On the boundary of almost isoperimetric domains

TL;DR

This paper demonstrates that nearly optimal isoperimetric domains in Euclidean space have boundaries closely approximating spheres, with refined closeness under curvature bounds, and addresses a question about almost extremal hypersurfaces.

Contribution

It establishes quantitative boundary closeness of near-isoperimetric domains to spheres and refines this relation with curvature bounds, also answering a specific open question.

Findings

Boundaries of small deficit domains are Hausdorff-close to spheres.

Refined closeness results under integral curvature bounds.

Resolved an open question about almost extremal hypersurfaces.

Abstract

We prove that finite perimeter subsets of $\mathbb{R}^{n+1}$ with small isoperimetric deficit have boundary Hausdorff-close to a sphere up to a subset of small measure. We also refine this closeness under some additional a priori integral curvature bounds. As an application, we answer a question raised by B. Colbois concerning the almost extremal hypersurfaces for Chavel's inequality.

Equations

$$I(\Omega)\geqslant I\bigl{(}\mathbb{B}^{n+1}\bigr{)}$$

$$\delta (\Omega ):=\frac{I(\Omega )}{I({\mathbb{B}}^{n+1})}-1$$

$$"distance"\; from\; \Omega \; to\; some\; ball⩽C\delta (\Omega {)}^{1/\alpha }for\; a\; given\; category\; of\; \Omega \subset {\mathbb{R}}^{n+1}$$

$$\mathcal{A}(\Omega ):=\underset{x\in {\mathbb{R}}^{n+1}}{\inf }\frac{|\Omega \Delta B_x(R_{\Omega })|}{|\Omega |}for\; \Omega \subset {\mathbb{R}}^{n+1}$$

$$\mathcal{A}(\Omega )⩽C(n)\delta (\Omega {)}^{1/\alpha (n)}$$

$$16\pi d_{H}^{2}({\mathcal{C}},\partial\Omega)\leqslant P(\Omega)^{2}-4\pi|\Omega|\leqslant 4\pi|\Omega|\delta(\Omega)\bigl{(}2+\delta(\Omega)\bigr{)}$$

$$\frac{d_H(\mathcal{C},\partial \Omega )}{\mathrm{d}iam\Omega }⩽\frac{\sqrt{3\pi }}{4}\delta (\Omega {)}^{\frac{1}{2}}$$

$$\underset{x\in {\mathbb{R}}^{n+1}}{\inf }\frac{d_H(\Omega ,B_x(R_{\Omega }))}{R_{\Omega }}⩽C(n)\delta (\Omega {)}^{\frac{2}{n+2}}.$$

$$A(\Omega )=\mathcal{F}(\Omega )\cap A_{\delta (\Omega {)}^{\frac{1}{4}}}$$

$$\Vert f{\Vert }_p={\left(\frac{1}{P(\Omega )}{\int }_{\partial \Omega }|f{|}^{p}d{\mathcal{H}}^n\right)}^{1/p}for\; any\; measurablef:\partial \Omega \to \mathbb{R}.$$

$$\displaystyle d_{H}\bigl{(}\partial\Omega,$$

$$\mathrm{C}ard(I)⩽C(n,p)P(\Omega )\Vert \mathrm{H}{\Vert }_p^n\delta (\Omega {)}^{\frac{p-n}{4p}}.$$

$$d_L(\partial \Omega ,{\mathbb{S}}_{x_{\Omega }}(R_{\Omega }))⩽C(n,p)\delta (\Omega {)}^{\frac{2(p-n)}{p(n+2)-2n}}$$

$$d_H(\partial \Omega ,S_{x_{\Omega }}(R_{\Omega }))⩽C(n,p,K)R_{\Omega }\delta (\Omega {)}^{\frac{2p-n}{2p-2n+np}}$$

$$d_H(\partial \Omega ,S_{x_{\Omega }}(R_{\Omega }))⩽C(p,K)R_{\Omega }(-\delta (\Omega )\ln \delta (\Omega ){)}^{\frac{1}{2}}$$

$$\lambda_{1}^{\Sigma}\leqslant\frac{n}{(n+1)^{2}}\Bigl{(}\frac{{\mathcal{H}}^{n}(\Sigma)}{|\Omega|}\Bigr{)}^{2}$$

$$\delta (\Omega )⩽C(n)\gamma (\Omega {)}^{1/2}$$

$$d_H(A,B)=\inf \{\epsilon \mid A\subset B_{\epsilon }\text{and}B\subset A_{\epsilon }\}$$

$$|\mu |(\Omega ):=\sup \left\{\sum _{k\in \mathbb{N}}\Vert \mu ({\Omega }_k)\Vert \mid {\Omega }_i\cap {\Omega }_j=\varnothing ,\bigcup _{k\in \mathbb{N}}{\Omega }_k\subset \Omega \right\}$$

$$\mathcal{F}(\Omega ):=\left\{x\in {\mathbb{R}}^{n+1}\mid \forall r>0,|D{\chi }_{\Omega }|(B_x(r))>0\text{and}\underset{r⟶0^{+}}{\lim }\frac{D{\chi }_{\Omega }(B_x(r))}{|D{\chi }_{\Omega }|(B_x(r))}\in {\mathbb{S}}^n\right\}$$

$${\partial }^{\star }\Omega :={\mathbb{R}}^{n+1}\setminus ({\Omega }^0\cup {\Omega }^1)$$

$$\displaystyle\int_{\mathcal{F}(G)}\bigl{|}|x-x_{\Omega}|-R_{G}\bigr{|}d{\mathcal{H}}^{n}\leqslant\frac{10(n+1)^{3}}{k(n)}|G|\delta(\Omega)^{1/2}$$

$$\bigl{(}1-C(n)\delta(\Omega)\bigr{)}{\mathcal{H}}^{n}\bigl{(}\mathcal{F}(\Omega)\bigr{)}\leqslant{\mathcal{H}}^{n}\bigl{(}\mathcal{F}(\Omega)\cap\mathcal{F}(G)\bigr{)}$$

$$\displaystyle{\mathcal{H}}^{n}\bigl{(}\mathcal{F}(G)\bigr{)}$$

$$\displaystyle\geqslant\bigl{(}1-C(n)\delta(\Omega)\bigr{)}{\mathcal{H}}^{n}\bigl{(}\mathcal{F}(\Omega)\bigr{)}$$

$${\mathcal{H}}^{n}\bigl{(}\mathcal{F}(F_{\infty})\bigr{)}\leqslant\bigl{(}1+k(n)\bigr{)}{\mathcal{H}}^{n}\bigl{(}\mathcal{F}(\Omega)\cap\mathcal{F}(F_{\infty})\bigr{)}.$$

$${\mathcal{H}}^n(\mathcal{F}(\Omega ))={\mathcal{H}}^n(\mathcal{F}(\Omega )\cap \mathcal{F}(G))+{\mathcal{H}}^n(\mathcal{F}(\Omega )\cap \mathcal{F}(F_{\infty }))$$

$$\displaystyle(1+k(n)){\mathcal{H}}^{n}\bigl{(}\mathcal{F}(\Omega)\bigr{)}$$

$$\displaystyle=2{\mathcal{H}}^{n}\bigl{(}\mathcal{F}(G)\cap\mathcal{F}(\Omega)\bigr{)}+(1+k(n)){\mathcal{H}}^{n}\bigl{(}\mathcal{F}(\Omega)\cap\mathcal{F}(F_{\infty})\bigr{)}$$

$$\displaystyle\geqslant 2{\mathcal{H}}^{n}\bigl{(}\mathcal{F}(G)\cap\mathcal{F}(\Omega)\bigr{)}+{\mathcal{H}}^{n}\bigl{(}\mathcal{F}(F_{\infty})\bigr{)}$$

$$\displaystyle={\mathcal{H}}^{n}\bigl{(}\mathcal{F}(G)\bigr{)}+{\mathcal{H}}^{n}\bigl{(}\mathcal{F}(\Omega)\bigr{)}$$

$$\displaystyle\geqslant\bigl{(}2-C(n)\delta(\Omega)\bigr{)}{\mathcal{H}}^{n}\bigl{(}\mathcal{F}(\Omega)\bigr{)}$$

$${\mathcal{H}}^{n}\bigl{(}\mathcal{F}(G)\cap\mathcal{F}(\Omega)\bigr{)}\geqslant\bigl{(}1-C(n)\delta(\Omega)\bigr{)}{\mathcal{H}}^{n}\bigl{(}\mathcal{F}(\Omega)\bigr{)}.$$

Full text

1. Introduction

In all the paper, $B_{x}(r)$ and $S_{x}(r)$ denote respectively the Euclidean ball and sphere with center $x$ and radius $r$ in $\mathbb{R}^{n+1}$. We also set $\mathbb{B}^{k}$ the unit ball centred at [math] in $\mathbb{R}^{k}$ and $\mathbb{S}^{k-1}$ the unit sphere centred at [math] in $\mathbb{R}^{k}$.

For any Borel set $\Omega$ of $\mathbb{R}^{n+1}$, we denote $|\Omega|$ its Lebesgue measure, $P(\Omega)$ its perimeter (see definition in section 2) and $I(\Omega)=\frac{P(\Omega)}{|\Omega|^{\frac{n}{n+1}}}$ its isoperimetric ratio. Then it satisfies the isoperimetric inequality

[TABLE]

with equality if and only if $\Omega$ is a Euclidean ball up to set of Lebesgue measure [math]. To study the stability of the isoperimetric inequality, we denote by

[TABLE]

the isoperimetric deficit of a Borel set $\Omega$ of finite perimeter and address the following question:

”How far from a ball are almost isoperimetric domains?(i.e. with small $\delta(\Omega)$)”

More quantitatively, by stability of the isoperimetric inequality, we understand the validity of an inequality of the form

[TABLE]

where the ”distance” need to be defined and where $C$ and $\alpha$ are some positive universal constants. Many authors have studied this stability problem with the Fraenkel asymmetry $\mathcal{A}(\Omega)$ as distance function. We recall that

[TABLE]

where $R_{\Omega}$ is given by $R_{\Omega}^{n+1}|\mathbb{B}^{n+1}|=|\Omega|$ and $U\Delta V=(U\setminus V)\cup(V\setminus U)$. So the isoperimetric inequality is said stable with respect to the Fraenkel asymmetry if there exists $C(n),\alpha(n)>0$ such that

[TABLE]

holds for a given category of domains $\Omega\subset\mathbb{R}^{n+1}$.

