Smooth approximation of Lipschitz domains, weak curvatures and isocapacitary estimates

Abstract

We provide a novel approach to approximate bounded Lipschitz domains via a sequence of smooth, bounded domains. The flexibility of our method allows either inner or outer approximations of Lipschitz domains which also possess weakly defined curvatures, namely, domains whose boundary can be locally described as the graph of a function belonging to the Sobolev space $W^{2,q}$ for some $q\ge 1$. The sequences of approximating sets is also characterized by uniform isocapacitary estimates with respect to the initial domain $\Omega$.

1 Introduction

In this paper we are concerned with inner and outer approximation of bounded Lipschitz domains $\Omega$ of the Euclidean space $\mathbb {R}^n$, $n\ge 2$. Specifically, we construct two sequences of $C^\infty$-smooth bounded domains $\{\omega _m\},\{\Omega _m\}$ such that $\omega _m\Subset \Omega \Subset \Omega _m$ for all $m\in {\mathbb {N}}$, which also satisfy natural covergence properties like, for instance, in the sense of the Lebesgue measure and in the sense of Hausdorff to $\Omega$.

Geometric quantities like a Lipschitz characteristic $\mathcal L_\Omega =(L_\Omega ,R_\Omega )$ and the diameter $d_\Omega$ of the domain $\Omega$ are comparable to the corresponding ones of its approximating sets $\omega _m, \Omega _m$. Here, the constant $R_\Omega$ stands for the radius of the ball domains on which the boundary can be described as a function of $(n-1)$-variables– i.e. the local boundary chart– and $L_\Omega$ is their Lipschitz constant– see Sect. 2 for the precise definition of a Lipschitz characteristic of $\Omega$.

Furthermore, the smooth charts locally describing the boundaries $\partial \omega _m,\partial \Omega _m$ are defined on the same reference systems as the local charts describing $\partial \Omega$, together with strong convergence in the Sobolev space $W^{1,p}$ for all $p\in [1,\infty )$.

If in addition the local charts describing $\partial \Omega$ belong to the Sobolev space $W^{2,q}$ for some $q\in [1,\infty )$, then we also have strong convergence in the $W^{2,q}$-sense. In a certain way, this means that the second fundamental forms $\mathcal {B}_{\omega _m}$ and $\mathcal {B}_{\Omega _m}$ of the regularized sets converge in $L^q$ to the “weak” curvature $\mathcal {B}_\Omega$ of the initial domain $\Omega$.

Smooth approximation of open sets, not necessarily having Lipschitzian boundary, has been object of study by many authors. To the best of our knowledge, the first author who provided an approximation of this kind is Nečas [20], followed by Massari & Pepe [15] and Doktor [6]. The underlying idea behind their proof is nowadays standard, and it is typically used to approximate sets of finite perimeter. This consists in regularizing the characteristic function of $\Omega$ via mollification and convolution, and then define the approximating set $\Omega _m$ as a suitable superlevel set of the mollified characteristic functions–see for instance [1, Theorem 3.42] or [14, Section 13.2]. We point out that Schmidt [21] and Gui, Hu & Li [8] constructed smooth approximating domains strictly contained in $\Omega$ under additional assumptions on the finite perimeter domain $\Omega$, whereas an outer approximation via smooth sets is given by Doktor [6] when the domain $\Omega$ is endowed with a Lipschitz continuous boundary.

A different kind of approach, which makes use of Stein’s regularized distance, has been recently developed by Ball & Zarnescu [4]. Here, the authors deal with $C^0$ domains, i.e. domains whose boundary can be locally described by merely continuous charts, and hence need not have finite perimeter. We mention that their regularized domains $\Omega _\varepsilon$ are defined as the $\varepsilon$-superlevel set of the regularized distance function, which in turn is obtained via mollification of the usual signed distance function. Here, the parameter $\varepsilon$ can be taken either positive or negative, according to the preferred method of approximation, whether from the inside or outside of $\Omega$.

The aforementioned techniques have thus been used to treat domains with “rough" boundaries; however, they do not seem suitable to approximate domains which possess weakly defined curvatures, even in the case of domains having bounded curvatures, e.g. $\partial \Omega \in C^{1,1}$. Namely, we do not recover any quantitative information or convergence property regarding the second fundamental forms $\mathcal {B}_{\Omega _m}$ from the original one $\mathcal {B}_{\Omega }$. This is because first-order estimates regarding $\Omega _m$ are proven by a careful pointwise analysis of the gradient of the local charts describing their boundaries. In order to obtain estimates about their second fundamental form $\mathcal {B}_{\Omega _m}$, such pointwise analysis needs to be extended to second-order derivatives, and this calls for the application of the implicit function theorem, for which $\Omega$ is required to be at least of class $C^2$.

This drawback is probably due to the fact that the above regularization procedures are global in nature, i.e. they are obtained via mollification of functions “globally” describing $\Omega$, like its characteristic function or signed distance, whereas the second fundamental form of hypersurfaces of $\mathbb {R}^n$ is defined via local parametrizations.

Comparatively, our proof relies on techniques which, in a sense, can be deemed as local in nature, since the starting point of our method is the regularization of the functions of $(n-1)$-variables which locally describe $\partial \Omega$. Thus, our approach seems more suitable when dealing with weak curvatures, though at the cost of requiring $\Omega$ to have a Lipschitz continuous boundary.

Regarding its applications, approximation via a sequence of smooth bounded domains has proven to be a powerful tool especially when dealing with boundary value problems in Partial Differential Equations. Indeed, by tackling the same boundary value problem (or its suitable regularization) on smoother domains, accordingly one obtains smoother solutions, hence it is possible to perform all the desired computations and infer a priori estimates which do not depend on the full regularity of the approximating sets $\Omega _m$, but only on their Lipschitz characteristics or other suitable quantities possibly depending on the second fundamental form $\mathcal {B}_{\Omega _m}$. For instance, various investigations such as [2, 3, 5, 17, 18] showed that global regularity of solutions to linear and quasilinear PDEs may depend on a weighted isocapacitary function for subsets $\partial \Omega$, the weight being the norm of the second fundamental form on $\partial \Omega$.

This function, which we denote by ${\mathcal {K}}_\Omega$, is defined as

(1.1)

and it was first introduced in [5]. Above, $\textrm{cap} (E, B_r(x))$ denotes the standard capacity of a compact set E relative to the ball $B_r(x)$, i.e.

$$\begin{aligned} \textrm{cap} (E, B_r(x))=\inf \bigg \{\int _{B_r(x)} |\nabla v|^2\,dx:\,v\in C_c^{0,1}(B_r(x)),\,v\ge 1\text { on } E\bigg \}, \end{aligned}$$

where $C^{0,1}_c(A)$ is the set of Lipschitz continuous functions with compact support in A.

We remark that, in order for $\mathcal {K}_\Omega (r)$ to be well defined, it suffices that $\partial \Omega$ is Lipschitz continuous and belongs to $W^{2,1}$, as it can be inferred from inequalities (2.10) below.

1.1 Plan of the paper

The rest of the paper is organized as follows: in Sect. 2, we explain some non-standard notation used throughout the paper, and provide the definitions of $\mathcal {L}_\Omega$-Lipschitz domain, of $W^{2,q}$-domain and of weak curvature.

In Sect. 3 we state in detail our main results, and we provide a few comments and an outline of their proofs.

In Sect. 4 we state and prove a useful convergence property of mollification and convolution, which will be used in the proof of the convergence properties of the approximating sets.

In Sect. 5 we introduce the notion of transversality of a unit vector $\textbf{n}$ to a Lipschitz function $\phi$, and we show a very interesting fact, i.e. this transversality property is equivalent to the graphicality of $\phi$ with respect to the coordinate system $(y',y_n)$ having $\textbf{n}=e_n$. We then close this section by showing that the transversality condition– hence the graphicality with respect to the reference system $(y',y_n)$– is inherited by the convoluted function $M_m(\phi )$.

As a byproduct, we will find an interesting, yet expected result: if $\partial \Omega \in W^{2,q}$, then any Lipschitz function locally describing $\partial \Omega$ is of class $W^{2,q}$. This means that second-order Sobolev regularity is an intrinsic property of the local charts describing $\partial \Omega$– see Corollary 1.

Finally, Sect. 6 is devoted to the proof of the main Theorem 1.

2 Basic notation and definitions

In this section, we provide the relevant definitions and notation of use throughout the rest of the paper.

• For $d\in {\mathbb {N}}$, $U\subset {\mathbb {R}}^d$ open, and a function $v:U\,\rightarrow {\mathbb {R}}$, we shall denote by $\nabla v$ its d-dimensional gradient, and $\nabla ^2 v$ its hessian matrix. We will often use the short-hand notation for its level and sublevel sets

  $$\begin{aligned} \begin{aligned}&\{v<0\}:=\{z\in U:\,v(z)<0\}. \\&\{v=0\}:=\{z\in U:\,v(z)=0\}. \end{aligned} \end{aligned}$$

• We denote by $W^{k, p}(\Omega )$ the usual Sobolev space of $L^p(\Omega )$ weakly differentiable functions having weak k-th order derivatives in $L^p(\Omega )$. For any $\alpha \in (0,1]$, the spaces $C^k(\Omega )$ and $C^{k,\alpha }(\Omega )$ will denote, respectively, the space of functions with continuous and $\alpha$-Hölder continuous derivatives up to order $k\in {\mathbb {N}}$.

• Point of ${\mathbb {R}}^n$ will be written as $x=(x',x_n)$, with $x'\in {\mathbb {R}}^{n-1}$ and $x_n\in {\mathbb {R}}$. We write $B_r(x)$ to denote the n-dimensional ball of radius $r>0$ and centered at $x\in {\mathbb {R}}^n$. Also, $B'_r(x')$ will denote the $(n-1)$-dimensional ball of radius $r>0$ and centered at $x'\in {\mathbb {R}}^{n-1}$—when the centers are omitted, the balls are assumed to be centered at the origin, i.e. $B_r:=B_r(0)$ and $B_r':= B_r'(0')$.

• For $d\in {\mathbb {N}}$, and for a given matrix $X\in \mathbb {M}_{d\times d}$, we shall denote by |X| its Frobenius Norm $|X|=\sqrt{\textrm{tr}(X^t X)}$, where $X^t$ is the transpose of X.

