus getting

$$\begin{aligned} |E\cap B_r(x)|\ge c_1r^N\,, \end{aligned}$$

for a suitable positive constant $c_1$ depending only on N.

We show that $\overline{\partial ^*E}\subset \partial ^eE$. To this aim note that (ii) holds for every $x\in \overline{\partial ^*E}$. Thus, if $x\in {\mathbb {R}}^N\setminus {\textbf{C}}$, since both E and ${\mathbb {R}}^N\setminus E$ are $\Lambda$-minimizers in a neighborhood of x we have that $|B_r(x)\setminus E|\ge c_1r^N$ for r small. Thus $x\not \in (E^{(0)}\cup E^{(1)})$, that is $x\in \partial ^eE$. If $x\in \partial {\textbf{C}}\cap \overline{\partial ^*E}$ then there exists a constant $c_2>0$, depending on x such that for r small $|{\textbf{C}}\cap B_r(x)|\ge c_2r^N$. This estimate, together with (ii) again implies that $x\in \partial ^eE$. Hence ${{{\mathcal {H}}}}^{N-1}(\overline{\partial ^*E}\setminus \partial ^*E)\le {{{\mathcal {H}}}}^{N-1}(\partial ^eE\setminus \partial ^*E)=0$, where the last equality follows from Theorem 16.2 in [15].

Set now $\Omega =E^{(1)}\setminus \partial E^{(1)}$. Recalling that $\partial E^{(1)}=\overline{\partial ^*E}$, see (2.2), we have that $\Omega$ is an open set equivalent to E such $\partial \Omega =\partial E^{(1)}$. Hence the conclusion follows. $\square$

1.2 A maximum principle

Next result is essentially the strong maximum principle proved in [9, Lemma 2.13]. However, we have to apply it in a slightly different situation and therefore we indicate the changes needed in the proof.

Lemma 5.3

Let $E\subset \{x_1>0\}$ be a set of locally finite perimeter such that

$$\begin{aligned} {{{\mathcal {H}}}}^{N-1}((\partial E\setminus \partial ^*E)\setminus \{{x_1}=0\})=0 \end{aligned}$$

(5.2)

satisfying the following minimality property: for every $r>0$ and every open set with Lipschitz boundary $V\subset \!\subset Q_r(0)$ such that ${{{\mathcal {H}}}}^{N-1}(\partial E\cap \partial V)=0$ we have

$$\begin{aligned} P(E; Q_r(0))\le P(E\setminus V; Q_r(0))\,. \end{aligned}$$

(5.3)

Assume also that ${{{\mathcal {H}}}}^{N-1}(\partial E\cap \{x_1=0\})=0$. Then $0\not \in \partial E$.⁴

The proof of lemma above is in turn based on the following variant of [9, Lemma 2.12]. To this aim, given $r>0$ we set $C_r:=(0,r)\times D_r$, where $D_r:=\{x'\in {\mathbb {R}}^{N-1}:\, |x'|<r\}$.

Lemma 5.4

Let E be as in Lemma 5.3, let ${{\bar{r}}}>0$ and let $u_0\in C^2(D_{{{\bar{r}}}})\cap \text {Lip}(D_{{{\bar{r}}}})$ with $0<u_0<{{{\bar{r}}}}$ on ${{\overline{D}}}_{{{\bar{r}}}}$. Assume also that

$$\begin{aligned}{} & {} E^{(1)}\cap [(0,{{{\bar{r}}}})\times \partial D_{\bar{r}}]\subset \{(x_1,x')\in (0,{{{\bar{r}}}})\times \partial D_{{{\bar{r}}}}:\,x_1\ge u_0(x')\}\,,\\{} & {} \textrm{div}\left( \frac{\nabla u_0}{\sqrt{1+|\nabla u_0|^2}}\right) =0\qquad \hbox { in}\ D_{{{\bar{r}}}} \end{aligned}$$

and

$$\begin{aligned} {{{\mathcal {H}}}}^{N-1}\big (\partial E\cap \partial \{(x_1,x')\in C_{\bar{r}}:\,x_1< u_0(x')\}\big )=0\,. \end{aligned}$$