Such inequalities were first obtained for domains of $\mathbb{R}^{2}$ by Bernstein ([4]) and Bonnesen ([5]). The first result in higher dimension was due to Fuglede ([11]) for convex domains. Without convexity assumption, the main contributions are due to Hall, Haymann, Weitsman (see [17] and [18]) who established this inequality with $\alpha(n)=4$, and later to Fusco, Maggi and Pratelli who proved this inequality with the sharp exponent $\alpha(n)=2$ in [15] (see also the paper of Figalli, Magelli and Pratelli ([10]) or [8] and [14] for other proofs of this last result).

To get more precise informations on the geometry of almost isoperimetric domains than a small Fraenkel asymmetry, we can take as ”distance” function the Hausdorff distance. The first result in that direction was the following inequality proved by Bonnesen ([5]) for convex curves and by Fuglede ([12]) in the general case: if $\partial\Omega$ is a $C^{1}$-piecewise closed curve there exists a Euclidean circle $\mathcal{C}$ such that

[TABLE]

where  $d_{H}$  denotes  the  Hausdorff  distance. Note that assuming $\delta(\Omega)\leqslant 1$ and using  the  isodiametric  inequality  $|\Omega|\leqslant\frac{\pi}{4}({\mathrm{d}iam}\,\Omega)^{2}$, we infer the following inequality

[TABLE]

However, this result is false for more general domains in $\mathbb{R}^{2}$, especially non connected one (consider for instance the disjoint union of a large ball and a tiny one far from each other). Moreover, in higher dimension $n\geqslant 2$, even for connected smooth domains, we cannot expect to control the Hausdorff distance from $\partial\Omega$ to a sphere by the isoperimetric deficit alone, as proves the sets obtained by adding or subtracting to a ball a thin tubular neighbourhood of a Euclidean subset of dimension not larger than $n-1$ (see for instance [6]). So to generalize this kind of stability result in higher dimension, it is necessary to assume additional informations on the geometry of the domains we consider. In [11] Fuglede proved that if $n\geqslant 3$, $\Omega$ is a convex set and $\delta(\Omega)$ small enough then

[TABLE]

($\delta(\Omega)^{\frac{2}{n+2}}$ is replaced by $\sqrt{\delta(\Omega)}$ for $n=1$ and by $(\delta(\Omega)\log[1/\delta(\Omega)])^{1/2}$ for $n=2$). Note that since $\Omega$ is convex, $\partial\Omega$ is also close to a sphere of radius $R_{\Omega}$. Actually, Fuglede deals with more general sets called nearly spherical domains and this Fuglede’s result has been generalized by Fusco, Gelli and Pisante ([13]) for any set of finite perimeter satisfying an interior cone condition.

In this paper, we prove generalizations of inequalities (1.4) and (1.5) to any smooth domain (even nonconvex) with integral control on the mean curvature of the boundary. We even get a weak Hausdorff control for almost isoperimetric domains that need no additional assumption on their boundary.

1.1. No assumption on the boundary

Let $\mathcal{F}(\Omega)$ be the reduced boundary of $\Omega$ (see the section 2 for the definition). When $\Omega$ is a smooth domain, we have $\mathcal{F}(\Omega)=\partial\Omega$.

Theorem 1**.**

Let $\Omega$ be a set of $\mathbb{R}^{n+1}$ with finite perimeter with $\delta(\Omega)\leqslant\frac{1}{C(n)}$. There exists $x_{\Omega}\in\mathbb{R}^{n+1}$ and $A(\Omega)\subset\mathcal{F}(\Omega)$ such that

1. (1)

$\displaystyle\frac{\mathcal{H}^{n}(\mathcal{F}(\Omega)\setminus A(\Omega))}{P(\Omega)}\leqslant C(n)\delta(\Omega)^{\frac{1}{4}}$, 2. (2)

\displaystyle\frac{d_{H}\bigl{(}A(\Omega),S_{x_{\Omega}}(R_{\Omega})\bigr{)}}{R_{\Omega}}\leqslant C(n)\delta(\Omega)^{\beta(n)}.

Here $\mathcal{H}^{n}$ denotes the $n$-dimensional Hausdorff measure and $\beta(n)=\min(\frac{1}{4n},\frac{1}{8})$.

In other words, the boundary $\mathcal{F}(\Omega)$ is Hausdorff close to a sphere up to a set of small measure. Note that we have

[TABLE]

where for any $\eta>0$ we set A_{\eta}=\bigl{\{}x\in\mathbb{R}^{n+1}/\,\bigl{|}|x-x_{\Omega}|-R_{\Omega}\bigr{|}\leqslant R_{\Omega}\eta\bigr{\}}.

Remark 1**.**

Note that the sets of the previous theorem also satisfy

1. (1)

$\displaystyle\frac{|\Omega\Delta B_{x_{\Omega}}(R_{\Omega})|}{|\Omega|}\leqslant C(n)\delta(\Omega)^{1/2}$* ,* 2. (2)

\displaystyle\frac{d_{H}\bigl{(}\Omega\cap B_{x_{\Omega}}(R_{\Omega}),B_{x_{\Omega}}(R_{\Omega})\bigr{)}}{R_{\Omega}}\leqslant C(n)\delta(\Omega)^{\frac{1}{2(n+1)}}* (see the end of the section 2).*

In other words $\Omega\cap B_{x_{\Omega}}(R_{\Omega})$ is Hausdorff close to the ball $B_{x_{\Omega}}(R_{\Omega})$ up to a set of small measure, which is a weak generalization of inequality (1.5).

Remark 2**.**

When $n=1$ or $\Omega$ convex Theorem 1 easily implies earlier results à la Bonnesen [5] and Fuglede [11] but with non optimal power $\beta(n)$.

Remark 3**.**

See also Theorem 8 in Section 4.4 that is a reformulation of Theorem 1 in term of Preiss distance between the normalized measures associated to $\mathcal{F}(\Omega)$ and $S_{x_{\Omega}}(R_{\Omega})$.

To get informations on the smooth domain $\Omega$ itself, and not up to a set of small measure, additional assumptions are required. A reasonable assumption is an integral control on the mean curvature ${\mathrm{H}}$. In the sequel, for any $p\geqslant 1$, we define

[TABLE]

Note that a upper bound on $\|{\mathrm{H}}\|_{p}$ with $p<n-1$ is not sufficient. Indeed, we can refer to examples constructed by the authors in [2, 3]: by adding small tubular neighbourhood of well chosen trees to $B_{0}(1)$, we get a set almost isoperimetric domains on which $\|{\mathrm{H}}\|_{p}$ is uniformly bounded for any $p<n-1$ and that is dense for the Hausdorff distance among all the closed set of $\mathbb{R}^{n+1}$ that contain $B_{0}(1)$.

1.2. Upper bound on $\|{\mathrm{H}}\|_{n-1}$

Theorem 2**.**

Let $\Omega$ be an open set with a smooth boundary $\partial\Omega$, finite perimeter and $\delta(\Omega)\leqslant\frac{1}{C(n)}$. There exists a subset $T$ of $\mathbb{R}^{n+1}$ which satisfies whose $1$-dimensional Hausdorff measure satisfies

1. (1)

$\mathcal{H}^{1}(T)\leqslant C(n)R_{\Omega}\displaystyle\int_{\partial\Omega\setminus A_{\delta(\Omega)^{1/4}}}|{\mathrm{H}}|^{n-1}d{\mathcal{H}}^{n}$, 2. (2)

d_{H}\bigl{(}\partial\Omega,S_{x_{\Omega}}(R_{\Omega})\cup T\bigr{)}\leqslant C(n)R_{\Omega}\delta(\Omega)^{\beta(n)}, 3. (3)

the set $A_{\delta(\Omega)^{1/4}}\cup T$ has at most $N+1$ connected components,

where $\mathcal{H}^{1}(T)$ denotes the 1-dimensional Hausdorff measure of $T$ and $N$ is the number of the connected components of $\partial\Omega$ that do not intercept $A_{\delta(\Omega)^{\frac{1}{4}}}$.

Note that by Theorem 1 at least one connected component of $\partial\Omega$ intercepts $A_{\delta(\Omega)^{\frac{1}{4}}}$ and so if $\partial\Omega$ is connected then we have $N=0$ and $A_{\delta(\Omega)^{1/4}}\cup T$ is connected. Moreover note that for $n=1$ we recover Fuglede’s result (1.3) for $C^{2}$-piecewise closed curves.

The case $N=\infty$ in Theorem 2 is trivial since the sets obtained by the union of a sphere and infinitely numebrable many points are dense for the Hausdorff distance among all the closed sets containing $S_{x_{\Omega}}(R_{\Omega})$.

Similarly to the case of curves, Theorem 2 is quite optimal as prove examples given by a domain \Omega_{\varepsilon}=\bigl{[}B_{0}(R)\setminus\displaystyle\bigcup_{i}T_{i,\varepsilon}\bigr{]}\cup\bigcup_{j}T_{j,\varepsilon}, where $(T_{i})$ and $(T_{j})$ are some families of Euclidean trees and the $T_{i,\varepsilon}$ denotes the $\varepsilon$-tubular neighbourhood of $T_{i}$. In these examples, the integral of $|{\mathrm{H}}|^{n-1}$ on $\partial\Omega_{\varepsilon}\setminus A_{\delta(\Omega)^{\frac{1}{4}}}$ will converge, up to a multiplicative constant $C(n)$, to the sum of the length of the trees as $\varepsilon$ tends to [math].