• Given a Lebesgue measurable set A, we shall write |A| for its Lebesgue measure. Also, given two open bounded sets A, B, we will denote by $\textrm{dist}_\mathcal {H}(A,B)$ their Hausdorff distance.

• For a given function $\phi :\,U\rightarrow {\mathbb {R}}$ with $U\subset {\mathbb {R}}^{n-1}$ open, we write $G_\phi$ and $S_\phi$ to denote its graph and subgraph in ${\mathbb {R}}^n$, i.e.

  $$\begin{aligned} G_\phi =\{x=\big (x',\phi (x')\big ):\,x'\in U\}\quad \text {and}\quad S_\phi =\{x=\big (x',x_n\big ):\,x'\in U,\,x_n<\phi (x')\}. \end{aligned}$$

• We will denote by $\rho =\rho (x')$ the standard convolution Kernel in ${\mathbb {R}}^{n-1}$, i.e.

  

  and we will write $\rho _m(x')=m^{n-1}\rho \big (m\,x'\big )$ for $m\in {\mathbb {N}}$. Given $h\in L^1_{loc}({\mathbb {R}}^{n-1})$, the convolution operator $M_m(h)$ is defined as

  $$\begin{aligned} M_m(h)(x')=h*\rho _m(x')=\int _{{\mathbb {R}}^{n-1}}h(y')\,\rho _m(x'-y')\,dy'. \end{aligned}$$

In the following, we specify the definition of Lipschitz domain and of Lipschitz characteristic.

Definition 1

(Lipschitz characteristic of a domain) An open, connected set $\Omega$ in $\mathbb {R}^n$ is called a Lipschitz domain if there exist constants $L_\Omega >0$ and $R_\Omega \in (0, 1)$ such that, for every $x_0\in \partial \Omega$ and $R\in (0, R_\Omega ]$ there exist an orthogonal coordinate system centered at $0\in \mathbb {R}^n$ and an $L_\Omega$-Lipschitz continuous function $\phi : B'_{R}\rightarrow (-\ell , \ell )$, where

$$\begin{aligned} \ell = R (1+L_\Omega ), \end{aligned}$$

(2.1)

satisfying $\phi (0')=0$, and

$$\begin{aligned} \begin{aligned}&\partial \Omega \cap \big (B'_{R}\times (-\ell ,\ell )\big )=\{(x', \phi (x')):\,x'\in B'_{R}\}, \\&\Omega \cap \big (B'_{R}\times (-\ell ,\ell )\big )=\{(x',x_n):\,x'\in B'_{R},\,-\ell<x_n<\phi (x')\}. \end{aligned} \end{aligned}$$

(2.2)

Moreover, we set

$$\begin{aligned} {\mathfrak {L}}_\Omega = (L_\Omega , R_\Omega ), \end{aligned}$$

(2.3)

and call ${\mathfrak {L}}_\Omega$ a Lipschitz characteristic of $\Omega$.

It is easily seen that the above definition coincides with the standard one for uniformly Lipschitz domains–see e.g. [9, Section 2.4]. Our definition has the advantage of pointing out $\mathfrak {L}_\Omega = (L_\Omega , R_\Omega )$ which appears in the characterization of our approximation sets.

We also remark that, in general, a Lipschitz characteristic $\mathfrak {L}_\Omega = (L_\Omega , R_\Omega )$ is not uniquely determined. For instance, if $\partial \Omega \in C^1$, then $L_\Omega$ may be taken arbitrarily small, provided that $R_\Omega$ is chosen sufficiently small.

The function $\phi$ in definition 1 is typically called local (boundary) chart. By Rademacher’s theorem, this function is differentiable for $\mathcal {H}^{n-1}$-almost every $x'$, with gradient $\nabla \phi$ bounded by $L_\Omega$. In particular, this implies that any Lipschitz domain $\Omega$ admits a tangent plane on $\mathcal {H}^{n-1}$-almost every point of its boundary.

The local chart $\phi$ naturally endows $\partial \Omega$ of a local parametrization

$$\begin{aligned} \iota _\phi (x')=\big (x',\phi (x') \big ) \end{aligned}$$

(2.4)

under which, whenever $\phi$ is differentiable at $x'$, a basis of the tangent space at the point $(x',\phi (x'))$ is given by

$$\begin{aligned} \mathcal {E}_\phi =\bigg \{e_i+\frac{\partial \phi (x')}{\partial x'_i}\bigg \}_{i=1,\dots , n-1} \end{aligned}$$

(2.5)

where $e_i=(0,\dots ,1,\dots ,0)$ is the i-th canonical unit vector of ${\mathbb {R}}^n$.

Moreover, via such parametrization $\iota _\phi (x')$, the first fundamental form $g=\{g_{ij}\}_{i,j=1}^{n-1}$ can be computed as

$$\begin{aligned} g_{ij}(x')=\delta _{ij}+\frac{\partial \phi (x')}{\partial x'_i}\,\frac{\partial \phi (x')}{\partial x'_j}, \end{aligned}$$

(2.6)

where $\delta _{ij}$ denotes the Kronecker’s delta, and $x'$ is a point of differentiability of $\phi$. Then, the inverse matrix $g^{-1}=\{g^{ij}\}_{i,j=1}^{n-1}$ can be explictly computed:

$$\begin{aligned} g^{ij}(x')=\delta _{ij}-\frac{1}{1+|\nabla \phi (x')|^2}\,\frac{\partial \phi (x')}{\partial x'_i}\,\frac{\partial \phi (x')}{\partial x'_j}. \end{aligned}$$

(2.7)

For such points $x_0=\big (x',\phi (x')\big )\in \partial \Omega$, we shall denote by $T_{x_0}\partial \Omega =T_{x'}\partial \Omega$ the tangent space at $x^0$. From the discussion above, $\partial \Omega$ admits a tangent plane $\mathcal {H}^{n-1}$-almost every point $x_0\in \partial \Omega$, hence we may want to define a notion of weak second fundamental form which extends the classical one for $C^\infty$-smooth domains of ${\mathbb {R}}^n$.

For this purpose, we need some additional regularity assumptions on $\phi$, and in particular on its second-order derivatives.

Definition 3

($W^{2,q}$ domains and weak curvature) Let $q\in [1,\infty )$. We say that a bounded Lipschitz domain $\Omega$ is of class $W^{2,q}$ if the local boundary chart $\phi$ satisfying (2.2) belongs to the Sobolev space $W^{2,q}(B'_R)$. If $\phi \in W^{2,\infty }(B'_R)$, we say that $\partial \Omega \in C^{1,1}$ (or $\partial \Omega \in W^{2,\infty }$).

If $\partial \Omega \in W^{2,1}$, the weak curvature $\mathcal {B}_\Omega$ of $\partial \Omega$ is a bilinear operator

$$\begin{aligned} \mathcal {B}_\Omega (x_0):\,T_{x_0}\partial \Omega \times T_{x_0}\partial \Omega \rightarrow {\mathbb {R}}\end{aligned}$$

defined for $\mathcal {H}^{n-1}$-almost every point $x_0\in \partial \Omega$. With the choice of local parametrization $\iota _\phi$ in (2.4), its components $\big \{\mathcal {B}_{ij}\big \}_{i,j=1}^{n-1}$ with respect to the basis $\mathcal {E}_\phi$ in (2.5) of $T_{x'}\partial \Omega$ are locally defined as

$$\begin{aligned} \mathcal {B}_{ij}(x')=-\frac{1}{\sqrt{1+|\nabla \phi (x')|^2}}\,\frac{\partial ^2\phi (x')}{\partial x'_i\partial x'_j}, \end{aligned}$$

(2.8)

for $\mathcal {H}^{n-1}$-almost every points $x'\in B'_R$ of differentiability of $\phi$. Its norm is then given by

$$\begin{aligned} |\mathcal {B}_\Omega (x')|= \frac{\sqrt{\textrm{trace}\big ((g^{-1}\,\nabla ^2\phi )^2 \big )}}{\sqrt{1+|\nabla \phi (x')|^2}}, \end{aligned}$$

(2.9)

where $g^{-1}$ is the inverse matrix of g given by (2.7).

The reader may verify that identities (2.6)-(2.9) concur with the usual ones when $\partial \Omega$ is a smooth hypersurface of $\mathbb {R}^n$–see e.g. [12, pp. 246-249]. However, these definitions also make sense when $\phi$ is merely Lipschitz continuous and belongs to the Sobolev space $W^{2,1}$. Indeed, the following inequalities hold true:

$$\begin{aligned} \frac{|\nabla ^2\phi (x')|}{(1+L_\Omega ^2)^{3/2}}\le |\mathcal {B}_\Omega (x')|\le |\nabla ^2\phi (x')|. \end{aligned}$$

(2.10)

In order to prove (2.10), we first recall that for all symmetric matrices X, Y, with X definite positive, we have the elementary linear algebra inequalities

$$\begin{aligned} \lambda ^2_{\min }|Y|^2\le \textrm{tr}\big ((XY)^2\big )\le \lambda ^2_{\max }\,|Y|^2, \end{aligned}$$

where $\lambda _{\min },\lambda _{\max }$ denote the smallest and largest eigenvalues of X–see e.g. [2, Lemma 3.6] and its proof. Then, owing to (2.7), we observe that the largest and smallest eigenvalues of the matrix $g^{-1}$ are respectively 1 and $(1+|\nabla \phi |^2)^{-1}$, and since $|\nabla \phi |\le L_\Omega$ we immediately infer (2.10). Inequalities (2.10) also show that (locally) second fundamental form $\mathcal {B}_\Omega$ is equivalent to the second-order derivatives of the local charts.

We close this section by pointing out that the above definitions can be easily extended to domains with boundary $\partial \Omega \in W^{k,q}$. Similarly, standard definitions follow for domains of class $C^k$ and $C^{k,\alpha }$.

3 Main results

Having dispensed of the necessary definitions and notation, we can now give a precise statement of our main results. This is the content of this section, coupled with a few comments and an outline of the proofs. Our first main result reads as follows.

Theorem 1

Let $\Omega \subset \mathbb {R}^n$ be a bounded, Lipschitz domain, with Lipschitz characteristic ${\mathcal {L}}_\Omega =(L_\Omega ,R_\Omega )$.