(5.4)

Then,

$$\begin{aligned} E^{(1)}\cap C_{\bar{r}}\subset \{(x_1,x')\in C_{{{\bar{r}}}}:\,x_1\ge u_0(x')\}\,. \end{aligned}$$

Proof

The proof goes exactly as the one of Lemma 2.12 in [9] as it is based on the comparison with he competitor $F=E\setminus V$, where $V=\{(x_1,x')\in C_{\bar{r}}:\,x_1< u_0(x')\}$. Observe that assumption (5.4) guarantees that such a competitor satisfies ${{{\mathcal {H}}}}^{N-1}(\partial E\cap \partial V)=0$, which is required in order (5.3) to hold. $\square$

Proof of Lemma 5.3

For reader’s convenience we reproduce the proof of Lemma 2.13 in [9] with the small changes needed in our case.

We choose ${{{\bar{r}}}}>0$ so that ${{{\mathcal {H}}}}^{N-1}(\partial E\cap \partial C_{\bar{r}})=0$ and ${{{\mathcal {H}}}}^{N-2}(\partial E\cap \partial D_{{{\bar{r}}}})=0$, where with a slight abuse of notation $\partial D_{{{\bar{r}}}}$ stands for the relative boundary of $D_{{{\bar{r}}}}$ in $\{x_1=0\}$. Note that a.e. $r>0$ satisfies these conditions thanks to (5.2) and to the assumption ${{{\mathcal {H}}}}^{N-1}(\partial E\cap \{x_1=0\})=0$. Define now a function $w_E : \overline{D_{{{\bar{r}}}}}\rightarrow [0,\infty ]$ by setting

$$\begin{aligned} w_E(x')=\inf \{x_1\in {\mathbb {R}}:\, (x_1,x')\in \overline{C_{{{\bar{r}}}}}\cap \partial E\}. \end{aligned}$$

Observe that $w_E$ is nonnegative and lower semicontinuous on $\overline{D_{{{\bar{r}}}}}$, with the property that

$$\begin{aligned} E^{(1)}\cap C_{{{\bar{r}}}}\subset \{(x_1,x'):\,x'\in D_{\bar{r}},\,x_1\ge w_E(x')\}\,. \end{aligned}$$

Recalling that ${{{\mathcal {H}}}}^{N-2}(\partial E\cap \partial D_{{{\bar{r}}}})=0$, we have that $w_E>0$ ${{{\mathcal {H}}}}^{N-2}$-a.e. on $\partial D_{{{\bar{r}}}}$. Therefore there exists a family $(\varphi _t)_{t\in (0,1)}\subset C^\infty (\partial D_{{{\bar{r}}}})$ such that

$$\begin{aligned} 0\le \varphi _{t_1}\le \varphi _{t_2}\le \min \Big \{w_E,\frac{\bar{r}}{2}\Big \}\quad \varphi _{t_1}\not \equiv \varphi _{t_2}\quad \text {for all}\ 0<t_1<t_2<1\,. \end{aligned}$$

By Lemma 2.11 in [9] for every $t\in (0,1)$ there exists $u_t\in C^{\infty }(D_{{{\bar{r}}}})\cap \textrm{Lip}(D_{{\bar{r}}})$ such that

$$\begin{aligned} {\left\{ \begin{array}{ll} \textrm{div}\bigg (\displaystyle \frac{\nabla u_t}{\sqrt{1+|\nabla u_t|^2}}\bigg )=0&{} \text {in}\ D_{{{\bar{r}}}}, \\ u_t=\varphi _t &{} \text {on}\ \partial D_{{{\bar{r}}}}\,. \end{array}\right. } \end{aligned}$$