We refer to Theorem 10 of Section 5.2 for a generalization of inequality (1.5) similar to Theorem 2.

1.3. Bound on $\|{\mathrm{H}}\|_{p}$ with $p>n-1$

If we assume some upper bound on the $L^{p}$ norm of $|{\mathrm{H}}|$ with $p>n-1$, then combining Theorem 2 and Lemma 2 with Hölder inequality readily gives the following improved result.

Theorem 3**.**

Let $p\geqslant n-1$ and $\Omega$ be an open set with a smooth boundary $\partial\Omega$, finite perimeter and $\delta(\Omega)\leqslant\frac{1}{C(n)}$. Let $(\partial\Omega_{i})_{i\in I}$ be the connected components of $\partial\Omega$ that do not intercept $A_{\delta(\Omega)^{\frac{1}{4}}}$. For any $i\in I$, there exists $x_{i}\in\partial\Omega_{i}$ such that

[TABLE]

Moreover if $p\geqslant n$ and if ${\mathrm{H}}$ is $L^{p}$-integrable then $I$ is finite and we have

[TABLE]

Remark 4**.**

We will see in the proof that the above estimates are more precise since as in Theorem 2, we can replaced $\|{\mathrm{H}}\|_{p}$ by \Bigl{(}\frac{1}{P(\Omega)}\displaystyle\int_{\partial\Omega\setminus A_{\delta(\Omega)^{1/4}}}|{\mathrm{H}}|^{p}d{\mathcal{H}}^{n}\Bigr{)}^{\frac{1}{p}}.

Remark 5**.**

If we assume that $\partial\Omega$ is connected, then Theorem 3 implies that $\partial\Omega$ is Hausdorff close to a sphere. If $\partial\Omega$ has N connected component, the it asserts that $\partial\Omega$ is Hausdorff close to a sphere union a finite set with at most $N-1$ points.

Note that in the case $p<n$ we can not control the cardinal of $I$ in terms of $\|{\mathrm{H}}\|_{p}$. Indeed, consider the sequence of domains $\Omega_{k}$ obtained by the union of $\mathbb{B}^{n+1}$ and $k$ balls $B_{x_{i}}(r_{i}/k)$ where $x_{i}$ are some points satisfying for instance ${\mathrm{d}ist}(0,x_{i})=2i$. If $\displaystyle\sum_{i\geqslant 0}r_{i}^{n-p}$ is convergent then $\displaystyle\lim_{k\longrightarrow\infty}\delta(\Omega_{k})=0$ and $P(\Omega_{k})\|{\mathrm{H}}_{k}\|_{p}$ (where ${\mathrm{H}}_{k}$ denotes the mean curvature of $\partial\Omega_{k}$) remains bounded when ${\mathrm{C}ard}(I)$ tends to infinity.

Here also we refer to Theorem 11 of Section 5.2 for a version of Theorem 3 generalizing inequality 1.5.

1.4. Bound on $\|{\mathrm{H}}\|_{p}$ with $p>n$

When $p>n$, it follows from 1.6 that if $\delta(\Omega)$ is small enough then $I=\emptyset$ and $\partial\Omega$ is Hausdorff close to $S_{x_{\Omega}}(R_{\Omega})$. More precisely we have that

Theorem 4**.**

Let $p>n$. There exists a constant $C(n,p)>0$ such that if $\Omega$ is an open set with smooth boundary $\partial\Omega$ such that ${\mathcal{H}}^{n}(\partial\Omega)\|{\mathrm{H}}\|_{p}^{n}\leqslant K$ and $\delta(\Omega)\leqslant\frac{1}{C(n,p,K)}$ then $\partial\Omega$ is diffeomorphic and quasi-isometric to $S_{x_{\Omega}}(R_{\Omega})$. Moreover the Lipschitz distance $d_{L}$ satisfies

[TABLE]

for any $n\geqslant 2$ and the Hausdorff distance

[TABLE]

when $n\geqslant 3$ and

[TABLE]

when $n=2$.

Remark 6**.**

Actually, under the assumption of the previous theorem, we show that $\partial\Omega=\{\varphi(w)w,w\in S_{x_{\Omega}}(R_{\Omega})\}$, where $\varphi\in W^{1,\infty}(S_{x_{\Omega}}(R_{\Omega}))\cap W^{2,p}(S_{x_{\Omega}}(R_{\Omega}))$, with $\|d\varphi\|_{\infty}\leqslant\frac{C(n,p,K)}{R_{\Omega}}\delta(\Omega)^{\frac{p-n}{2p-2n+pn}}$ and $\|\nabla d\varphi\|_{p}\leqslant C(n,p,K)/R_{\Omega}^{2}$. So $\Omega$ is a nearly spherical domain in the sense of Fuglede and is the graph over $S_{x_{\Omega}}(R_{\Omega})$ of a $C^{1,1-\frac{n}{p}}(S_{x_{\Omega}}(R_{\Omega}))$ function. It implies that any sequence of domain $(\Omega_{k})_{k}$ with $\delta(\Omega_{k})\to 0$ and ${\mathcal{H}}^{n}(\partial\Omega_{k})\|{\mathrm{H}}_{k}\|_{p}^{n}\leqslant K$ converges to $S_{x_{\Omega}}(R_{\Omega})$ in $C^{1,q}$ topology for any $q<1-\frac{p}{n}$.

Remark 7**.**

The estimates on $d_{L}$ and $d_{H}$ in Theorem 4 are sharp with respect of the exponent of $\delta(\Omega)$ involved, but not for what concern the constant $C(n,p,K)$. We show it by constructing example at the end of section 6. Note moreover that in the case $p=\infty$ we recover the same exponent as in the convex case.

1.5. Stability of the Chavel Inequality

In the last part of this paper we answer a question asked by Bruno Colbois concerning the almost extremal hypersurfaces for the Chavel’s inequality: if we set $\lambda_{1}^{\Sigma}$ the first nonzero eigenvalue of a compact hypersurface $\Sigma$ that bounds a domain $\Omega$, Chavel’s inequality says that

[TABLE]

Moreover equality holds if and only if $\Sigma$ is a geodesic sphere. Now if we denote by $\gamma(\Omega)$ the deficit of Chavel’s inequality (i.e. \gamma(\Omega)=\frac{n}{\lambda_{1}^{\Sigma}(n+1)^{2}}\Bigl{(}\frac{{\mathcal{H}}^{n}(\Sigma)}{|\Omega|}\Bigr{)}^{2}-1), we have

Theorem 5**.**

Let $\Sigma$ be an embedded compact hypersurface bounding a domain $\Omega$ in $\mathbb{R}^{n+1}$. If $\gamma(\Omega)\leqslant\frac{1}{C(n)}$ then we have

[TABLE]

Consequently, $\delta(\Omega)$ can be replaced by $\gamma(\Omega)^{\frac{1}{2}}$ in all the previous theorems, which gives the stability of the Chavel’s inequality. Note moreover that $\gamma$ small implies readily that $\Sigma=\partial\Omega$ is connected and so we have $N=0$ and $I=\emptyset$ is this case.

2. Preliminaries

2.1. Definitions

First let us introduce some notations and recall some definitions used in the paper. Throughout the paper we adopt the notation that $C(n,k,p,\cdots)$ is function which depends on $p$, $q$, $n$, $\cdots$. It eases the exposition to disregard the explicit nature of these functions. The convenience of this notation is that even though $C$ might change from line to line in a calculation it still maintains these basic features.

Given two bounded sets $A$ and $B$ the Hausdorff distance between $A$ and $B$ is defined by

[TABLE]

where for any subset $E$, $E_{\varepsilon}=\{x\in\mathbb{R}^{n+1}\mid{\mathrm{d}ist}(x,E)\leqslant\varepsilon\}$.

Let $\mu$ be a $\mathbb{R}^{n+1}$-valued Borel measure on $\mathbb{R}^{n+1}$. Its total variation is the nonnegative measure $|\mu|$ defined on any Borel set $\Omega$ by

[TABLE]

Given a Borel set $\Omega$ of $\mathbb{R}^{n+1}$, we say that $\Omega$ is of finite perimeter if the distributional gradient $D\chi_{\Omega}$ of its characteristic function is a $\mathbb{R}^{n+1}$-valued Borel measure such that $|D\chi_{\Omega}|(\mathbb{R}^{n+1})<\infty$. The perimeter of $\Omega$ is then $P(\Omega):=|D\chi_{\Omega}|(\mathbb{R}^{n+1})$. Of course if $\Omega$ is a bounded domain with a smooth boundary we have $P(\Omega)={\mathcal{H}}^{n}(\partial\Omega)$. For any set $\Omega$ with finite perimeter, we have P(\Omega)={\mathcal{H}}^{n}\bigl{(}\mathcal{F}(\Omega)\bigr{)} where $\mathcal{F}(\Omega)$ is the reduced boundary defined by

[TABLE]

Moreover Federer (see [1]) proved that $\mathcal{F}(\Omega)\subset\partial^{\star}\Omega$ where $\partial^{\star}\Omega$ is the essential boundary of $\Omega$ defined by

[TABLE]

where $\Omega^{t}:=\displaystyle\left\{x\in\mathbb{R}^{n+1}\ \mid\ \lim_{r\longrightarrow 0}\frac{|\Omega\cap B_{x}(r)|}{|B_{x}(r)|}=t\right\}$.

2.2. Some results proved in [10]

Now we gather some results proved in [10] about almost isoperimetric sets, that will be used in this paper.