(i) There exist sequences of bounded domains $\{\omega _m\},\{\Omega _m\}$, such that $\partial \omega _m\in C^\infty ,\,\partial \Omega _m\in C^\infty$, and

$$\begin{aligned} \omega _m\Subset \Omega \Subset \Omega _m\quad \text {for all }m\in {\mathbb {N}}. \end{aligned}$$

Their diameters satisfy

$$\begin{aligned} d_{\Omega _m}\le c(n)\,d_\Omega ,\quad d_{\omega _m}\le c(n)\,d_\Omega , \end{aligned}$$

(3.1)

the following convergence property hold true

$$\begin{aligned} \lim _{m\rightarrow \infty } |\Omega _m\setminus \Omega |=0,\quad \lim _{m\rightarrow \infty }|\Omega \setminus \omega _m|=0, \end{aligned}$$

(3.2)

the Hausdorff distances safisfy

$$\begin{aligned} \textrm{dist}_\mathcal {H}(\omega _m,\Omega )+\textrm{dist}_\mathcal {H}(\Omega _m,\Omega )\le \frac{12\,L_\Omega \sqrt{1+L_\Omega ^2}}{m}\quad \text {for all } m\in {\mathbb {N}}, \end{aligned}$$

(3.3)

and we may choose their Lipschitz characteristics $\mathcal {L}_{\Omega _m}=(L_{\Omega _m},R_{\Omega _m})$ and $\mathcal {L}_{\omega _m}=(L_{\omega _m},R_{\omega _m})$ such that

$$\begin{aligned} \begin{aligned} L_{\Omega _m}\le c(n)(1+L_\Omega ^2),&\quad R_{\Omega _m}\ge R_\Omega /\big (c(n)(1+L_\Omega ^2)\big ) \\ L_{\omega _m}\le c(n)(1+L_\Omega ^2),&\quad R_{\omega _m}\ge R_\Omega /\big (c(n)(1+L_\Omega ^2)\big ),\quad \text {for all } m\in {\mathbb {N}}. \end{aligned} \end{aligned}$$

(3.4)

Moreover, the smooth boundaries $\partial \omega _m, \partial \Omega _m$ are described with the help of the same co-ordinate systems as $\partial \Omega$, i.e. there exist finite number of local boundary charts $\{\phi ^i\}_{i=1}^N,\{\psi ^i_m\}_{i=1}^N$ and $\{\varphi ^i_m\}_{i=1}^N$ which describe $\partial \Omega ,\,\partial \Omega _m$ and $\partial \omega _m$ respectively, such that for each $i=1,\dots ,N$ the functions $\psi ^i_m,\varphi ^i_m\in C^\infty$ are defined on the same reference system as $\phi ^i$, and

$$\begin{aligned} \psi ^i_m\xrightarrow {m\rightarrow \infty } \phi ^i\quad \text {and}\quad \varphi ^i_m \xrightarrow {m\rightarrow \infty } \phi ^i\quad \text {in } W^{1,p}(B'_{R_\Omega -\varepsilon _0}), \end{aligned}$$

(3.5)

for all $p\in [1,\infty )$, for all $i=1,\dots ,N$, and any fixed constant $\varepsilon _0\in (0,R_\Omega /2)$.

(ii) If in addition $\partial \Omega \in W^{2,q}$ for some $q\in [1,\infty )$, then

$$\begin{aligned} \psi ^i_m\xrightarrow {m\rightarrow \infty } \phi ^i\quad \text {and}\quad \varphi ^i_m \xrightarrow {m\rightarrow \infty } \phi ^i\quad \text {in } W^{2,q}(B'_{R_\Omega -\varepsilon _0}), \end{aligned}$$

(3.6)

and there exists a constant $\widehat{c}=\widehat{c}(n,\mathcal {L}_\Omega , d_\Omega )$ such that

$$\begin{aligned} \mathcal {K}_{\Omega _m}(r)+\mathcal {K}_{\omega _m}(r)\le {\left\{ \begin{array}{ll} \widehat{c}\,\Big \{ \mathcal {K}_{\Omega }\big ( \widehat{c}\,(r+\tfrac{1}{m})\big )+r\Big \}\quad &{} \text {if }n\ge 3 \\ \\ \widehat{c}\,\Big \{ \mathcal {K}_{\Omega }\big (\widehat{c}\,(r+\tfrac{1}{m})\big )+r\,\log (1+\tfrac{1}{r})\Big \}\quad &{} \text {if } n=2 \end{array}\right. } \end{aligned}$$

(3.7)

for all $m\in {\mathbb {N}}$ and $r\le r_0(n,\mathcal {L}_\Omega )$.

Let us briefly comment on our result. Part (i) of Theorem 1 is mostly analogous to [6, Theorem 5.1]; as expected from domains $\Omega$ with Lipschitz continuous boundary, the local charts of $\partial \Omega _m,\partial \omega _m$ converge to the corresponding local charts of $\partial \Omega$ in $W^{1,p}$ for all $p\in [1,\infty )$. In particular, by the classical Morrey-Sobolev’s embedding Theorems, this entails an “almost Lipschitz convergence”, i.e. the local charts $\psi ^i_m$ and $\varphi ^i_m$ converge to $\phi ^i$ in every Hölder space $C^{0,\alpha }$ with $\alpha \in (0,1)$.

The main novelty of our result is given in Part (ii), where information about the second fundamental forms $\mathcal {B}_{\omega _m}$ and $\mathcal {B}_{\Omega _m}$ (or equivalently $\nabla ^2\varphi ^i_m$ and $\nabla ^2 \psi ^i_m$) is retrieved when $\partial \Omega$ is endowed with a weak curvature. For instance, by definition (2.8) and from the results of Theorem 1, via a standard covering argument it is easy to show that

$$\begin{aligned}{} & {} \int _{\partial \Omega _m}|\mathcal {B}_{\Omega _m}|^qd\mathcal {H}^{n-1}\nonumber \\{} & {} \quad \rightarrow \int _{\partial \Omega }|\mathcal {B}_\Omega |^q d\mathcal {H}^{n-1}\quad \text {and}\quad \int _{\partial \omega _m}|\mathcal {B}_{\omega _m}|^q d\mathcal {H}^{n-1}\rightarrow \int _{\partial \Omega }|\mathcal {B}_\Omega |^qd\mathcal {H}^{n-1}, \end{aligned}$$

(3.8)

for all $q\in [1,\infty )$ such that $\partial \Omega \in W^{2,q}$.

Other than this, we obtain the isocapacitary estimate (3.7), where $\mathcal {K}_{\Omega }(r)$ and $\mathcal {K}_{\Omega _m},\mathcal {K}_{\omega _m}$ are the functions defined in (1.1) relative to $\Omega ,\Omega _m$ and $\omega _m$, respectively. In the proof of (3.7), we will also explicitly write the constant $\widehat{c}$ appearing therein.

Finally, the boundaries $\partial \Omega ,\,\partial \Omega _m$ and $\partial \omega _m$ all share the same coordinate cylinders $\{K^i_{\varepsilon _0}\}_{i=1}^N$ which are, up to an isometry, equal to $B'_{R_\Omega -\varepsilon _0}\times (-\ell ,\ell )$, with $\ell =(1+L_\Omega )\,R_\Omega$.

This means that their local boundary charts, $\phi ^i,\psi ^i_m$ and $\varphi ^i_m$ respectively, are defined on the same $(n-1)$-dimensional ball $B'_{R_\Omega -\varepsilon _0}$, independently on $i=1,\dots N$ and $m\in {\mathbb {N}}$—see Fig. f1 below.

Here, the fixed parameter $\varepsilon _0\in (0,R_\Omega /2)$, also appearing in (3.5) and (3.6), is purely technical and does not affect the validity of the convergence results. Indeed, from our construction in Sect. 6, the boundaries $\partial \Omega ,\,\partial \Omega _m$ and $\partial \omega _m$ will be covered by smaller coordinate cylinders of the kind $B'_{R_\Omega /2}\times (-\ell ,\ell )$.

Fig. 1

The local boundary charts of $\partial \Omega$ and $\partial \Omega$, $\phi ^i$ and $\psi ^i_m$ respectively, are defined on the same reference system

3.1 Outline of the proof

We fix a covering of $\partial \Omega$, with corresponding partition of unity $\{\xi _i\}_{i}$ and local boundary charts $\{\phi ^i\}_{i}$, which are $L_\Omega$-Lipschitz continuous.

Then we regularize each function $\phi ^i$ via convolution, and add (or subtract) a suitable constant, so that we obtain $C^\infty$-smooth functions $\{\phi ^i_m\}_{i}$ such that $\phi ^i_m>\phi ^i$ ( or $\phi ^i_m<\phi ^i$).

However, in the original reference system, the graphs of these smooth functions $G_{\phi ^i_m}$ are not “glued" together, and thus their union is not the boundary of a domain, unlike the graphs $G_{\phi ^i}$ whose union describes $\partial \Omega$—see Fig. 2 below.

To overcome this problem, we define a suitable $C^\infty$-smooth function $F_m$, built upon $\{\phi ^i_m\}_i$ and $\{\xi _i\}_i$– see equation (6.14) below– and define the regularized set $\Omega _m$ as the sublevel set $\{F_m<0\}$, so that

$$\begin{aligned} \partial \Omega _m=\{F_m=0\}, \end{aligned}$$

and by construction we will have $\omega _m\Subset \Omega \Subset \Omega _m$.

The function $F_m$ is called boundary defining functions of $\Omega _m$—see [11, Section 5.4].

In order to show that $\partial \Omega _m$ is a smooth manifold, we prove that the gradient of $F_m$ along the directions of graphicality of $\phi ^i$ is greater than a positive constant depending on $L_\Omega$—see estimate (6.20). This property of $F_m$ will be proven by exploiting the so-called transversality condition of $\phi ^i$, which is inherited via convolution by $\phi ^i_m$ as well. Therefore, $F_m$ is strictly monotone along these directions, which entails that its zero-level set $\partial \Omega _m$ is a smooth manifold with local boundary charts $\psi ^i_m$ defined on the same reference system as $\phi ^i$.

Thanks to the properties of convolution, we show that $F_m$ converge to the boundary defining function F of $\Omega$ built upon $\{\phi ^i\}_{i}$ and $\{\xi _i\}_i$– see equations (6.9) and (6.10)– and thus $\psi ^i_m$ converge uniformly to $\phi ^i$.