Note that by the Strong Maximum Principle Theorem 2.5 we have that $0<u_{t_1}<u_{t_2}<{{{\bar{r}}}}/2$ in $D_{{{\bar{r}}}}$ for every $0<t_1<t_2<1$. Therefore the graphs $\Gamma _t$ of $u_t$ are mutually disjoint in $C_{{{\bar{r}}}}$ and so ${{{\mathcal {H}}}}^{N-1}(\Gamma _t\cap \partial E)=0$ for all but countably many $t\in (0,1)$. In particular there exists ${{\bar{t}}}$ such that (5.4) holds with $u_0$ replaced by $u_{{{\bar{t}}}}$. Therefore we may apply Lemma 5.4 to conclude that $E^{(1)}\cap C_{{{\bar{r}}}}\subset \{(x_1,x')\in C_{{{\bar{r}}}}:\,x_1\ge u_{{{\bar{t}}}}(x')\}$ so that in particular $w_E(0)\ge u_{{{\bar{t}}}}(0)>0$, hence $0\not \in \partial E$. $\square$

1.3 A regularity result

The following proposition is a slight variant of a result contained in [21].

Proposition 5.5

Let $\Omega \subset {\mathbb {R}}^N$ be a bounded open set and let $x_0\in \partial \Omega$ be such that $\partial \Omega$ is of class $C^2$ in a neighborhood U of $x_0$. Let $E\subset \Omega$ satisfy

$$\begin{aligned} P(E)\le P(F)\quad \text {for all}\ F\subset \Omega , |F|=|E|, \text {s.t.}\ E\Delta F\subset \!\subset U\,. \end{aligned}$$

(5.5)

If there exists a support hyperplane $\Pi$ to E at $x_0$ such that $\partial E\cap \Pi =\{x_0\}$, then $\partial E$ is of class $C^{1,1}$ in a neighborhood V of $x_0$. Moreover if $\partial ^*E\cap \Omega \cap V\not =\emptyset$, then for ${{{\mathcal {H}}}}^{N-1}$-a.e. $x\in \partial E\cap \partial \Omega \cap V$

$$\begin{aligned} H_{\partial \Omega }(x)\le H\,, \end{aligned}$$

(5.6)

where H denotes the constant curvature of $\partial ^*E\cap \Omega \cap V$.

Proof

Observe that by a standard argument (5.5), together with the assumption that $\partial \Omega$ of class $C^2$, implies that E is a $(\Lambda ,r_0)$-minimizer in a possibly smaller naeighborhood $U'$ of $x_0$. Hence, since there exists a support hyperplane to $\partial E$ at $x_0$, by Theorem 2.6$\partial E$ is of class $C^{1,\alpha }$ in a neighborhood of $x_0$. Moreover, up to a change of coordinate system, we may assume that the support hyperplane to E at $x_0$ is the horizontal hyperplane $\{x_N=0\}$ and $E\subset \{x_N>0\}$. Since $\{x_N=0\}\cap \partial E=\{x_0\}$, there exists $\varepsilon >0$ sufficiently small such that $E\cap \{x_N=\varepsilon \}$ is an $(N-1)$-dimensional relatively open set, denoted by $\omega$, and there exist $\beta \in C^2(\omega )$, $u\in C^{1,\alpha }(\omega )$ whose graphs coincide with $\partial \Omega \cap (\omega \times (-r,r))$ and $\partial E\cap (\omega \times (-r,r))$ respectively, for some $r>0$, with $u=0$ on $\partial \omega$ and $\beta \le u\le 0$. The $C^{1,1}$ regularity of $\partial E$ then follows arguing exactly as in the proof at p. 658 of [21]. Finally, inequality (5.6) is also a byproduct of the same proof, see (3.5) in [21]. $\square$