Theorem 6**.**

(A. Figalli, F. Maggi, A. Pratelli, [10]) Let $\Omega$ be a set of $\mathbb{R}^{n+1}$ of finite perimeter, with $0<|\Omega|<\infty$ and $\delta(\Omega)\leqslant\min\left(1,\displaystyle\frac{k(n)^{2}}{8}\right)$ where $k(n):=\displaystyle\frac{2-2^{\frac{n}{n+1}}}{3}$. Then there exists a domain $G\subset\Omega$ such that

1. (1)

$0\leqslant|\Omega|-|G|\leqslant|\Omega\setminus G|\leqslant\frac{\delta(\Omega)}{k(n)}|\Omega|$, 2. (2)

$P(G)\leqslant P(\Omega)$, 3. (3)

$\delta(G)\leqslant\frac{3}{k(n)}\delta(\Omega)$, 4. (4)

There exists a point $x_{\Omega}\in\mathbb{R}^{n+1}$ such that

[TABLE]

where $X$ is the vector position of $\mathbb{R}^{n+1}$, 5. (5)

$|G\Delta B_{x_{\Omega}}(R_{G})|\leqslant\frac{20(n+1)^{3}}{k(n)}|G|\delta(\Omega)^{1/2}$.

The following property is important for our purpose and derive easily from [10], but since it is not proved nor stated in [10], we give a proof of it for sake of completeness.

Lemma 1**.**

There exists a constant $C(n)>0$ such that under the assumptions and notations of the previous theorem, we have

[TABLE]

Proof.

We reuse the notations of [10]. First of all, by the previous theorem, we have

[TABLE]

and by the construction made in [10], $\Omega$ is the disjoint union of $G$ and a set $F_{\infty}$ which satisfy

[TABLE]

Then we have

[TABLE]

and

[TABLE]

where we have used Inequality (2.1). We infer that

[TABLE]

∎

2.3. Proof of remark 1

Up to a translation we can assume that $x_{\Omega}=0$ and from the Theorem 6 we have :

[TABLE]

Since

[TABLE]

we deduce immediately that $|\Omega\Delta B_{0}(R_{\Omega})|\leqslant C(n)|\Omega|\delta(\Omega)^{1/2}$ which proves the point (1) of the remark.

On the other hand let $x\in B_{0}(R_{\Omega})$ and $R_{\Omega}\geqslant\varepsilon>0$ such that

[TABLE]

We then have $B_{x}(\varepsilon)\cap B_{0}(R_{\Omega})\subset\Omega\Delta B_{0}(R_{\Omega})$ and since $B_{x}(\varepsilon)\cap B_{0}(R_{\Omega})$ contains the ball with diameter $\mathbb{R}x\cap B_{x}(\varepsilon)\cap B_{0}(R_{\Omega})$ whose length is larger than $\varepsilon$, we get

[TABLE]

Since $\Omega\cap B_{0}(R_{\Omega})\subset B_{0}(R_{\Omega})$, it suffices to get the point (2) that is

[TABLE]

3. Concentration in a tubular neighborhood of a sphere

The main result of this section is the following theorem :

Lemma 2**.**

Let $\Omega$ be a set of $\mathbb{R}^{n+1}$ with finite perimeter and let

[TABLE]

If $\delta(\Omega)\leqslant\frac{1}{C(n)}$ then for any $\alpha\in(0,\frac{1}{2})$, we have

[TABLE]

Proof.

By inequalities (4) and (1) of Theorem 6, we get

[TABLE]

where we have used that {\mathcal{H}}^{n}\bigl{(}\mathcal{F}(G)\bigr{)}=P(G)=C(n)\bigl{(}1+\delta(G)\bigr{)}|G|^{\frac{n}{n+1}}\leqslant C(n)|\Omega|^{\frac{n}{n+1}} (by Theorem 6 (1) and (3)).

Now by Lemma 1 and Inequality (2) of Theorem 6, we have

[TABLE]

And so

[TABLE]

Then choosing $\eta:=\delta(\Omega)^{\alpha}$ and $\delta(\Omega)\leqslant 1$ we get the desired result. ∎

4. Domains with small deficit without assumption on the boundary

In this section, we gather the proofs of several geometric-measure properties of the boundary of almost isoperimetric domains.

4.1. Proof of Theorem 1

By Lemma 2, we have Inequality (1) with $A(\Omega)=\mathcal{F}(\Omega)\cap A_{\delta(\Omega)^{1/4}}$. Inequality (2) will be a consequence of the following density theorem.

Theorem 7**.**

Let $\Omega$ be a set of $\mathbb{R}^{n+1}$ with finite perimeter and \rho\in\bigl{[}C(n)\delta(\Omega)^{\frac{1}{8}}R_{\Omega},R_{\Omega}\bigr{]}. Then for any $x\in S_{x_{\Omega}}(R_{\Omega})$ we have

[TABLE]

Let $x\in S_{x_{\Omega}}(R_{\Omega})$ and $\rho=C_{1}(n)R_{\Omega}\delta(\Omega)^{\beta(n)}$ with $\beta(n):=\min(\frac{1}{8},\frac{1}{4n})$. Then for $C_{1}(n)$ large enough and $\delta(\Omega)\leqslant(1/C_{1}(n))^{1/\beta(n)}$, \rho\in\bigl{[}C(n)\delta(\Omega)^{\frac{1}{8}}R_{\Omega},R_{\Omega}\bigr{]} and the estimate of Theorem 7 combined to the fact that there exists a constant $C_{2}(n)$ such that

[TABLE]

gives for $C_{1}(n)$ great enough

[TABLE]

Moreover from the lemma 2 we have

[TABLE]

If $C_{1}(n)$ is large enough. This implies that $\mathcal{F}(\Omega)\cap A_{\delta(\Omega)^{1/4}}\cap B_{x}(\rho)$ has non-zero measure, hence is non-empty for any $x\in S_{x_{\Omega}}(R_{\Omega})$. Putting $A(\Omega)=\mathcal{F}(\Omega)\cap A_{\delta(\Omega)^{1/4}}$, we obtain that $d_{H}(A(\Omega),S_{x_{\Omega}}(R_{\Omega}))\leqslant\rho$ for $C_{1}(n)$ large enough which gives the fact (2) of Theorem 1.

Note that Theorem 7 implies that density of $\mathcal{F}(\Omega)$ near each point of $\mathbb{S}^{n}(R_{\Omega})$ converges to 1 at any fixed scale. It will be combined with Allard’s regularity theorem in Section 6 to prove Theorem 4.

4.2. Proof of Theorem 7

It will be a consequence of the following fundamental proposition.

Proposition 1**.**

Let $\Omega$ be a set of $\mathbb{R}^{n+1}$ of finite perimeter, with $\delta(\Omega)\leqslant\frac{1}{C(n)}$. For any $f\in C^{1}_{c}(\mathbb{R}^{n+1})$, we have

[TABLE]

where we denote $\|df\|_{\infty}=\displaystyle\sup_{y}|d_{y}f(y)|$.

Proof.

Up to translation, we can assume that $x_{\Omega}=0$ subsequently. Let $G\subset\Omega$ be the subset associated to $\Omega$ in Theorem 6. We note $X$ the field $X_{x}=x$ for any $x\in\Omega$. We have ${\mathrm{d}iv}_{x}(fX)=df_{x}(X_{x})+(n+1)f(x)$ and so we get

[TABLE]

Where we have used Inequality (5) of Theorem 6. Now we have

[TABLE]

Now a straightforward computation shows that \delta(G)=\frac{R_{G}{\mathcal{H}}^{n}\bigl{(}\mathcal{F}(G)\bigr{)}}{(n+1)|G|}-1. Consequently

[TABLE]

Combining Inequalities (4.1) and (4.2) gives

[TABLE]

Where we have used Inequality (1) of Theorem 6 to get

[TABLE]

We have

[TABLE]

with

[TABLE]

Note that $A_{2}$ is controlled by Inequality (4.3). Let us now estimate $A_{1}$. By Lemma 1 we have

[TABLE]

where once again we have used the estimates of Theorem 6. ∎

Proof of the theorem 7: Up to translation, we can assume that $x_{\Omega}=0$. Let $\rho\leqslant R_{\Omega}$. By Lemma 2, we have

[TABLE]

We set $\eta=\delta(\Omega)^{\frac{1}{4}}\leqslant\frac{1}{2}$ and $\varphi:[0,+\infty)\to[0,1]$ be a $C^{1}$ function with compact support in $(0,2R_{\Omega})$, $\frac{2}{R_{\Omega}}$-Lipschitz and such that $\varphi(t)=1$ on \bigl{[}R_{\Omega}(1-\eta),R_{\Omega}(1+\eta)\bigr{]}. For any function v\in C^{1}\bigl{(}S_{0}(R_{\Omega})\bigr{)}, we set $f(x)=\varphi(|x|)v(\frac{R_{\Omega}x}{|x|})$. Then $|df_{x}(x)|\leqslant 4\|v\|_{\infty}$ and applying Proposition 1 to $f$, we get

[TABLE]

Let $x\in S_{0}(R_{\Omega})$ and $v_{r}$ be the characteristic function of the geodesic ball of center $x$ and radius $r$ in $S_{0}(R_{\Omega})$. By convolution, we can approximate $v_{r}$ in $L^{1}(S_{0}(R_{\Omega}))$ by $C^{1}$ functions $u_{k}$ such that $\|u_{k}\|_{\infty}\leqslant 1$. Applying Inequality (4.6) to $v=u_{k}$ and letting $k$ tends to $\infty$, we get

[TABLE]

where $f_{r}=\varphi_{R_{\Omega}}(\|x\|)v_{r}(\frac{R_{\Omega}x}{\|x\|})$ and where \mathcal{C}_{\alpha}=\bigl{\{}y\in\mathbb{R}^{n+1}\setminus\{0\}/\,\langle\frac{y}{\|y\|},\frac{x}{\|x\|}\rangle\geqslant\cos\alpha\bigr{\}}. Now, since $\|f_{r}\|_{\infty}\leqslant 1$, Lemma 2 gives us

[TABLE]

By construction of $f_{r}$, we have

[TABLE]