Then, as in the proof of the implicit function theorem, we differentiate the identity $F_m\big (y',\psi ^i_m(y')\big )=0$, so that we may express the gradient $\nabla \psi ^i_m$ (and its Hessian $\nabla ^2 \psi ^i_m$) in terms of $\{\phi ^j_m, \nabla \phi ^j_m\}_j$ (and $\{\nabla ^2 \phi ^j_m\}_{j}$), and then (3.4), (3.5) (and (3.6)) will be obtained by exploiting the convergence properties of convolution.

Finally, in order to get the isocapacitary estimate (3.7), we make use of the estimates on $|\nabla ^2 \psi ^i_m|$ obtained in the previous steps, as to evaluate weighted Poincaré type quotients of the kind

$$\begin{aligned} \frac{\int _{\partial \Omega _m}v^2\,|\mathcal {B}_{\Omega _m}|\,d\mathcal {H}^{n-1}}{\int _{{\mathbb {R}}^n}|\nabla v|^2 dx},\quad v\in C^{\infty }_c\big (B_r(x^0_m)\big ),\,x^0_m\in \partial \Omega _m \end{aligned}$$

in terms of the corresponding quotient with weight $|\mathcal {B}_\Omega |$, and then (3.7) will follow from the celebrated isocapacitary equivalency Theorem of Maz’ya [16, 19, Theorem 2.4.1].

Our next and final result shows the flexibility of our approximation method, which takes into account even higher regularity of the domain $\Omega$.

Theorem 2

Under the same notations as Theorem 1, we have that

• (1) if $\partial \Omega \in C^k$ for some $k\in {\mathbb {N}}$, then

  $$\begin{aligned} \psi ^i_m\xrightarrow {m\rightarrow \infty } \phi ^i\quad \text {and}\quad \varphi ^i_m \xrightarrow {m\rightarrow \infty } \phi ^i\quad \text {in } C^k(B'_{R_\Omega -\varepsilon _0}); \end{aligned}$$

• (2) if $\partial \Omega \in C^{k,\alpha }$ for some $k\in {\mathbb {N}}$ and $\alpha \in (0,1)$, then

  $$\begin{aligned} \psi ^i_m\xrightarrow {m\rightarrow \infty } \phi ^i\quad \text {and}\quad \varphi ^i_m \xrightarrow {m\rightarrow \infty } \phi ^i\quad \text {in } C^{k,\alpha '}(B'_{R_\Omega -\varepsilon _0}), \end{aligned}$$

  for all $0<\alpha '<\alpha$;

• (3) if $\partial \Omega \in W^{k,q}$ for some $k\in {\mathbb {N}}$ and $q\in [1,\infty )$, then

  $$\begin{aligned} \psi ^i_m\xrightarrow {m\rightarrow \infty } \phi ^i\quad \text {and}\quad \varphi ^i_m \xrightarrow {m\rightarrow \infty } \phi ^i\quad \text {in } W^{k,q}(B'_{R_\Omega -\varepsilon _0}). \end{aligned}$$

• (4) if $\partial \Omega \in C^{k,1}$ for some $k\in {\mathbb {N}}$, then

  $$\begin{aligned} \psi ^i_m\xrightarrow {m\rightarrow \infty } \phi ^i\quad \text {and}\quad \varphi ^i_m \xrightarrow {m\rightarrow \infty } \phi ^i\quad \text {weakly- }*\text { in } W^{k,\infty }(B'_{R_\Omega -\varepsilon _0}). \end{aligned}$$

The proof of Theorem 2 can be easily carried out by extending the proof and estimates of Theorem 1 to higher order derivatives, and by using standard compactness theorems such as Ascoli-Arzelá’s and weak-$*$ compactness. For this very reason, we decided to omit the proof.

Fig. 2

In red: the graphs of the regularized local charts (up to isometry)

4 Auxiliary results

In this section, we state and prove a useful convergence property regarding the convolution of functions composed with a suitable family of bi-Lipschitz maps.

Proposition 1

Let $U\subset {\mathbb {R}}^{n-1}$ be a bounded domain, $K>0$ be a constant, and $\{\Psi _m\}_{m\in {\mathbb {N}}}$ be a family of bi-Lipschitz maps on U such that

$$\begin{aligned} \sup _{m\in {\mathbb {N}}}\Vert \nabla \Psi _m^{-1}\Vert _{L^\infty }\le K , \end{aligned}$$

(4.1)

and there exists a bi-Lipschitz map $\Psi :U\rightarrow \Psi (U)$ such that

$$\begin{aligned} \Vert \Psi _m-\Psi \Vert _{L^\infty (U)}\le \frac{K}{m}\quad \text {for all } m\in {\mathbb {N}}. \end{aligned}$$

(4.2)

Let $\mathcal {O}\subset {\mathbb {R}}^{n-1}$ open be such that $\Psi (U)\Subset \mathcal {O}$, and $\phi \in L^p(\mathcal {O})$ for some $p\in [1,\infty )$. Then

$$\begin{aligned} M_m(\phi )\circ \Psi _m\xrightarrow {m\rightarrow \infty } \phi \circ \Psi \quad \mathcal {H}^{n-1} \hbox {-a.e. in} U \hbox { and in }L^p(U). \end{aligned}$$

(4.3)

Proof

Set

$$\begin{aligned} U_\phi :=\big \{x'\in U:\, \Psi (x')\text { is a Lebesgue point of } \phi \big \} \end{aligned}$$

By Lebesgue differentiation theorem and since $\Psi$ is a bi-Lipschitz map, we have that $U_\phi$ is a subset of U with full measure. Also, thanks to (4.2) and the fact that $\Psi \big (U\big )\Subset \mathcal {O}$, we have that $\phi$ and $M_m(\phi )$ are well defined on a neighbourhood of $\Psi _m(U)$ for $m>m_0$ large enough. Then, for all $x'\in U_\phi$ we have

$$\begin{aligned} \begin{aligned} \big |M_m(\phi )\big ( \Psi _m (x')\big ) -\phi \big (\Psi (x')\big )\big |=\bigg |\int _{B'_{\frac{1}{m}}(\Psi _m(x'))}\Big [\phi (z')-\phi \big (\Psi (x')\big )\Big ] \rho _m\big (\Psi _m(x')-z'\big )\,dz\bigg |\\ \le \big (\sup _{{\mathbb {R}}^{n-1}}\rho \big )\,m^{n-1}\int _{B'_{\frac{(K+1)}{m}} (\Psi (x'))}\big |\phi (z')-\phi \big (\Psi (x')\big ) \big |\,dz'\xrightarrow {m\rightarrow \infty }0. \end{aligned} \end{aligned}$$

Above we used the fact that $\Psi (x')$ is a Lebesgue point of $\phi$, and $B'_{\frac{1}{m}}(\Psi _m(x'))\subset B'_{\frac{(K+1)}{m}}(\Psi (x'))$ as a consequence of (4.2).

Now fix $\varepsilon >0$, and take a function $\widetilde{\phi } \in C^\infty _c({\mathbb {R}}^{n-1})$ satisfying

$$\begin{aligned} \Vert \phi -\widetilde{\phi }\Vert ^p_{L^p(\mathcal {O})}\le \varepsilon . \end{aligned}$$

(4.4)

Standard properties of convolutions ensure that

$$\begin{aligned} \Vert M_m(\widetilde{\phi })-\widetilde{\phi }\Vert _{L^\infty (\mathcal {O})}\xrightarrow {m\rightarrow \infty }0. \end{aligned}$$

(4.5)

Then we have

$$\begin{aligned} \begin{aligned} \int _U&\big |M_m(\phi )\big ( \Psi _m (x')\big ) -\phi \big (\Psi (x')\big )\big |^p\,dx'\le c(p)\,\int _U\big | M_m(\phi -\widetilde{\phi })\big ( \Psi _m (x')\big )\big |^p\,dx' \\&+c(p)\,\int _U \big |M_m(\widetilde{\phi })\big ( \Psi _m (x')\big )- \widetilde{\phi }\big ( \Psi (x')\big )\big |^p\,dx'+c(p)\,\int _U \big | \widetilde{\phi }\big ( \Psi (x')-\phi \big ( \Psi (x')\big )\big |^p\,dx' \end{aligned} \end{aligned}$$

(4.6)

By applying Jensen inequality, the change of variables $w'=\Psi _m (x')-z'$ and Fubini-Tonelli’s Theorem we obtain

$$\begin{aligned} \begin{aligned}&\int _U\big | M_m(\phi -\widetilde{\phi })\big ( \Psi _m (x')\big )\big |^p\,dx'\\&\le \int _U\int _{B'_{1/m}}\big | \phi \big ( \Psi _m(x')-z'\big )-\widetilde{\phi }\big ( \Psi _m(x')-z'\big )\big |^p \rho _m(z')\,dz'\,dx' \\&\le c(n)\,K^{n-1}\,\int _{{\mathbb {R}}^{n-1}}\rho _m(z')\,dz'\,\int _{\mathcal {O}}\big |\phi (w')-\widetilde{\phi }(w') \big |^p\,dw'\le c(n)\,K^{n-1}\,\varepsilon , \end{aligned} \end{aligned}$$

where we also used estimates (4.1) and (4.4).

Then, by using (4.2) and (4.5), it is immediate to verify that

$$\begin{aligned} \lim _{m\rightarrow \infty }\int _U \big |M_m(\widetilde{\phi })\big ( \Psi _m (x')\big )- \widetilde{\phi }\big ( \Psi (x')\big )\big |^p\,dx'= 0, \end{aligned}$$

and finally, via a change of variables $y'=\Psi (x')$, and (4.4) we get

$$\begin{aligned} \int _U \big | \widetilde{\phi }\big ( \Psi (x')-\phi \big ( \Psi (x')\big )\big |^p\,dx'\le c(n)\,\Vert \nabla \Psi ^{-1}\Vert _{L^\infty }^{n-1}\,\varepsilon . \end{aligned}$$

Henceforth, by plugging the last three estimates into (4.6), we find

$$\begin{aligned} \limsup _{m\rightarrow \infty }\int _U \big |M_m(\phi )\big ( \Psi _m (x')\big ) -\phi \big (\Psi (x')\big )\big |^p\,dx'\le c(n,p,L,\Psi )\,\varepsilon , \end{aligned}$$

and thus (4.3) follows by the arbitrariness of $\varepsilon$. $\square$

We close this section recalling a variant of Lebesgue dominated convergence Theorem which will be useful later on. Since we could not find a precise reference, we provide a proof.