Appendix B: some steps of the proof of Theorem 1.2

1.1 Proof of the claim of Step 1

We argue by contradiction assuming that there exist a sequence $\Lambda _h\rightarrow +\infty$, $\eta _h\in [0,{{\tilde{\eta }}}]$, $\eta _h\rightarrow \eta _0$, $m_h\in [m',m'']$ converging to some m, and a sequence $E_h\subset \Omega _0\setminus {\textbf{C}}_{\eta _h}$ such that each $E_h$ is a minimizer of (4.8) with $\Lambda ', m$ and $\eta$ replaced by $\Lambda _h, m_h$ and $\eta _h$ respectively, and $|E_h|\not =m_h$. Since $P(E_h;{\mathbb {R}}^N\setminus {\textbf{C}}_{\eta _h})\le P(\Omega _{\eta _h,m_h};{\mathbb {R}}^N\setminus {\textbf{C}}_{\eta _h})$, from (4.5) we have that the perimeters of $E_h$ are equibounded perimeters. Therefore, without loss of generality we may assume that $E_h$ converges in $L^1$ to some set $F\subset \Omega _0\setminus {\textbf{C}}_{\eta _0}$ such that $|F|=m$. We assume also that $|E_h|<m_h$ for all h, the other case being analogous. Note also that, since $\Lambda _h\rightarrow +\infty$ we have $m_h-|E_h|\rightarrow 0$.

Observe now that (4.6) implies that there exists a point ${x_0}\in \partial ^*F\cap (\Omega _0\setminus {\textbf{C}}_{\eta _0})$. Arguing as in Step 1 of Theorem 1.1 in [10], given $\varepsilon >0$ sufficiently small, we can find nearby $x_0$ a point $x'$ and $r>0$ such that $B_r(x')\subset \subset \Omega _0\setminus {\textbf{C}}_{\eta _0}$ and

$$\begin{aligned} | F\cap B_{r/2}(x')|<\varepsilon r^N\,, \quad | F \cap B_r(x')|>\frac{\omega _N}{2^{N+2}}r^N\,. \end{aligned}$$

Therefore, for h sufficiently large, we also have

$$\begin{aligned} | E_h\cap B_{r/2}(x')|<\varepsilon r^N\,, \quad | E_h\cap B_r(x')|>\frac{\omega _N}{2^{N+2}}r^N\,. \end{aligned}$$

We can now continue as in the proof of [10, Theorem 1]. We recall the main construction for the reader’s convenience. For a sequence $0<\sigma _h<1/2^N$ to be chosen, we introduce the following bilipschitz maps:

$$\begin{aligned} \Phi _x'(x):= {\left\{ \begin{array}{ll} x'+(1-\sigma _h(2^N-1))(x-x') &{} \text {if}\ |x-x'|\le \frac{r}{2}, \\ x+\sigma _h\left( 1-\displaystyle \frac{r^N}{|x-x'|^N}\right) (x-x') &{} {\frac{r}{2}\le |x-x'|<r,}\\ x &{} {|x-x'|\ge r.} \end{array}\right. } \end{aligned}$$

Setting ${{\widetilde{E}}}_h:=\Phi _h(E_h)$, arguing as for the proof of [10, formula (14)], we have

$$\begin{aligned} {{{\mathcal {H}}}}^{N-1}\big (\partial ^*E_h\setminus {\textbf{C}}_{\eta _h}\big )- {{{\mathcal {H}}}}^{N-1}(\partial ^*{{\widetilde{E}}}_h\setminus {\textbf{C}}_{\eta _h})\ge -2^NN\sigma _h {{{\mathcal {H}}}}^{N-1}(\partial ^*E_h\setminus {\textbf{C}}_{\eta _h})\,. \end{aligned}$$

(6.1)

Moreover, arguing exactly as in Step 4 of the proof of [10, Theorem 1] we have

$$\begin{aligned} |{{\widetilde{E}}}_h|-|E_h|\ge \sigma _hr^N(c-\varepsilon C) \end{aligned}$$

for suitable universal constants $c, C>0$. If we fix $\varepsilon$ so that the negative term in the brackets does not exceed half the positive one, then we have

$$\begin{aligned} |\widetilde{E}_h|-|E_h|\ge \frac{c}{2}\sigma _hr^N\,. \end{aligned}$$