Combining Inequalities (4.7), (4.8) and (4.9), we get

[TABLE]

We now assume that $\delta(\Omega)^{\frac{1}{4}}\leqslant\frac{\rho^{2}}{2R_{\Omega}^{2}}$. The following angles

[TABLE]

satisfy the following property (see figure 1)

[TABLE]

where we have set $C_{\mathrm{i}nt}=C_{{\alpha}_{\mathrm{i}nt}}$ and $C_{\mathrm{e}xt}=C_{{\alpha}_{\mathrm{e}xt}}$, so we get the following inequalities

[TABLE]

Since we have $B_{x}(\rho)\cap S_{0}(R_{\Omega})=\mathcal{C}_{{\alpha}_{\rho}}\cap S_{0}(R_{\Omega})$ for ${\alpha}_{\rho}=\arccos(1-\frac{\rho^{2}}{2R_{\Omega}^{2}})$, we infer the estimate

[TABLE]

Now, by the Bishop’s and Bishop-Gromov’s theorems, we have

[TABLE]

and

[TABLE]

that is

[TABLE]

where $B_{x}^{S_{0}(R_{\Omega})}(r)$ denotes the ball of center $x$ and radius $r$ in $S_{0}(R_{\Omega})$. These inequalities give

[TABLE]

Since by assumption $\delta(\Omega)^{\frac{1}{4}}\leqslant\frac{\rho^{2}}{2R_{\Omega}^{2}}$, we get $|{\alpha}_{\mathrm{e}xt}-{\alpha}_{\mathrm{i}nt}|\leqslant C(n)\delta(\Omega)^{\frac{1}{4}}$ which gives

[TABLE]

Finally, by Lemma 2, we have

[TABLE]

which gives

[TABLE]

4.3. A control of the unit normal to $\mathcal{F}(\Omega)$

In this subsection, we prove a result that we will use latter. It gives a weak control of the oscillation of the tangent planes of $\mathcal{F}(\Omega)$. Note that another proof of this result is proposed in [14].

Lemma 3**.**

Let $\Omega$ be a set of finite perimeter such that $\delta(\Omega)\leqslant\frac{1}{C(n)}$. Then we have

[TABLE]

Proof.

By Lemma 1 and the fact that $\nu_{\Omega}=\nu_{G}$ ${\mathcal{H}}^{n}$-almost everywhere in $\mathcal{F}(G)\cap\mathcal{F}(\Omega)$, we have

[TABLE]

Now, we have

[TABLE]

and by inequality (4.2), we have that

[TABLE]

where the last inequality comes from fact (4) of Theorem 6. ∎

4.4. A stability result involving the Preiss distance

First we recall the definition of the Preiss distance on Radon measures of $\mathbb{R}^{n+1}$.

Definition 1**.**

Let $\mu$ and $\nu$ be two Radon measures on $\mathbb{R}^{n+1}$, for any $i\in\mathbb{N}$, we set

[TABLE]

and

[TABLE]

it gives a distance on the Radon measure of $\mathbb{R}^{n+1}$ whose converging sequences are the weakly⋆ converging sequences.

For almost isoperimetric domains we have a control on the boundary in term of Preiss distance

Theorem 8**.**

Let $\Omega$ be a set of $\mathbb{R}^{n+1}$ with finite perimeter. Then there exists $x_{\Omega}\in\mathbb{R}^{n+1}$ such that

[TABLE]

where $d_{P}$ is the Preiss distance on Radon measures of $\mathbb{R}^{n+1}$.

Proof.

Note that if $f$ has support in $B_{0}(i)$ and is $1$-Lipschitz, then by convolution, it can be uniformly approximated by a sequence of $1$-Lipschitz, $C^{1}$ and compactly supported functions $(f_{k})$. We then have $\displaystyle\lim_{k}\|f_{k}\|_{\infty}=\|f\|_{\infty}\leqslant i$ and $\displaystyle\lim_{k}|d_{X}f_{k}(X)|\leqslant\displaystyle\lim_{k}\|df_{k}\|_{\infty}i\leqslant i$ and applying Proposition 1 to $f_{k}$ and letting $k$ tends to $\infty$ gives us

[TABLE]

and so

[TABLE]

Hence we get that if $\delta(\Omega)\leqslant\frac{1}{C(n)}$, then we have

[TABLE]

Since for any couple of measures $\mu,\nu$ we have $d_{P}(\mu,\nu)\leqslant 2$, we infer that we can leave the condition $\delta(\Omega)\leqslant\frac{1}{C(n)}$ as soon as we consider a larger $C(n)$. ∎

5. Domains with small deficit and $\|{\mathrm{H}}\|_{p}$ bounded in the case $p\leqslant n$

5.1. Proof of Theorems 2 and 3

These theorems are consequence of the following.

Theorem 9**.**

(E. Aubry, J.-F. Grosjean, [3]) There exists a (computable) constant $C=C(m)$ such that, for any compact submanifold $M^{m}$ of $\mathbb{R}^{n+1}$ and any closed subset $A\subset M$ that intercepts any connected component of $M$, there exists a finite family $(T_{i})_{\in I}$ of geodesic trees in $M$ with $A\cap T_{i}\neq\emptyset$ for any $i\in I$, d_{H}\bigl{(}A\cup\displaystyle\bigcup_{i\in I}T_{i},M\bigr{)}\leqslant C\bigl{(}{\mathrm{V}ol}\,(M\setminus A)\bigr{)}^{\frac{1}{m}} and $\displaystyle\sum_{i\in I}{\mathcal{H}}^{1}(T_{i})\leqslant C^{m(m-1)}\displaystyle\int_{M\setminus A}|{\mathrm{H}}|^{m-1}$.

Remark 8**.**

Note that by construction the $A\cup\displaystyle\bigcup_{i\in I}T_{i}$ has the same number of connected components than $A$.

Proof Theorems 2 and 3 : We set $\partial_{r}\Omega$ the union of the connected components of $\partial\Omega$ that intercept $A_{\delta(\Omega)^{\frac{1}{4}}}$ and we apply Theorem 9 to the hypersurface $\partial_{r}\Omega$ and the set $A_{0}=\partial\Omega\cap A_{\delta(\Omega)^{1/4}}=\partial_{r}\Omega\cap A_{\delta(\Omega)^{1/4}}$. We set $T_{0}$ the union of the trees given by the theorem. Then we get ${\mathcal{H}}^{1}(T_{0})\leqslant C(n)\displaystyle\int_{\partial_{r}\Omega\setminus A_{0}}|{\mathrm{H}}|^{n-1}$, the set $A_{\delta(\Omega)^{1/4}}\cup T_{0}$ is connected and by the first point of Theorem 1 (or Lemma 2) and Theorem 9, we have

[TABLE]

If we now apply Theorems 9 and Theorem 1 to each connected component $C_{i}$ of $\partial\Omega\setminus\partial_{r}\Omega$ with $A_{i}=\{x_{i}\}\subset C_{i}$, we get a connected union of trees $T_{i}$ such $d_{H}(T_{i},C_{i})\leqslant C(n)R_{\Omega}\delta(\Omega)^{\frac{1}{4n}}$ and ${\mathcal{H}}^{1}(T_{i})\leqslant C(n)\displaystyle\int_{C_{i}}|{\mathrm{H}}|^{n-1}d{\mathcal{H}}^{n}$. If we set $T=T_{0}\cup\displaystyle\bigcup_{i\in I}T_{i}$, then we have ${\mathcal{H}}^{1}(T)\leqslant C(n)\displaystyle\int_{\partial\Omega\setminus A_{\delta(\Omega)^{\frac{1}{4}}}}|{\mathrm{H}}|^{n-1}d{\mathcal{H}}^{n}$ and

[TABLE]

the last inequality comes from Theorem 1. This completes the proof of Theorem 2.

Now to prove Theorem 3, we have

[TABLE]

To finish the proof of Theorem 3 we just have to use Hölder’s inequality and Lemma 2. For what concerns cardinality of $I$, remark that the Michael-Simon Inequality applied to the function $f=1$ and to any connected component $C$ of $\partial\Omega\setminus\partial_{r}\Omega$ gives us

[TABLE]

and so $\int_{C}|{\mathrm{H}}|^{n}d{\mathcal{H}}^{n}\geqslant\frac{1}{C(n)}$ for any connected component of $\partial\Omega\setminus\partial_{r}\Omega$. We infer that

[TABLE]

we conclude for any $p\geqslant n$ by Hölder inequality and Lemma 2.

5.2. Variants of Theorems 2 and 3 that generalize inequality (1.5)

Theorem 10**.**

Let $\Omega$ be an open set with smooth boundary, finite perimeter and $\delta(\Omega)\leqslant\frac{1}{C(n)}$. There exists a subset $T\subset\mathbb{R}^{n+1}$ with

1. (1)

\mathcal{H}^{1}(T)\leqslant C(n)R_{\Omega}\int_{\partial\Omega\setminus B_{x_{\Omega}}\bigl{(}R_{\Omega}(1+\delta(\Omega)^{\frac{1}{4}})\bigr{)}}|{\mathrm{H}}|^{n-1}d{\mathcal{H}}^{n}, 2. (2)

d_{H}\bigl{(}\Omega,B_{x_{\Omega}}(R_{\Omega})\cup T\bigr{)}\leqslant C(n)R_{\Omega}\delta(\Omega)^{\beta(n)}, 3. (3)

the set B_{x_{\Omega}}\bigl{(}R_{\Omega}(1+\delta(\Omega)^{\frac{1}{4}})\bigr{)}\cup T has at most $N+1$ connected components,

where $N$ is the number of connected components of $\partial\Omega$ that do not intercept the ball B_{x_{\Omega}}\bigl{(}R_{\Omega}(1+\delta(\Omega)^{\frac{1}{4}})\bigr{)}.