Theorem 3

(Dominated convergence Theorem) Let $\{f_k\}_{k\in {\mathbb {N}}}$ be a sequence of measurable functions on $E\subset {\mathbb {R}}^{n-1}$ such that

• (i) $f_k\rightarrow f$ almost everywhere on E;

• (ii) $|f_k|\le g_k$ almost everywhere on E, with $g_k\in L^q(E)$ for some $q\in [1,\infty )$;

• (iii) there exists $g\in L^q(E)$ such that $g_k\rightarrow g$ a.e. on E, and $\int _E g_k^q\,dx\rightarrow \int _E g^q\,dx$.

Then $f\in L^q(E)$, and

$$\begin{aligned} \int _E |f_k-f|^q\,dx\rightarrow 0. \end{aligned}$$

Proof

Set

$$\begin{aligned} F_k=|f_k-f|^q,\quad F=0,\quad G_k=2^{q-1}\big \{g_k+g\big \},\quad \text {and }\quad G=2^q\,g. \end{aligned}$$

Observe that, by hypothesis, $F_k\rightarrow F$ and $G_k\rightarrow G$ almost everywhere on E as $k\rightarrow \infty$, $0\le F_k\le G_k$ almost everywhere, with $G_k,G\in L^1(E)$, and

$$\begin{aligned} \int _E G_k\,dx\rightarrow \int _E G\,dx. \end{aligned}$$

The thesis then follows from a standard generalization of dominated convergence theorem–see for instance [7, Exercise 20, pp. 59]. $\square$

5 Transversality and graphicality

Throughout this section, we shall consider an isometry T of ${\mathbb {R}}^n$, such that

$$\begin{aligned} Tx=\mathcal {R}x+x^0,\quad x\in {\mathbb {R}}^n, \end{aligned}$$

(5.1)

where $\mathcal {R}=\big \{\mathcal {R}_{ij}\big \}_{i,j=1}^n$ is an orthogonal matrix of ${\mathbb {R}}^n$, and $x^0\in {\mathbb {R}}^n$. Let

$$\begin{aligned} \textbf{n}=\mathcal {R}^t e_n\in \mathbb {S}^{n-1}, \end{aligned}$$

where $e_n$ denotes the n-th canonical vector of ${\mathbb {R}}^n$, i.e. $e_n=(0,\dots ,0,1)$, $\mathcal {R}^t$ is the transpose matrix of $\mathcal {R}$, and $\mathbb {S}^{n-1}$ is the unit sphere on ${\mathbb {R}}^n$.

Here we introduce the geometric notion of transversality, which was already used in [10] in a wider sense. The definition given here suffices to our purposes.

Definition 1

(Transversality) Let $\phi :U\rightarrow {\mathbb {R}}$ be a Lipschitz continuous function on $U\subset {\mathbb {R}}^{n-1}$ open. We say that a unit vector $\textbf{n}\in \mathbb {S}^{n-1}$ is transversal to $\phi$ if there exists $\kappa >0$ such that

$$\begin{aligned} \textbf{n}\cdot \nu (x')\ge \kappa \quad \text {for } \mathcal {H}^{n-1}\text {-a.e. }x'\in U, \end{aligned}$$

where $\nu$ denotes the outward normal to $G_\phi$ with respect to the subgraph $S_\phi$.

The next proposition shows a very interesting feature: the transversality of $\textbf{n}\in \mathbb {S}^{n-1}$ to a Lipschitz function $\phi$ is equivalent to the graphicality (and subgraphicality) of $\phi$ with respect to any reference system having $e_n=\textbf{n}$, that is after performing a rotation of the axes through $\mathcal {R}$, the graph and subgraph of $\phi$ are mapped onto the graph and subgraph of another function $\psi$– see identities (5.2) below.

Proposition 2

Let $U\subset {\mathbb {R}}^{n-1}$ be open, $\phi :\,U\rightarrow {\mathbb {R}}$ be a Lipschitz function, let T be an isometry of the form (5.1), and let $\textbf{n}= \mathcal {R}^te_n$.

(i) If there exists an L-Lipschitz function $\psi :V\rightarrow {\mathbb {R}}$ such that

$$\begin{aligned} TG_\phi =G_{\psi }\quad \text {and}\quad TS_\phi =S_{\psi }\cap T(U\times {\mathbb {R}}), \end{aligned}$$

(5.2)

then we have the transversality condition

$$\begin{aligned} \textbf{n}\cdot \nu (x')\ge \frac{1}{\sqrt{1+L^2}}\quad \text {for } \mathcal {H}^{n-1}\hbox {-a.e. } x'\in U. \end{aligned}$$

(5.3)

(ii) Viceversa, if $\phi \in C^k(U)$ for some $k\in {\mathbb {N}}$ and (5.3) holds, then there exist $V\subset {\mathbb {R}}^{n-1}$ open, and a function $\psi \in C^k(V)$ such that $\Vert \nabla \psi \Vert _{L^\infty (V)}\le L$ and (5.2) holds true.

Let us comment on this result. Part (i) states that if $G_\phi$ and $S_\phi$ are, respectively, the graph and subgraph of an L-Lipschitz function $\psi$ with respect to the reference system $z=(z',z_n)$ having $\textbf{n}=e_n$, then the quantitative transversality estimate (5.3) holds true.

Part (ii) states the opposite in the $C^k$ case: the transversality condition (5.3) implies the graphicality and subgraphicality of $\phi$ with respect to the coordinate system $z=(z',z_n)$, and it also provides a Lipschitz estimate to $\psi$.

Before starting the proof, we need to introduce the so-called transition map ${\mathcal {C}}$ from $\phi$ to $\psi$. Under the same notation as Proposition 2, the transition map ${\mathcal {C}}:U\rightarrow V$ is defined as

$$\begin{aligned} {\mathcal {C}}x':=\Pi \,T\big (x',\phi (x')\big ). \end{aligned}$$

Here $\Pi :{\mathbb {R}}^n\rightarrow {\mathbb {R}}^{n-1}$ is the projection map $\Pi (x',x_n)=x'$. Observe that, when identities (5.2) hold true, by the very definition of ${\mathcal {C}}$ we have the equation

$$\begin{aligned} T\big (x',\phi (x')\big )=\big ({\mathcal {C}}x',\psi \big ({\mathcal {C}}x'\big )\big ) \end{aligned}$$

In particular, this implies that ${\mathcal {C}}$ is a bijection, with inverse function ${\mathcal {C}}^{-1}:V\rightarrow U$ given by

$$\begin{aligned} {\mathcal {C}}^{-1}z'=\Pi \,T^{-1}\big (z',\psi (z')\big ). \end{aligned}$$

Also, since $\phi ,\psi$ are Lipschitz continuous, then ${\mathcal {C}}$ is a bi-Lipschitz tranformation from U to V.

Fig. 3

The new graph after the rigid motion T

Proof of Proposition 2

(i) By Rademacher’s theorem, the normal vector $\nu$ to $G_\phi$ outward with respect to $S_\phi$ is well defined $\mathcal {H}^{n-1}$-almost everywhere, and thanks to (5.2) and the definition of ${\mathcal {C}}$, we may write (Fig.  3)

$$\begin{aligned} \nu (x')=\frac{(-\nabla \phi (x'),1)}{\sqrt{1+|\nabla \phi (x')|^2}}=\mathcal {R}^t\Bigg (\frac{(-\nabla \psi ({\mathcal {C}}x'),1)}{\sqrt{1+|\nabla \psi ({\mathcal {C}}x')|^2}}\Bigg )\quad \mathcal {H}^{n-1}\text { -a.e. } x'\in U. \end{aligned}$$

(5.4)

Therefore, since $\mathcal {R}\textbf{n}=e_n$ and $|\nabla \psi |\le L$, from (5.4) we infer

$$\begin{aligned} \textbf{n}\cdot \nu (x')=e_n\cdot \mathcal {R}\nu (x')=\frac{1}{\sqrt{1+|\nabla \psi ({\mathcal {C}}x')|^2}}\ge \frac{1}{\sqrt{1+L^2}}\quad \text {for } \mathcal {H}^{n-1}\text { -a.e. } x'\in U. \end{aligned}$$

(5.5)

(ii) Assume $\phi \in C^k(U)$ and that (5.3) is in force.

Consider the $C^k$-function $f:U\times {\mathbb {R}}\rightarrow {\mathbb {R}}$, defined as $f(x):=x_n-\phi (x')$, so that

$$\begin{aligned} \{f=0\}=G_\phi \quad \text {and}\quad \{f<0\}=S_\phi . \end{aligned}$$

(5.6)

Now let $\tilde{f}:T(U\times {\mathbb {R}})\rightarrow {\mathbb {R}}$ be the function defined as $\tilde{f}(z)=f(x)$ for $z=Tx$. Recalling $\mathcal {R}\textbf{n}=e_n$, via the chain rule we compute

$$\begin{aligned} \frac{\partial \tilde{f}(z)}{\partial z_n}=\mathcal {R}_{nn}-\sum _{k=1}^{n-1}\frac{\partial \phi (x')}{\partial x'_k}\,\mathcal {R}_{nk}=(-\nabla \phi (x'),1)\cdot \textbf{n}. \end{aligned}$$

(5.7)

Thus, from expression (5.4) of $\nu (x')$ and estimate (5.3), we obtain

$$\begin{aligned} \frac{\partial \tilde{f}(z)}{\partial z_n}=\sqrt{1+|\nabla \phi (x')|^2}\,\nu (x')\cdot \textbf{n}\ge \frac{1}{\sqrt{1+L^2}}\quad \text {for } z=Tx. \end{aligned}$$

(5.8)

Therefore, owing to (5.8) and the implicit function theorem, we immediately infer the existence of a function $\psi \in C^k(V)$, with $V\subset {\mathbb {R}}^{n-1}$ open, such that

$$\begin{aligned} \{\tilde{f}=0\}=G_\psi \quad \text {and}\quad \{\tilde{f}=0\}=S_\psi \cap T(U\times {\mathbb {R}}). \end{aligned}$$

Thereby, (5.2) follows from the very definition of $\tilde{f}$ and (5.6).