(6.2)

In particular from this inequality it is clear that we can choose $\sigma _h$ so that $|{{\widetilde{E}}}_h|=m_h$; this implies $\sigma _h\rightarrow 0$. With this choice of $\sigma _h$, recalling that $\Lambda _h\rightarrow +\infty$ and that the perimeters of $E_h$ are equibounded, it follows from (6.1) and (6.2) that

$$\begin{aligned} P({{\widetilde{E}}}_h;{\mathbb {R}}^N\setminus {\textbf{C}}_{\eta _h})+\Lambda _h||\widetilde{E}_h|-m_h|&\le P( E_h;{\mathbb {R}}^N\setminus {\textbf{C}}_{\eta _h})+\Lambda _h|| E_h|-m_h| \\&\qquad +2^NN\sigma _h {{{\mathcal {H}}}}^{N-1}(\partial ^*E_h\setminus {\textbf{C}}_{\eta _h})-\Lambda _h \frac{c}{2}\sigma _hr^N\\&<P( E_h;{\mathbb {R}}^N\setminus {\textbf{C}}_{\eta _h})+\Lambda _h|| E_h|-m_h| \end{aligned}$$

for h large, thus contradicting the minimality of $E_h$.

1.2 Proof of the claim of Step 3

We start by showing that the functions $I_\eta$ are strictly increasing in $[0,m_\eta ]$ for all $\eta \in [0,{{\bar{\eta }}}]$. To this end we fix $m\in (0,m_\eta ]$ and a point $x\in \pi _{{\textbf{C}}_\eta }(\Omega _{\eta ,m})$, where $\pi _{{\textbf{C}}_\eta }$ is the orthogonal projection on ${\textbf{C}}_\eta$. Let $\Pi$ be the tangent hyperplane to ${\textbf{C}}_\eta$ at x. Define $\Pi _t=\Pi +t\nu _{{\textbf{C}}_\eta }(x)$ for $t\in {\mathbb {R}}$ and set

$$\begin{aligned} {{\bar{t}}}=\max \left\{ t\ge 0:\,\Pi _t\cap \overline{\Omega _{\eta ,m}}\not =\emptyset \right\} . \end{aligned}$$

Note that ${{\bar{t}}}>0$ and that $\Pi _{{{\bar{t}}}}$ is a support hyperplane for $\Omega _{\eta ,m}$ with dist$(\Pi _{{{\bar{t}}}},{\textbf{C}}_\eta )={{\bar{t}}}$. For all $t\in (0,{{\bar{t}}})$ we denote by $\Omega _{\eta ,m,t}$ the intersection of $\Omega _{\eta ,m}$ with the half space with boundary $\Pi _t$ containing ${\textbf{C}}_\eta$. Then $I_{\eta }(|\Omega _{\eta ,m,t}|)\le P(\Omega _{\eta ,m,t};{\mathbb {R}}^N\setminus {\textbf{C}}_\eta )<P(\Omega _{\eta ,m};{\mathbb {R}}^N\setminus {\textbf{C}}_\eta )=I_{\eta }(m)$. Since the function $t\rightarrow |\Omega _{\eta ,m,t}|$ is increasing and continuous in a left neighborhood of ${{\bar{t}}}$ and $|\Omega _{\eta ,m,t}|<|\Omega _{\eta ,m}|$ if $t<{{\bar{t}}}$, it follows that

$$\begin{aligned} \text {for every}\ m\in (0,m_\eta ]\ \text {there exists}\ \varepsilon >0\ \text {s.t.}\ I_{\eta }(s)<I_{\eta }(m)\ \text {for all}\ s\in (m-\varepsilon ,m)\,. \end{aligned}$$

(6.3)