Theorem 11**.**

Let $p\geqslant n-1$ and $\Omega$ be an open set with smooth boundary $\partial\Omega$, finite perimeter and $\delta(\Omega)\leqslant\frac{1}{C(n)}$. Let $(\partial\Omega_{i})_{i\in I}$ be the connected components of $\partial\Omega$ that do not intercept the ball B_{x_{\Omega}}\bigl{(}R_{\Omega}(1+\delta(\Omega)^{\frac{1}{4}})\bigr{)}. For any $i\in I$, there exists $x_{i}\in\partial\Omega_{i}$ such that

[TABLE]

Moreover if $p\geqslant n$ and ${\mathrm{H}}$ is $L^{p}$-integrable then $I$ is of finite cardinal $N$ and we have

[TABLE]

Remark 9**.**

The norm $\|{\mathrm{H}}\|_{p}$ can be replaced by \left(\frac{1}{P(\Omega)}\displaystyle\int_{\partial\Omega\setminus B_{x_{\Omega}}\bigl{(}R_{\Omega}(1+\delta(\Omega)^{\frac{1}{4}})\bigr{)}}|{\mathrm{H}}|^{p}d{\mathcal{H}}^{n}\right)^{\frac{1}{p}}.

Proof of Theorems 10 and 11 : We set $\partial_{r}\Omega$ the union of the connected components of $\partial\Omega$ that intercept B_{x_{\Omega}}\bigl{(}R_{\Omega}(1+\delta(\Omega)^{\frac{1}{4}}) and then we construct $T$ as in the previous section. Arguing as in the previous subsection, we get that the $C(n)R_{\Omega}\delta(\Omega)^{\beta(n)}$-tubular neighbourhood of $B_{x_{\Omega}}(R_{\Omega})\cup T$ contains \partial\Omega\setminus B_{x_{\Omega}}\bigl{(}R_{\Omega}(1+\delta(\Omega)^{\frac{1}{4}})\bigr{)}. We set with $R^{\prime}_{\Omega}=R_{\Omega}(1+2C(n)\delta(\Omega)^{\frac{1}{2(n+1)}})$ (where $C(n)$ is the constant of Remark 1 (2)). Then for any $x\in\Omega$, either we have $x\in B_{x_{\Omega}}(R^{\prime}_{\Omega})$ and then d\bigl{(}x,B_{x_{\Omega}}(R_{\Omega})\cup T\bigr{)}\leqslant 2C(n)R_{\Omega}\delta(\Omega)^{\frac{1}{2(n+1)}}, either we have $x\in\Omega\setminus B_{x_{\Omega}}(R^{\prime}_{\Omega})$, and then $x\in\Omega\Delta B_{x_{\Omega}}(R_{\Omega})$. From the Remark 1 (1), we infer (as in the proof of Remark 1 (2)) that

[TABLE]

and even more precisely, d\bigl{(}x,\partial\Omega\setminus B_{x_{\Omega}}\bigl{(}R_{\Omega}(1+\delta(\Omega)^{\frac{1}{4}}))\bigr{)}\leqslant C(n)R_{\Omega}\delta(\Omega)^{\frac{1}{2(n+1)}}. We infer that we have

[TABLE]

On the other hand, for any $x\in B_{x_{\Omega}}(R_{\Omega})\cup T$, either $x\in B{x_{\Omega}}(R_{\Omega})$ and then $d(x,\Omega)\leqslant C(n)R_{\Omega}\delta(\Omega)^{\frac{1}{2(n+1)}}$ by Remark 1 (2), either $x\in T$ and then $d(x,\Omega)=0$. We then get

[TABLE]

which gives the result as in the proofs of Theorems 2 and 3.

6. A quasi-isometry result : proof of Theorem 4

Let us first remind Duggan’s version of Allard’s local regularity theorem about hypersurface of suitably bounded mean curvature.

Theorem 12** (J.P. Duggan [9]).**

If $p>n$ is arbitrary, then there are $\eta=\eta(n,p)$, $\gamma=\gamma(n,p)\in(0,1)$ and $c=c(n,p)$ such that if $M\subset\mathbb{R}^{n+1}$ is a hypersurface, $x\in M$ and $\rho>0$ satisfy the hypotheses

1. (1)

${\mathcal{H}}^{n}(B_{x}(\rho)\cap M)\leqslant(1+\eta)\rho^{n}|\mathbb{B}^{n}|$ 2. (2)

$\rho^{p-n}\displaystyle\int_{B_{x}(\rho)\cap M}|{\mathrm{H}}|^{p}d{\mathcal{H}}^{n}\leqslant\eta^{p}$**

then there exists a linear isometry $q$ of $\mathbb{R}^{n+1}$ and $u\in W^{2,p}(B_{0}^{\mathbb{R}^{n}}(\gamma\rho))$ with $u(0)=0$, M\cap B_{x}(\gamma\rho)=\bigl{(}x+q(graph\ u)\bigr{)}\cap B_{x}(\gamma\rho) and

[TABLE]

So the Morrey-Campanato says that for any $v\in W^{1,p}(B_{0}^{\mathbb{R}^{n}}(1))$ we have

[TABLE]

Up to a normalization and under the assumptions of Theorem 12, the Morrey-Campanato theorem gives us that

[TABLE]

Now let $\Phi:B_{0}^{\mathbb{R}^{n}}(\gamma\rho)\rightarrow\mathbb{R}^{n+1}\ ,\ a\mapsto q(a,u(a))$. Then $d\Phi_{a}(h)=q(h,du_{a}(h))$. Since $q$ is an isometry, a unit normal is given by \nu_{\Phi(a)}=q\bigl{(}\frac{((\nabla u)\mid_{a},-1)}{\sqrt{1+|\nabla u\mid_{a}|^{2}}}\bigr{)} which gives for any $x\in\partial\Omega$

[TABLE]

Lemma 4**.**

Let $p>n$. There exist $3$ positive constants $C_{1}(n,p)$, $C_{2}(n,p)$ and $C_{3}(n,p)$ such that for any domain $\Omega$ with smooth boundary $\partial\Omega$ satisfying $P(\Omega)\left\|{\mathrm{H}}\right\|_{p}^{n}\leqslant K$, and $\delta(\Omega)\leqslant\frac{1}{C_{1}(n,p)K^{\alpha(n,p)}}$, we have

[TABLE]

and the assumptions of Theorem 12 are satisfied for $\bar{\rho}=\frac{R_{\Omega}}{C_{3}(n,p)K^{\frac{p}{n(p-n)}}}$. Moreover we have

[TABLE]

Where $Z_{x}=\frac{x-x_{\Omega}}{|x-x_{\Omega}|}-\nu_{x}$. Here we have set $\alpha(n,p)=\frac{8p}{p-n}$.

Proof.

Since the computations are a bit messy, we organize them in several steps:

1. (1)

For what concern the point (2) of Theorem 12, we have for any $\rho>0$

[TABLE]

From (1.1) and the definition of $R_{\Omega}$, we have $R_{\Omega}^{n}\leqslant C(n)P(\Omega)$ and so

[TABLE]

From this we deduce that there exists a constant $C_{3}(n,p)$ large enough such that $\partial\Omega$ satisfies assumption (2) of Theorem 12 for $\rho=\bar{\rho}=\frac{R_{\Omega}}{C_{3}(n,p)K^{\frac{p}{n(p-n)}}}$. 2. (2)

Let $x\in\mathbb{S}^{n}$ then there exists a $r(n,p)\in]0,1]$ such that we have $\frac{{\mathcal{H}}^{n}(B_{x}(r)\cap\mathbb{S}^{n})}{|\mathbb{B}^{n}|r^{n}}\in[1/2,1+\eta(n,p)/4]$, for any $r<r(n,p)$, where $\eta(n,p)$ is the constant of Theorem 12. By Michael-Simon Sobolev inequality, we have $K\geqslant P(\Omega)\|{\mathrm{H}}\|_{p}^{n}\geqslant k(n)$, and so we can assume $C_{3}(n,p)$ large enough to have $\bar{\rho}/R_{\Omega}\leqslant r(n,p)\leqslant 1$.

From now on $C_{3}(n,p)$ is fixed so that it satisfies both the two previous conditions. 3. (3)

Since $K\geqslant k(n)$, we can assume $C_{1}(n,p)$ large enough for $\delta(\Omega)\leqslant\frac{1}{C_{1}(n,p)K^{\alpha(n,p)}}$ to imply that \delta(\Omega)\leqslant\min\bigl{(}\frac{\eta}{\eta+4},(\frac{|\mathbb{B}^{n}|(C(n))^{n-1}\eta}{8{\mathcal{H}}^{n}(\mathbb{S}^{n})})^{8}\bigr{)}\leqslant 1 in what follows, where $C(n)$ is the constant of Theorem 7. 4. (4)

From Theorem 3, the number $N$ of connected components of $\partial\Omega$ that do not intercept $A_{\delta(\Omega)^{1/4}}$ satisfies

[TABLE]

So, when $\delta(\Omega)\leqslant\frac{1}{(2C(n,p)K)^{\alpha(n,p)}}$, we have $N=0$. We infer by Theorem 3

[TABLE]

which gives inequality (6.3) for any $C_{1}(n,p)\geqslant(2C(n,p))^{\alpha(n,p)}$ such that the previous condition (3) also holds. Note that we have used $\delta(\Omega)\leqslant 1$. At this stage, $C_{2}(n,p)$ is fixed, and does not depends on $C_{1}(n,p)$. 5. (5)

Similarly for $C_{1}(n,p)$ large enough and $\delta(\Omega)\leqslant\frac{1}{C_{1}K^{\alpha}}$, we have from the previous point that

[TABLE]

From this we deduce that

[TABLE]

which gives with 6.2 the inequality 6.4. 6. (6)

We want to apply Theorem 7 to $\partial\Omega$ and $B_{x}(\bar{\rho})$ and so need $\bar{\rho}\in[C(n)\delta(\Omega)^{1/8}R_{\Omega},R_{\Omega}]$. Note that $\bar{\rho}\leqslant R_{\Omega}$ was already obtained in (2). On the other hand, we have that