Finally, by using (5.5) we infer that $|\nabla \psi ({\mathcal {C}}x')|\le L$ for all $x'\in U$, whence $\Vert \nabla \psi \Vert _{L^\infty (V)}\le L$ since the transition map ${\mathcal {C}}$ is a bijection.

Remark 1

We point out that inequality (5.8), when evaluated at points $z=T\big (x',\phi (x')\big )$, holds true if $\phi$ and $\psi$ are merely Lipschitz continuous and satisfy (5.2).

Indeed, since ${\mathcal {C}}$ is a bi-Lipschitz map, by Rademacher’s Theorem and the chain rule we may perform the same computations as (5.7)-(5.8) and get

$$\begin{aligned} \mathcal {R}_{nn}-\sum _{k=1}^{n-1}\frac{\partial \phi (x')}{\partial x'_k}\,\mathcal {R}_{nk}\ge \frac{1}{\sqrt{1+L^2}}\quad \text {for } \mathcal {H}^{n-1}\text {-a.e. } x'\in U. \end{aligned}$$

(5.9)

By making use of this information, we now show that the transversality condition (5.3) is inherited by the regularized function $M_m(\phi )$. This is the content of the following proposition

Proposition 3

Let $U,\,V\subset {\mathbb {R}}^{n-1}$ be open bounded, let T be an isometry of the form (5.1), and $\textbf{n}=\mathcal {R}^t e_n$. Let $\phi :\,U\rightarrow {\mathbb {R}}$ and $\psi :\,V\rightarrow {\mathbb {R}}$ be L-Lipschitz functions satisfying (5.2). If we set

$$\begin{aligned} U_m:=\big \{x'\in U:\,\textrm{dist}(x',\partial U)>\tfrac{1}{m}\big \} \end{aligned}$$

and for some sequence $\{c_m\}_{m\in {\mathbb {N}}}\subset {\mathbb {R}}$ we define

$$\begin{aligned} \phi _m(x'):=M_m(\phi )(x')+c_m\quad \text {for } x'\in U_m, \end{aligned}$$

then $\phi _m$ is L-Lipschitz continuous on $U_m$ and

$$\begin{aligned} \Vert \phi _m-\phi \Vert _{L^\infty (U_m)}\le \frac{L}{m}+|c_m|. \end{aligned}$$

(5.10)

In addition, we have the transversality condition

$$\begin{aligned} \mathcal {R}_{nn}-\sum _{k=1}^{n-1}\frac{\partial \phi _m}{\partial x'_k}(x')\mathcal {R}_{nk}=\big (-\nabla \phi _m(x'),1\big )\cdot \textbf{n}\ge \frac{1}{\sqrt{1+L^2}}\quad \text {for all }x'\in U_m, \end{aligned}$$

(5.11)

and

$$\begin{aligned} \textbf{n}\cdot \nu _m(x')\ge \frac{1}{1+L^2}\quad \text {for all } x'\in U_m, \end{aligned}$$

(5.12)

where $\nu _m$ is the outward unit normal to $G_{\phi _m}$ with respect to the subgraph $S_{\phi _m}$.

Proof

Let $x'_0\in U_m$. By multiplying (5.9) with $\rho _m(x'_0-x')$ and integrating in $x'$ we immediately obtain

$$\begin{aligned} \mathcal {R}_{nn}-\sum _{k=1}^{n-1}\frac{\partial M_m(\phi )(x_0')}{\partial x'_k}\,\mathcal {R}_{nk}\ge \frac{1}{\sqrt{1+L^2}}\quad \text {for all } x'_0 \in U_m, \end{aligned}$$

and (5.11) holds true.

Next, from the L-Lipschitz continuity of $\phi$, we have

$$\begin{aligned} \begin{aligned} \big |M_m(\phi )(x')-M_m(\phi )(y')\big |&\le \int _{{\mathbb {R}}^{n-1}} \big |\phi (x'-z')-\phi (y'-z')\big |\,\rho _m(z')\,dz'\\&\le L\,|x'-y'|\,\int _{{\mathbb {R}}^{n-1}}\rho _m(z')\,dz'=L\,|x'-y'| \end{aligned} \end{aligned}$$

for all $x',y'\in U_m$, hence $\phi _m$ is L-Lipschitz continuous as well. From this and (5.11), we get

$$\begin{aligned} \textbf{n}\cdot \nu _m(x')=\textbf{n}\cdot \frac{\big (-\nabla M_m(\phi )(x'),1\big )}{\sqrt{1+|\nabla M_m(\phi )(x')|^2}}\ge \frac{1}{1+L^2}\quad \text {for all }x' \in U_m, \end{aligned}$$

that is (5.12). Next, since $\rho _m$ is radially symmetric and $\phi$ is L-Lipschitz continuous, for all $x'\in U_m$ we get

$$\begin{aligned} \big |M_m(\phi )(x')-\phi (x')\big |&\le \int _{B'_{1/m}}\big | \phi (x'+y')-\phi (x')\big |\,\rho _m(y')\,dy'\\&\le \int _{B'_{1/m}}L\,|y'|\,\rho _m(y')\,dy'\le \frac{L}{m}\,, \end{aligned}$$

and thus (5.10) follows. $\square$

Since we have proven that the regularized function $M_m(\phi )$ satisfies the transversality condition, Part (ii) of Proposition 2 entails its “graphicality” with respect to the coordinate system having $\textbf{n}=e_n$.

Proposition 4

Under the same assumptions of Proposition 3, there exist $V_m\subset {\mathbb {R}}^{n-1}$ open bounded such that

$$\begin{aligned} \textrm{dist}_\mathcal {H}(V_m,V)\le \frac{2\sqrt{1+L^2}}{m}+|c_m|, \end{aligned}$$

(5.13)

and a function $\psi _m\in C^\infty (V_m)$ satisfying

$$\begin{aligned}{} & {} \Vert \nabla \psi _m\Vert _{L^\infty (V_m)}\le 2(1+L^2), \end{aligned}$$

(5.14)

$$\begin{aligned}{} & {} TG_{\phi _m}=G_{\psi _m}\quad \text {and}\quad TS_{\phi _m}=S_{\psi _m}\cap T\big (U_m\times {\mathbb {R}}\big ). \end{aligned}$$

(5.15)

If in addition $V_m\cap V\ne \emptyset$, then

$$\begin{aligned} \Vert \psi _m-\psi \Vert _{L^\infty (V_m\cap V)}\le \frac{L(1+L)}{m}+(1+L)\,|c_m|, \end{aligned}$$

(5.16)

and if ${\mathcal {C}}_m$ is the transition map of $\phi _m$, we have that

$$\begin{aligned} \Vert {\mathcal {C}}_m-{\mathcal {C}}\Vert _{L^\infty (U_m)}+\Vert {\mathcal {C}}^{-1}_m-{\mathcal {C}}^{-1}\Vert _{L^\infty (V_m\cap V)}\le c(n)\,(1+L^2)\Big (\frac{1}{m}+|c_m|\Big ). \end{aligned}$$

(5.17)

Proof

From the results of Part (ii) of Proposition 2 and (5.12), there exist $V_m\subset {\mathbb {R}}^{n-1}$ open bounded, and a function $\psi _m\in C^\infty (V_m)$ such that (5.15) holds. Also, owing to (5.3), we immediately obtain (5.14).

Now we recall that the transition map of $\phi _m$ is the function ${\mathcal {C}}_m:\, U_m\rightarrow V_m$ defined as ${\mathcal {C}}_m x'=\Pi \,T\big (x',\phi _m(x')\big )$, and for all $x'\in U_m$ we have

$$\begin{aligned} T\big (x',\phi (x')\big )=\big ( {\mathcal {C}}x',\psi ({\mathcal {C}}x')\big )\quad \text {and}\quad T\big (x',\phi _m(x')\big )=\big ( C_m x',\psi _m({\mathcal {C}}_m x')\big ), \end{aligned}$$

so that from (5.10) we infer

$$\begin{aligned} \begin{aligned}&|c_m|+ \frac{L}{m}\ge |\phi _m(x')-\phi (x')|\\&\quad =\big |\big (x',\phi _m(x')\big )-\big (x',\phi (x')\big )\big |= \big |\big ({\mathcal {C}}_m x',\psi _m({\mathcal {C}}_m x')\big )-\big ({\mathcal {C}}x',\psi ({\mathcal {C}}x') \big )\big |, \end{aligned} \end{aligned}$$

for all $x'\in U_m$. In particular

(5.18)

The first inequality in (5.18) entails $\textrm{dist}_\mathcal {H}\big (V_m, {\mathcal {C}}(U_m)\big )\le \frac{L}{m}+|c_m|$.

On the other hand, by definition of $U_m$, for any $x'\in U$ we may find $x'_m\in U_m$ such that $|x'-x'_m|\le \frac{1}{m}$. Since $\Pi$ and T are 1-Lipschitz continuous, and $\phi$ is L-Lipschitz continuous, it follows that

$$\begin{aligned} |{\mathcal {C}}x'-{\mathcal {C}}x'_m|\le \big |\big ( x',\phi (x')\big )-\big ( x'_m,\phi (x'_m)\big )\big |\le \frac{\sqrt{1+L^2}}{m}, \end{aligned}$$

which implies $\textrm{dist}_\mathcal {H}\big ({\mathcal {C}}(U_m),V\big )\le \frac{\sqrt{1+L^2}}{m}$ since ${\mathcal {C}}(U)=V$. Hence, by using the triangle inequality we get

$$\begin{aligned} \textrm{dist}_\mathcal {H}\big (V_m,V\big )\le \textrm{dist}_\mathcal {H}\big (V_m,{\mathcal {C}}(U_m)\big )+\textrm{dist}_\mathcal {H}\big ({\mathcal {C}}(U_m),V\big )\le \frac{2\sqrt{1+L^2}}{m}+|c_m|, \end{aligned}$$

that is (5.13).