Let $I=\{0<s<m:\,I_{\eta }(\sigma )\le I_{\eta }(m)$ for all $\sigma \in [s,m)\}$. We claim that $I=(0,m)$. Indeed if ${{\bar{m}}}=\inf I>0$, then there exist $m_n\in I$, with $m_n\rightarrow {\bar{m}}^+$. Since the minimizers $\Omega _{\eta , m_n}$ are equibounded sets with equibounded perimeters, see (4.5), up to a subsequence we may assume that $\Omega _{\eta , m_n}$ converge to a set $E\subset \Omega _0\setminus {\textbf{C}}_\eta$ with $|E|={{\bar{m}}}$. Then, by the lower semicontinuity of the perimeter we conclude that $I_{\eta }({\bar{m}})\le P(E;{\mathbb {R}}^N\setminus {\textbf{C}}_\eta )\le \liminf _{n}I_{\eta }(m_n)\le I_\eta (m)$. In turn, (6.3) implies that there exists exists a left neighborhood $({{\bar{m}}}-\varepsilon ,{{\bar{m}}})$ such that $I_{\eta }(s)<I_{\eta }({{\bar{m}}})\le I_\eta (m)$ for all $s\in (\bar{m}-\varepsilon ,{{\bar{m}}})$ which is a contradiction to the fact that $\bar{m}=\inf I$. This contradiction proves that $I_{\eta }$ is increasing. The strict monotonicity now follows from (6.3).

Finally if, $m_1,m_2\in [m',m'']$, from (4.8) we have for $\eta \in [0,{{\tilde{\eta }}}]$

$$\begin{aligned} I_\eta (m_2)=P\left( \Omega _{\eta ,m_2};{\mathbb {R}}^N\setminus {\textbf{C}}_\eta \right) \le P\left( \Omega _{\eta ,m_1};{\mathbb {R}}^N\setminus {\textbf{C}}_\eta \right) +\Lambda '|m_2-m_1|\,. \end{aligned}$$

This proves the $\Lambda '$-Lipschitz continuity of $I_\eta$.

1.3 Proof of claim (4.17)

Let us start by assuming also that

$$\begin{aligned} {{{\mathcal {H}}}}^{N-1}(\partial E_h\cap \partial V\cap {\mathscr {H}})=0 \qquad \text {for all}\ h\in \mathbb N\,. \end{aligned}$$

(6.4)

To this aim we fix $\delta >0$ and set $\mathscr {H}_\delta :=\{x\in {\mathscr {H}}:\, x_N>\delta \}$ and $(E)_\delta =E+B_\delta (0)$. Then we denote by $\Phi _h:\overline{Q_r(0)}\rightarrow \overline{Q_r(0)}$ a sequence of $C^1$ diffeomorphisms converging in $C^1$ to the identity map as $h\rightarrow + \infty$ with the property that $\Phi _h(\overline{Q_r(0)\cap \mathscr {H}})=\overline{Q_r(0)\setminus {\textbf{C}}_{\eta ,h}}$, $\Phi _h(\partial \mathscr {H}\cap Q_r(0))=\partial {\textbf{C}}_{\eta ,h}\cap Q_r(0)$ and $\Phi _h(x)=x$ if $x\in {\mathscr {H}}_\delta$. Recalling the $\Lambda$-minimality property (4.14), we have using (6.4) and observing that $\Phi _h(V)\subset \!\subset Q_r(0)$ for h sufficiently large

$$\begin{aligned}&P(E_h; Q_r(0)\setminus {\textbf{C}}_{\eta ,h})\le P(E_h\setminus \Phi _h(V); Q_r(0)\setminus {\textbf{C}}_{\eta ,h})+\frac{\Lambda }{h}|\Phi _h(V)|\\&\qquad \le P(E_h; (Q_r(0)\setminus {\textbf{C}}_{\eta ,h})\setminus \overline{\Phi _h(V)}) + P(\Phi _h(V); (Q_r(0)\setminus {\textbf{C}}_{\eta ,h})\cap E_h)\\&\qquad \qquad +{{{\mathcal {H}}}}^{N-1}(\partial \Phi _h(V)\cap \partial E_h\cap \{x_N\le \delta \}\cap (Q_r(0)\setminus {\textbf{C}}_{\eta ,h}))+\frac{\Lambda }{h}|\Phi _h(V)|\,. \end{aligned}$$