[TABLE]

Now it is clear that for $C_{1}(n,p)$ large enough, we have $\frac{\bar{\rho}}{R_{\Omega}}\geqslant C(n)\delta(\Omega)^{\frac{1}{8n}}\geqslant C(n)\delta(\Omega)^{\frac{1}{8}}$. 7. (7)

Now we prove that for $C_{1}(n,p)$ large enough, $\partial\Omega$ satisfies (1) for $\bar{\rho}=\frac{R_{\Omega}}{C_{3}(n,p)K^{\frac{p}{n(p-n)}}}$ with $C_{3}(n,p)$ fixed in (2). Let $x\in S_{x_{\Omega}}(R_{\Omega})$. Then Theorem 7 gives us

[TABLE]

where $x^{\prime}=x_{\Omega}+\frac{1}{R_{\Omega}}(x-x_{\Omega})$. Now by the condition (2) above, we have

[TABLE]

and so

[TABLE]

where we have used the fact that $\frac{P(\Omega)}{R_{\Omega}^{n}{\mathcal{H}}^{n}(\mathbb{S}^{n})}=1+\delta(\Omega)\leqslant 2$, and $\dfrac{R_{\Omega}^{n}}{\bar{\rho}^{n}}\leqslant\dfrac{1}{C(n)^{n}\delta(\Omega)^{1/8}}$ proved in (5). Now from the condition \delta(\Omega)\leqslant\min\bigl{(}\frac{\eta}{\eta+4},(\frac{|\mathbb{B}^{n}|(C(n))^{n-1}\eta}{8{\mathcal{H}}^{n}(\mathbb{S}^{n})})^{8}\bigr{)} we deduce that

[TABLE]

∎

Now, using Duggan’s regularity theorem, we can show a Calderon-Zygmund property of almost isoperimetric manifolds with $L^{p}$ bounded mean curvature:

Lemma 5**.**

Let $p>n$. There exists $C(n,p)>0$ such that for any domain $\Omega$ with smooth boundary $\partial\Omega$ satisfying $P(\Omega)\left\|{\mathrm{H}}\right\|_{p}^{n}\leqslant K$ and $\delta(\Omega)\leqslant\frac{1}{C(n,p)K^{\alpha(n,p)}}$ we have

[TABLE]

Remark 10**.**

We can improve the proof below to get $P(\Omega)\|{\mathrm{B}}\|_{p}^{n}\leqslant C(n,p)K^{\frac{p}{p-n}}$.

Proof.

Let $(x_{i})_{i}$ be a maximal family of points of $\partial\Omega$ such that the balls $B_{x_{i}}(\gamma\bar{\rho}/2)$ are disjoints in $\mathbb{R}^{n+1}$. Then the family \bigl{(}\partial\Omega\cap B_{x_{i}}(\gamma\bar{\rho})\bigr{)}_{i} covers $\partial\Omega$. By (6.3), all the balls $B_{x_{i}}(\gamma\bar{\rho}/2)$ are included in $B_{x_{i}}\left(\left(\frac{\gamma}{2C_{3}k^{\frac{p}{p-n}}}+\frac{C_{2}}{C_{1}^{\beta}k^{\alpha\beta}}+1\right)R_{\Omega}\right)$ and for $C_{1}$ and $C_{3}$ large enough, they are included in $B_{x_{\Omega}}(3R_{\Omega})$. And so the family has at most $(\frac{6R_{\Omega}}{\gamma\bar{\rho}})^{n+1}\leqslant C(n,p)K^{\frac{(n+1)p}{n(p-n)}}$ elements (note that using the fact that $\partial\Omega$ is Hausdorff close to $S_{x_{\Omega}}(R_{\Omega})$ we could replace $K^{\frac{(n+1)p}{n(p-n)}}$ by the better $K^{\frac{p}{p-n}}$).

By Theorem 12, denoting by $u_{i}$ each corresponding function we then have $|B|\leqslant\sqrt{n}\frac{|d^{2}u_{i}|}{\sqrt{1+|du_{i}|^{2}}}$ on $\partial\Omega\cap B_{x_{i}}(\gamma\bar{\rho})$ and

[TABLE]

from which we get

[TABLE]

∎

Using Duggan’s Theorem we now improve the $L^{2}$ smallness of $Z$ given by Lemma 3 in an $L^{\infty}$ one.

Lemma 6**.**

Let $p>n$. There exists $C(n,p)>0$ such that for any domain $\Omega$ with smooth boundary $\partial\Omega$ satisfying $P(\Omega)\left\|{\mathrm{H}}\right\|_{p}^{n}\leqslant K$ and $\delta(\Omega)\leqslant\frac{1}{C(n,p)K^{\alpha(n,p)}}$, we have

[TABLE]

Here $\alpha(n,p)$ is the same as in Lemma 4.

Proof.

Let

[TABLE]

where $C(n)$ is the constant of Theorem 7 and $C_{2}(n,p)$ is the constant of Lemma 4. We set $\rho^{\prime}=2\gamma C_{4}(n,p)\delta(\Omega)^{\frac{1}{8n}}R_{\Omega}$.

Assume now that $\delta(\Omega)\leqslant\frac{1}{C_{1}^{\prime}(n,p)K^{\alpha(n,p)}}$ where $C_{1}^{\prime}\geqslant C_{1}$. For $C_{1}^{\prime}(n,p)$ large enough we have $\delta(\Omega)\leqslant\frac{1}{C_{1}^{\prime}(n,p)K^{\alpha(n,p)}}\leqslant\frac{1}{(2C_{4}C_{3})^{8n}K^{\alpha(n,p)}}$ and $\rho^{\prime}\leqslant\gamma\bar{\rho}$. As explained in the point (2) of the proof of Lemma 4, we can assume $C_{1}^{\prime}(n,p)$ large enough to get that

[TABLE]

where $C_{3}(n,p)$ is the constant used in the proof of Lemma 4. For any $x\in\partial\Omega$ and for any $y,z\in\partial\Omega\cap B_{x}(\rho^{\prime})$, Inequality (6.4) and the value of $\bar{\rho}$ give us

[TABLE]

Since, $K\geqslant k(n)$, we can assume $C_{1}^{\prime}(n,p)$ large enough so that Lemma 3 applies and then for any $x\in\Omega$

[TABLE]

Now let $x^{\prime}=x_{\Omega}+R_{\Omega}\frac{x-x_{\Omega}}{|x-x_{\Omega}|}\in S_{x_{\Omega}}(R_{\Omega})$. From (6.3), an easy computation shows that B_{x^{\prime}}\bigl{(}\frac{\rho^{\prime}}{2}\bigr{)}\subset B_{x}(\rho^{\prime}). Indeed if y\in B_{x^{\prime}}\bigl{(}\frac{\rho^{\prime}}{2}\bigr{)}, then

[TABLE]

From the choices made in (6.7) and (6.8) we have $\delta(\Omega)^{\beta}\leqslant\frac{\gamma C_{4}}{C_{2}}\delta(\Omega)^{1/8n}$ and $|x-y|\leqslant\rho^{\prime}$. We then get

[TABLE]

Now (6.7) and (6.8) imply that $\delta(\Omega)^{1/8n}\leqslant 1/\gamma C_{4}$ and $C_{4}\geqslant C(n)/\gamma$ which gives $\frac{\rho^{\prime}}{2}\in[C(n)\delta(\Omega)^{1/8}R_{\Omega},R_{\Omega}]$. So we can apply Theorem 7, and since we have $\frac{\rho^{\prime}}{R_{\Omega}}\leqslant r(n,p)$ (see (2) in the proof of the previous lemma), we get {\mathcal{H}}^{n}\left(B_{x}\left(\frac{\rho^{\prime}}{2R_{\Omega}}\right)\cap\mathbb{S}^{n}\right)\geqslant\frac{|\mathbb{B}^{n}|}{2}\bigl{(}\frac{\rho^{\prime}}{2R_{\Omega}}\bigr{)}^{n} and

[TABLE]

where $x^{\prime\prime}=x_{\Omega}+\frac{1}{R_{\Omega}}(x^{\prime}-x_{\Omega})$ and in the last inequality we used again (6.7). Reporting this in (6.9) we obtain

[TABLE]

which gives the desired inequality by putting $C(n,p)=\max(C_{1}^{\prime}(n,p),C_{5}(n,p))$. ∎

Since we have an upper bound on the second fundamental form, we could also perform a Moser iteration as in [3] to prove the previous lemma.

Let $\Omega$ be an almost isoperimetric domain. We consider the map $F:\partial\Omega\longrightarrow S_{x_{\Omega}}(R_{\Omega})$ defined by

[TABLE]

Proof of Theorem 4 : In this proof, $C(n,p)$ is the constant of the Lemma 6. For more convenience up to a translation we can assume $x_{\Omega}=0$. Under the assumptions of Lemma 4, we have $|x|\geqslant\frac{1}{2}R_{\Omega}$. Hence $F$ is well defined on $\partial\Omega$. Moreover, for any $x\in\partial\Omega$ and $u\in T_{x}\partial\Omega$, we have $dF_{x}(u)=\frac{R_{\Omega}}{|x|}\left(u-\frac{\langle x,u\rangle}{|x|^{2}}x\right)=\frac{R_{\Omega}}{|x|}\left(u-\langle Z_{x},u\rangle\frac{x}{|x|}\right)$ and we have

[TABLE]

Let $D(n,p)\geqslant C(n,p)$ large enough and assume $\delta(\Omega)^{1/2}\leqslant\frac{1}{DK^{\alpha}}$. Since by Inequality (6.6) of Lemma 6 we have

[TABLE]

Hence we can assume $\|Z\|_{\infty}<1/2$ for $D(n,p)$ large enough, which infer that $F$ is a local diffeomorphism form $\partial\Omega$ into $S_{0}(R_{\Omega})$. Let $\partial\Omega_{0}$ be a connected component of $\partial\Omega$. Since $\partial\Omega_{0}$ is compact and $S_{0}(R_{\Omega})$ is simply connected, we get that $F$ is a diffeomorphism. Moreover since $||x|-R_{\Omega}|\leqslant C_{2}R_{\Omega}\delta(\Omega)^{\beta}$ we have $\left|\frac{R_{\Omega}}{|x|}-1\right|\leqslant 2C_{2}\delta(\Omega)^{\beta}$ and

[TABLE]

Now if $\partial\Omega$ as at least $2$ connected components $\partial\Omega_{0}$ and $\partial\Omega_{1}$ we have for any $i\in\{0,1\}$

[TABLE]

and

[TABLE]

Where we have used the fact that $\frac{{\mathcal{H}}^{n}(\partial\Omega)}{{\mathcal{H}}^{n}(S_{0}(R_{\Omega}))}=\frac{I(\Omega)}{I(\mathbb{B}^{n+1})}=1+\delta(\Omega)$. Now we can prove easily that $\frac{(1-C_{2}\delta(\Omega)^{\beta})^{n}}{1+\delta(\Omega)}>1/2$ for $D$ great enough and we deduce that $\partial\Omega$ has one connected component.