Next, on assuming that $V_m\cap V\ne \emptyset$, and ${\mathcal {C}}_m$ being a bijection between $U_m$ and $V_m$, we may take a point $y'\in V_m\cap V$ such that $y'={\mathcal {C}}_m x'$ for some $x'\in U_m$ From (5.18) we find

$$\begin{aligned} |{\mathcal {C}}_m x'-{\mathcal {C}}x'|=|y'-{\mathcal {C}}{\mathcal {C}}_m^{-1} y'|\le \frac{L}{m}+|c_m|, \end{aligned}$$

and

$$\begin{aligned} \big |\psi \big ( {\mathcal {C}}x'\big )-\psi _m\big ({\mathcal {C}}_m x'\big )\big |=\big | \psi \big ( {\mathcal {C}}{\mathcal {C}}^{-1}_m y'\big )-\psi _m(y')\big |\le \frac{L}{m}+|c_m|. \end{aligned}$$

By using these two estimates and the L-Lipschitz continuity of $\psi$, we obtain

$$\begin{aligned} \begin{aligned} |\psi (y')&-\psi _m(y')| \le |\psi (y')-\psi ({\mathcal {C}}{\mathcal {C}}_m^{-1}y')|+ |\psi ({\mathcal {C}}{\mathcal {C}}_m^{-1}y')-\psi _m(y')| \\&\le L\,|y'-{\mathcal {C}}{\mathcal {C}}_m^{-1} y'|+\frac{L}{m}+|c_m|\le \frac{L(1+L)}{m}+(1+L)\,|c_m|\quad \text {for all }y'\in V_m\cap V, \end{aligned} \end{aligned}$$

that is (5.16). Finally, by making use of (5.16) and a similar argument as in the proof of (5.18), we obtain (5.17). $\square$

The next proposition shows that if $\phi \in W^{2,q}$, then $\psi \in W^{2,q}$ as well. Namely, graphicality preserves Sobolev second-order regularity for Lipschitz functions.

Proposition 5

Under the same assumptions of Propositions 3-4, if in addition $\phi \in W^{2,q}_{loc}(U)$ for some $q\in [1,\infty ]$, then $\psi \in W^{2,q}_{loc}(V)$.

Proof

In the following proof, we will make use of Propositions 3-4 with $c_m\equiv 0$.

Fix $U_0\Subset U$ open, and set $V_0={\mathcal {C}}(U_0)$. Since $\textrm{dist}_\mathcal {H}(V_m,V)\rightarrow 0$ due to (5.13), from [9, Proposition 2.2.17] we may find $m_0>0$ large enough such that

$$\begin{aligned} V_0\Subset V\cap V_m \quad \text {for all }m>m_0. \end{aligned}$$

Now let

$$\begin{aligned} f_m(x)=x_n-M_m(\phi )(x')\quad \text {for } x\in U_m\times {\mathbb {R}}, \end{aligned}$$

and set $\widetilde{f}_m(y)\equiv f_m(x)$ for $y=Tx$. Then owing to (5.15), we have that $\widetilde{f}_m\big (y',\psi _m(y')\big )=0$ for all $y'\in V_m$. By differentiating this expression, we obtain

$$\begin{aligned} \frac{\partial \psi _m}{\partial y'_k}(y')=-\bigg (\frac{\partial \widetilde{f}_m}{\partial y_n}\big (y',\psi _m(y') \big )\bigg )^{-1}\bigg (\frac{\partial \widetilde{f}_m}{\partial y'_k}\big (y',\psi _m(y')\big )\bigg ), \end{aligned}$$

(5.19)

and from the chain rule, equation $\textbf{n}=\mathcal {R}^t e_n$, the definition of ${\mathcal {C}}_m^{-1}$ and (5.11), we have

$$\begin{aligned} \begin{aligned} \frac{\partial \widetilde{f}_m}{\partial y'_k}\big (y',\psi _m(y') \big )&=\mathcal {R}_{kn}-\sum _{l=1}^{n-1}\frac{\partial M_m(\phi )}{\partial x'_l}({\mathcal {C}}^{-1}_m y')\,\mathcal {R}_{kl} \\ \frac{\partial \widetilde{f}_m}{\partial y_n}\big (y',\psi _m(y') \big )&=\mathcal {R}_{nn}-\sum _{l=1}^{n-1}\frac{\partial M_m(\phi )}{\partial x'_l}({\mathcal {C}}^{-1}_m y')\mathcal {R}_{nl}\ge \frac{1}{\sqrt{1+L^2}}, \end{aligned} \end{aligned}$$

(5.20)

Moreover, thanks to (5.14) and the L-Lipschitz continuity of $M_m(\phi )$, the maps ${\mathcal {C}}_m$ are uniformly bi-Lipschitz, i.e.

$$\begin{aligned} \Vert \nabla {\mathcal {C}}_m\Vert _{L^\infty }+\Vert \nabla {\mathcal {C}}_m^{-1}\Vert _{L^\infty }\le C(n,L). \end{aligned}$$

Thanks to this piece of information and (5.17), we may apply Proposition 1 and get

$$\begin{aligned} \nabla M_m(\phi )({\mathcal {C}}^{-1}_m y')\rightarrow \nabla \phi ({\mathcal {C}}^{-1} y')\quad \hbox {for} \mathcal {H}^{n-1} \hbox {-a.e.} y'\in V_0 \end{aligned}$$

(5.21)

By combining (5.19)-(5.21), and by using dominated convergence theorem, we find that $\nabla \psi _m$ converges in $L^p(V_0)$ to some vector-valued function G for all $p\in [1,\infty )$. It then follows from (5.16) and the uniqueness of the distributional limit that $G=\nabla \psi$, hence

$$\begin{aligned} \nabla \psi _m\rightarrow \nabla \psi \quad \mathcal {H}^{n-1} \text { -a.e. in }V_0\text { and in } L^p(V_0). \end{aligned}$$

(5.22)

Next, we differentiate twice identity $\widetilde{f}_m\big (y',\psi _m(y')\big )=0$, and for $k,r=1,\dots ,n-1$ we obtain

$$\begin{aligned} \begin{aligned}&\frac{\partial ^2 \psi _m}{\partial y'_k \partial y'_r}(y')=-\bigg (\frac{\partial \widetilde{f}}{\partial y_n}\big (y',\psi _m(y')\big ) \bigg )^{-1}\\ \bigg \{&\frac{\partial ^2 \widetilde{f}}{\partial y'_k \partial y'_r}\big ( y',\psi _m(y')\big )+\frac{\partial ^2 \widetilde{f}}{\partial y'_k \partial y_n}\big ( y',\psi _m(y')\big )\,\frac{\partial \psi _m}{\partial y'_r}(y') \\&+\frac{\partial ^2 \widetilde{f}}{\partial y'_r \partial y_n}\big ( y',\psi _m(y')\big )\,\frac{\partial \psi _m}{\partial y'_k}(y') \\&+\frac{\partial ^2 \widetilde{f}}{\partial y_n \partial y_n}\big ( y',\psi _m(y')\big )\,\frac{\partial \psi _m}{\partial y'_k}(y')\, \frac{\partial \psi _m}{\partial y'_r}(y')\bigg \}, \end{aligned} \end{aligned}$$

(5.23)

while from the chain rule and the properties of ${\mathcal {C}}_m$, we obtain

$$\begin{aligned} \frac{\partial ^2 \widetilde{f}}{\partial y'_k \partial y'_r}\big ( y',\psi _m(y')\big )=-\sum _{l,t=1}^{n-1}\frac{\partial ^2 M_m(\phi )}{\partial x'_l\partial x'_t}({\mathcal {C}}^{-1}_m y')\,\mathcal {R}_{kl}\mathcal {R}_{rt}. \end{aligned}$$

(5.24)

Then, another application of Proposition 1 entails that

$$\begin{aligned} \nabla ^2 M_m(\phi )({\mathcal {C}}^{-1}_m y')\rightarrow \nabla ^2 \phi ({\mathcal {C}}^{-1} y')\quad \text {for } \mathcal {H}^{n-1}\text { -a.e. }y'\in V_0 \text { and in } L^q(V_0), \end{aligned}$$

in the Case $q\in [1,\infty )$. From this, (5.20), (5.22)-(5.24) and by using dominated convegence Theorem 3, we find that $\nabla ^2 \psi _m$ converges in $L^q(V_0)$ to some matrix valued function H. Whence $H=\nabla ^2 \psi$ due to the uniqueness of the distributional limit, and the proof in the Case $q\in [1,\infty )$ is complete due to the arbitrariness of $U_0$.

In the Case $q=\infty$, from (5.20), (5.23) and (5.24) we infer that $\{\psi _m\}_m$ is a sequence uniformly bounded in $W^{2,\infty }(V_0)$ with respect to m. Therefore, up to a subsequence, we have that $\psi _m$ weakly-$*$ converge in $W^{2,\infty }(V_0)$ to $\psi$, thus completing the proof. $\square$

At last, we close this section with the following intrinsic property of $W^{2,q}$ domains.

Corollary 1

Let $\Omega$ be a bounded Lipschitz domains such that $\partial \Omega \in W^{2,q}$ for some $q\in [1,\infty ]$. Then any Lipschitz local chart $\psi$ of $\partial \Omega$ is of class $W^{2,q}$.

Proof

From Definition 2, there exists a Lipschitz local chart $\phi \in W^{2,q}$ and an isometry T such that (5.2) holds. The thesis then follows from Proposition 5. $\square$

As a final remark, let us mention that both Proposition 5 and Corollary 1 can be easily extended to the $W^{k,q}$ Case.

6 Proof of Theorem 1

This section is devoted to the proof of Theorem 1, which is divided into a few steps.

From here onward, $m_0$ and $k_0$ will denote positive integers, possibly changing from line to line.