Since

$$\begin{aligned} P(E_h; Q_r(0)\setminus {\textbf{C}}_{\eta ,h})\ge P(E_h; (Q_r(0)\setminus {\textbf{C}}_{\eta ,h})\setminus \overline{\Phi _h(V)})+P(E_h; (Q_r(0)\setminus {\textbf{C}}_{\eta ,h})\cap \Phi _h(V))\,, \end{aligned}$$

and using the fact that ${\mathscr {H}}_\delta \cap V=\mathscr {H}_\delta \cap \Phi _h(V)\subset (Q_r(0)\setminus {\textbf{C}}_{\eta ,h})\cap \Phi _h(V)$, the inequality above yields

$$\begin{aligned}&P(E_h;{\mathscr {H}}_\delta \cap V)\le P(\Phi _h(V); (Q_r(0)\setminus {\textbf{C}}_{\eta ,h})\cap E_h)\nonumber \\&\qquad +{{{\mathcal {H}}}}^{N-1}(\partial \Phi _h(V)\cap \{x_N\le \delta \}\cap (Q_r(0)\setminus {\textbf{C}}_{\eta ,h}))+\frac{\Lambda }{h}|\Phi _h(V)| \nonumber \\&\quad \le P(V; Q_r(0)\cap {\mathscr {H}}_\delta \cap E_h)\nonumber \\&\qquad +2{{{\mathcal {H}}}}^{N-1}(\partial \Phi _h(V)\cap \{x_N\le \delta \}\cap (Q_r(0)\setminus {\textbf{C}}_{\eta ,h}))+\frac{\Lambda }{h}|\Phi _h(V)|\nonumber \\&\quad \le P(V; Q_r(0)\cap {\mathscr {H}}_\delta \cap (E)_\delta )\nonumber \\&\qquad +2(\text {Lip}(\Phi _h))^{N-1}P(V;\{0<x_N\le \delta \})+\frac{\Lambda }{h}|\Phi _h(V)|\,, \end{aligned}$$

(6.5)

where in the last inequality we used the fact that $\Phi _h^{-1}((Q_r(0)\setminus {\textbf{C}}_{\eta ,h})\cap \{x_N\le \delta \})= Q_r(0)\cap \{0<x_N\le \delta \}$ and the fact that $\overline{E_h}$ converge in the Kuratowski sense to ${\overline{E}}$ in $\mathscr {H}_\delta$, see Remark 2.3. By the lower semicontinuity of the perimeter, passing to the limit in (6.5)

$$\begin{aligned} P(E;{\mathscr {H}}_\delta \cap V)\le P(V; Q_r(0)\cap \mathscr {H}_\delta \cap (E)_\delta )+2P(V; \{0<x_N\le \delta \})\,. \end{aligned}$$

In turn, by letting $\delta \rightarrow 0$ we have

$$\begin{aligned} P(E;{\mathscr {H}}\cap V)\le P(V; Q_r(0)\cap E)\,, \end{aligned}$$

(6.6)

which is equivalent to (4.17) thanks to first condition in (4.17). To remove (6.4) it is enough to consider a sequence of smooth sets $V_j\subset \!\subset Q_r(0)$, $V\subset \!\subset V_j$, satisfying the first condition in (4.17) and (6.4), and such that $V_j\rightarrow V$ in $L^1$ and $P(V_j;Q_r(0))\rightarrow P(V;Q_r(0))$. The conclusion then follows by applying (4.17) with V replaced by $V_j$ and passing to the limit thanks to the first condition in (4.17).