Actually  Inequality 6 gives for $D$ great enough that $d_{L}(\partial\Omega,S_{0}(R_{\Omega}))\leqslant C_{6}(n,p)\delta(\Omega)^{\frac{1}{2n\alpha}}$ $=C_{6}(n,p)\delta(\Omega)^{\frac{16pn}{p-n}}$. But we can improve this bound in order to have sharp estimates with respect to the powers of $\delta(\Omega)$ involved in the estimates on $d_{L}$ and $d_{H}$.

Let $\varphi:S_{0}(R_{\Omega})\to\mathbb{R}$ given by $\varphi(w)=\|F^{-1}(w)\|/R_{\Omega}$. Then we have $\partial\Omega=\{\varphi(w)w,w\in S_{0}(R_{\Omega})\}$, $\varphi\geqslant 1/2$ and from 6.3 $\|\varphi-1\|_{\infty}\leqslant C_{2}\delta(\Omega)^{\beta}$. Moreover for any $u\in T_{w}S_{0}(R_{\Omega})$ we have :

[TABLE]

Consequently $R_{\Omega}|d\varphi_{w}|\leqslant|dF^{-1}_{w}||Z_{F^{-1}(w)}|$ and for $D$ great enough we deduce from 6 that $|dF^{-1}_{w}|^{2}\leqslant\frac{1}{1-C_{6}\delta(\Omega)^{1/2n\alpha}}\leqslant\frac{1}{2}$ and from 6.6 we get

[TABLE]

Now the second fundamental form ${\mathrm{B}}$ of the boundary can be expressed by the formulae

[TABLE]

which gives

[TABLE]

On the other hand

[TABLE]

Now $\frac{{\mathcal{H}}^{n}(\partial\Omega)}{{\mathcal{H}}^{n}(S_{0}(R_{\Omega}))}=\frac{I(\Omega)}{I(\mathbb{B}^{n+1})}=1+\delta(\Omega)\leqslant 2$ which gives with Lemma 5 and the fact that $P(\Omega)^{1/n}=R_{\Omega}|\mathbb{B}^{n+1}|^{1/n+1}I(\Omega)^{1/n}$

[TABLE]

If we set $u:\mathbb{S}^{n}\to\mathbb{R}$ defined by $u(x)=\varphi(R_{\Omega}x)-1$ we have for $D$ large enough $\|u\|_{\infty}=\|\varphi-1\|_{\infty}\leqslant C_{2}\delta(\Omega)^{\beta}\leqslant\frac{3}{20(n+1)}$ and $\|du\|_{\infty}=R_{\Omega}\|d\varphi\|_{\infty}\leqslant CK^{1/n}\delta(\Omega)^{1/n\alpha}\leqslant 1/2$ and so $\Omega$ is a nearly spherical domain in the sense of Fuglede.

Moreover since $\|\nabla du\|_{p}=R_{\Omega}^{2}\|\nabla d\varphi\|_{p}\leqslant C(n,p,K)$ we have $\|du\|_{W^{1,p}}\leqslant C(n,p,K)$ and by the Campanato-Morrey estimate, we then get for any $x,x_{0}\in\mathbb{S}^{n}$ that

[TABLE]

and choosing $x_{0}$ such that $|du_{x_{0}}|=\|du\|_{\infty}$ we have

[TABLE]

Let r_{0}:=\bigl{(}\frac{\|du\|_{\infty}}{C(n,p,K)}\bigr{)}^{\frac{1}{1-\frac{n}{p}}}. We can assume $r_{0}<\frac{\pi}{2}$ by taking $C(n,p,K)$ large enough. Integrating the above inequality on the ball of $\mathbb{S}^{n}$ of center $x_{0}$ and radius $r_{0}$ and using the estimates of [11] and then Inequality (I.a) of [11] we get that

[TABLE]

From which we infer that

[TABLE]

Using Inequalities (I.b) of [11] and the above inequality (6.12), we get that

[TABLE]

for $n\geqslant 3$ and $d_{H}(\partial\Omega,S_{x_{\Omega}}\leqslant C(p,K)R_{\Omega}(-\delta(\Omega)\ln\delta(\Omega))^{\frac{1}{2}}$ for $n=2$.

Now since for any $x\in\partial\Omega$, $F^{-1}(x)=x\varphi(x)$, and so $|dF_{R_{\Omega}x}^{-1}(v)|^{2}=(1+u(x))^{2}|v|^{2}+(du_{x}(v))^{2}$, we can use the previous estimates on $u$ to obtain

[TABLE]

From this and the definition of the Lipschitz distance we conclude that for $\delta(\Omega)$ small enough

[TABLE]

where for any diffeomorphism $f$ from $\partial\Omega$ into $S_{0}(R_{\Omega})$, ${\mathrm{d}il}(f)=\displaystyle\sup_{x\in\partial\Omega}|df(x)|$ (for more details on the Lipschitz distance see [16]).

We end this section by the construction of simple examples that prove the sharpness of Theorem 4 with respect of the power of delta involved in our estimates:

The sharpness in the case $n=3$ is already contained in Fuglede’s work [11]. In the case $n\geqslant 3$, let $\varphi:\mathbb{R}^{n}\to\mathbb{R}$ the function defined by

[TABLE]

$\varphi$ is a $C^{1,1-\frac{n}{p}}$ function on $\mathbb{R}^{n}$ with

[TABLE]

from which we infer that $\|\varphi\|_{\infty}=C(n,p)\delta^{\frac{2p-n}{2p-2n+pn}}$, $\|d\varphi\|_{\infty}\leqslant C(n,p)\delta^{\frac{p-n}{2p-2n+pn}}$ and $\frac{1}{C(n,p)}\delta\leqslant\int_{\mathbb{R}^{n}}|d\varphi|^{2}d{\mathcal{H}}^{n}\leqslant C(n,p)\delta$. $\varphi$ can be transposed to a function defined on $\mathbb{S}^{n}$ (via the exponential map at a fixed point of $\mathbb{S}^{n}$) for $\delta$ small enough. The previous estimates will be preserved and the surface $S_{\varphi}=\{(1+\varphi(x))x,x\in\mathbb{S}^{n}\}$ will be an almost spherical surface in the sense of Fuglede. In particular, according to the inequality (I.a) of [11], the isoperimetric deficit of the domain $\Omega_{\varphi}$ bounded by $S_{\varphi}$ satisfies $\frac{\delta}{C(n,p)}\leqslant\delta(\Omega_{\varphi})\leqslant C(p,n)\delta$. Since $d_{H}(S_{\varphi},S_{x_{\Omega}}(R_{\Omega}))=\|\varphi\|_{\infty}$, and for any $q<p$ there exists $K(n,q)$ such that $\|\nabla d\varphi\|_{q}\leqslant K(n,q)$ for any $\delta>0$, we infer that $\|B\|_{q}\leqslant K(n,q)$ for any $\delta>0$. These examples prove that the estimate of Theorem 4 are sharp with respect to the powers of $\delta$ involved in the estimate on $d_{H}$. An easy computation show that it is the same way for the estimate on $d_{L}$.

7. Almost extremal domains for Chavel’s inequality

Proof of Theorem 5 Let $\Sigma$ be an embedded compact hypersurface bounding a domain $\Omega$ in $\mathbb{R}^{n+1}$ and let $X$ be the vector position. Up to a translation we can assume that $\displaystyle\int_{\Sigma}Xd{\mathcal{H}}^{n}=0$ which allows us to use the variational characterization. Then

[TABLE]

Let us put $\rho_{\Omega}:=(n+1)\frac{|\Omega|}{{\mathcal{H}}^{n}(\Sigma)}$. From the inequalities above we deduce that

[TABLE]

which gives for $\gamma(\Omega)<1$

[TABLE]

Now by the divergence theorem to the field $Z=(|X|-\rho_{\Omega})\frac{X}{|X|}$, we get

[TABLE]

Now since $\rho_{\Omega}=\frac{1}{1+\delta(\Omega)}R_{\Omega}\leqslant R_{\Omega}$ and $|B_{0}(R_{\Omega})|=|\Omega|$ we have

[TABLE]

It follows that \bigl{|}|\Omega|-|B_{0}(\rho_{\Omega})|\bigr{|}\leqslant 2(3^{1/2}){\mathcal{H}}^{n}(\Sigma)\rho_{\Omega}\gamma(\Omega)^{1/2}. From the expression of $\rho_{\Omega}$, $I(\Omega)$ and the fact that $I(B_{0}(1))=(n+1)|B_{0}(1)|^{\frac{1}{n+1}}$, the last inequality can be rewritten as

[TABLE]

which gives the desired result.