6.1 Covering of $\partial \Omega$

By Definition 1, for any $x_0\in \partial \Omega$, we may find an $L_\Omega$-Lipschitz function $\phi ^{x_0}:B'_{R_\Omega }\rightarrow {\mathbb {R}}$, and an isometry $T^{x_0}$ of $\mathbb {R}^n$ such that $T^{x_0}x_0=0$, and

$$\begin{aligned} \begin{aligned}&T^{x_0} \partial \Omega \cap \big (B'_{R_\Omega }\times (-\ell ,\ell )\big )=\big \{(y', \phi ^{x_0} (y')):\,y'\in B'_{R_\Omega }\big \}, \\&T^{x_0} \Omega \cap \big (B'_{R_\Omega }\times (-\ell ,\ell )\big )=\big \{(y',y_n):\,x'\in B'_{R_\Omega },\,-\ell<y_n<\phi ^{x_0} (y')\big \}, \end{aligned} \end{aligned}$$

where $\ell =R_\Omega (1+L_\Omega )$. Let us consider the open covering $\{B_{R_\Omega /8}(x_0)\}_{x_0\in \partial \Omega }$ of $\partial \Omega$.¹ By compactness, we may find a finite sequence of points $\{x^i\}_{i=1}^N\subset \partial \Omega$ such that

$$\begin{aligned} \partial \Omega \Subset \bigcup _{i=1}^N B_{\frac{R_\Omega }{8}}(x^i), \end{aligned}$$

(6.1)

as well as $L_\Omega$-Lipschitz functions $\phi ^i$ and isometries $T^i$ satisfying

$$\begin{aligned} \begin{aligned}&T^i \partial \Omega \cap \big (B'_{R_\Omega }\times (-\ell ,\ell )\big )=\big \{(y', \phi ^i(y')):\,y'\in B'_{R_\Omega }\big \}, \\&T^i \Omega \cap \big (B'_{R_\Omega }\times (-\ell ,\ell )\big )=\big \{(y',y_n):\,y'\in B'_{R_\Omega },\,-\ell<y_n<\phi ^i(y')\big \}. \end{aligned} \end{aligned}$$

(6.2)

We denote by $\mathcal {R}^i$ the orthogonal matrix of $T^i$, i.e. $T^i$ can be written as

$$\begin{aligned} T^i x=\mathcal {R}^i(x-x^i)\quad x\in {\mathbb {R}}^n. \end{aligned}$$

Notice also that the cardinality N of this covering of $\partial \Omega$ may be chosen satisfying

$$\begin{aligned} N\le c(n)\,\bigg (\frac{d_\Omega }{R_\Omega }\bigg )^n. \end{aligned}$$

(6.3)

We then set

$$\begin{aligned} \Omega _t:=\{x\in \Omega :\, \textrm{dist}(x,\partial \Omega )>t\}, \end{aligned}$$

so that by (6.1) we have

$$\begin{aligned} \overline{\Omega }\Subset W:=\bigcup _{i=1}^N B_{\frac{R_\Omega }{8}}(x^i)\cup \Omega _{\frac{R_\Omega }{32}}. \end{aligned}$$

(6.4)

Starting from this point, we construct a suitable partition of unity: let

$$\begin{aligned} \eta _i:=\tilde{\rho }_{\frac{R_\Omega }{32}}*\chi _{B_{\frac{3R_\Omega }{16}}(x^i)}\quad \text {and}\quad \eta _0:=\tilde{\rho }_{\frac{R_\Omega }{64}}*\chi _{\Omega _{\frac{3R_\Omega }{64}}}, \end{aligned}$$

where $\tilde{\rho }_{t}$ is the standard, radially symmetric convolution kernel on ${\mathbb {R}}^n$, and $\chi _A$ denotes the indicator function of a set A.

Standard properties of convolution ensure that $\eta _i\in C^\infty _c(B_{\frac{R_\Omega }{4}}(x^i))$, $\eta _0\in C^\infty _c(\Omega _{\frac{R_\Omega }{16}})$, $0\le \eta _i\le 1$,

$$\begin{aligned} \eta _i\ge 1 \quad \text {on }B_{\frac{R_\Omega }{8}}(x^i),\quad \eta _0\ge 1 \quad \text {on }\Omega _{\frac{R_\Omega }{32}}, \end{aligned}$$

and

$$\begin{aligned} |\nabla ^k \eta _i|\le \frac{c(n,k)}{R_\Omega ^k},\quad \text {for all } k\in {\mathbb {N}}. \end{aligned}$$

Therefore, by defining $\xi _i:\,W\rightarrow [0,1]$ as

$$\begin{aligned} \xi _i:=\frac{\eta _i}{\sqrt{\sum _{j=0}^N \eta _j}},\quad i=0,\dots ,N, \end{aligned}$$

then we have that $\xi _i\in C^\infty _c(B_{\frac{R_\Omega }{4}}(x^i))$ for $i=1,\dots ,N$, $\xi _0\in C^\infty _c(\Omega _{\frac{R_\Omega }{16}})$,

$$\begin{aligned} \sum _{i=0}^N\xi _i(x)=1\quad \text {for all }x\in W, \end{aligned}$$

(6.5)

and

$$\begin{aligned} |\nabla ^k \xi _i|\le \frac{c(n,k)}{R_\Omega ^k}\quad \text {on }W,\text { for all }k\in {\mathbb {N}}. \end{aligned}$$

(6.6)

6.2 Boundary defining function

Starting from the partition of unity $\{\xi _i\}_{i=0}^N$, and the local charts $\{\phi ^i\}_{i=1}^N$, we can construct the boundary defining function of $\partial \Omega$ as in [11, Proposition 5.43].

For any $\varepsilon \in [0,R_\Omega )$ and $j=1,\dots ,N$, we define the rotated cylinders

$$\begin{aligned} K^j_\varepsilon :=(T^j)^{-1}\big (B'_{R_\Omega -\varepsilon }\times (-\ell ,\ell )\big ), \end{aligned}$$

(6.7)

where $\ell =R_\Omega (1+L_\Omega )$. Let $f^j:K^j_0\rightarrow {\mathbb {R}}$ be the functions defined as

$$\begin{aligned} f^j(x):=z_n-\phi ^j(z'),\quad z=T^j x, \end{aligned}$$

and observe that from (6.2) we have

$$\begin{aligned} \begin{aligned} \{f^j=0\}&=\partial \Omega \cap K^j_0 \\ \{f^j<0\}&= \Omega \cap K^j_0 \end{aligned} \end{aligned}$$

(6.8)

A boundary defining function of $\overline{\Omega }$ is the function $F:W\rightarrow {\mathbb {R}}$ defined as

$$\begin{aligned} F(x):=\sum _{j=1}^N f^j(x)\,\xi _j(x)-\xi _0(x), \end{aligned}$$

(6.9)

where the product $f^j(x)\,\xi _j(x)$ is set equal to zero if $x\not \in \textrm{supp}\,\xi _j$. Since each $f^j$ is Lipschitz continuous, so is the function F.

Thanks to the properties of $\{\xi _j\}_{j=0}^N$, (6.2) and (6.8), it is easily seen that

$$\begin{aligned} \Omega =\{x\in W:\,F(x)<0\}\quad \text {and}\quad \partial \Omega =\{x\in W:\,F(x)=0\}. \end{aligned}$$

(6.10)

6.3 Regularization and definition of the smooth approximating sets $\omega _m,\Omega _m$

For $i=1,\dots ,N$, we can define the smooth functions $\phi ^i_m,\widetilde{\phi }^i_m:B'_{R_\Omega -\frac{1}{m}}\rightarrow {\mathbb {R}}$ as

$$\begin{aligned}&\phi ^i_m:=M_m(\phi ^i)+\Vert M_m(\phi ^i)-\phi ^i\Vert _{L^\infty (B'_{R_\Omega -1/m})}+\frac{L_\Omega }{m} \nonumber \\&\text {and}\nonumber \\&\widetilde{\phi }^i_m:=M_m(\phi ^i)-\Vert M_m(\phi ^i)-\phi ^i\Vert _{L^\infty (B'_{R_\Omega -1/m})}-\frac{L_\Omega }{m}\,. \end{aligned}$$

(6.11)

From the results of Proposition 3, we deduce that $\phi ^i_m,\widetilde{\phi }^i_m\in C^\infty$ are $L_\Omega$-Lipschitz functions, and

$$\begin{aligned} \begin{aligned} \frac{L_\Omega }{m}&\le \phi ^i_m(y')-\phi ^i(y')\le \frac{3\,L_\Omega }{m} \\ \frac{L_\Omega }{m}&\le \phi ^i(y')-\widetilde{\phi }^i_m(y')\le \frac{3\,L_\Omega }{m}, \end{aligned} \end{aligned}$$

(6.12)

for all $y'\in B'_{R_\Omega -1/m}$ and $i=1,\dots ,N$. Taking inspiration from (6.8) and (6.10), we are led to define the functions

$$\begin{aligned} \begin{aligned}&f^j_m(x):=z_n-\phi ^j_m(z') \\&\tilde{f}^j_m(x):=z_n-\widetilde{\phi }^j_m (z'),\quad z=T^j x\in B'_{R_\Omega -\frac{1}{m}}\times (-\ell ,\ell ), \end{aligned} \end{aligned}$$

(6.13)

and functions $F_m,\widetilde{F}_m:W\rightarrow {\mathbb {R}}$ defined as

$$\begin{aligned} \begin{aligned} F_m(x):=\sum _{j=1}^N f^j_m(x)\,\xi _j(x)-\xi _0(x) \\ \widetilde{F}_m(x):=\sum _{j=1}^N \tilde{f}^j_m(x)\,\xi _j(x)-\xi _0(x), \end{aligned} \end{aligned}$$

(6.14)

where the products $f^j_m(x)\,\xi _j(x)$ and $\tilde{f}^j_m(x)\,\xi _j(x)$ have to be interpreted equal to zero when $x\not \in \textrm{supp}\,\xi _j$.

Clearly, $F_m$ and $\widetilde{F}_m$ are $C^\infty$-smooth functions on W, and since

$$\begin{aligned} \frac{L_\Omega }{m}\le f^j(x)-f^j_m(x)<\frac{3\,L_\Omega }{m},\quad \frac{L_\Omega }{m}\le \tilde{f}^j_m(x)-f^j(x)<\frac{3\,L_\Omega }{m} \end{aligned}$$

(6.15)

for all $x\in K^j_{1/m}$ thanks to (6.12), we then have

$$\begin{aligned} \frac{L_\Omega }{m}\le F(x)-F_m(x)\le \frac{3\,L_\Omega }{m},\quad \frac{L_\Omega }{m}\le \widetilde{F}_m(x)-F(x)\le \frac{3\,L_\Omega }{m}\quad \text {for all }x\in W. \end{aligned}$$

(6.16)

The approximating open sets $\Omega _m,\,\omega _m$ are thus defined as follows

$$\begin{aligned} \Omega _m:=\{x\in W:\,F_m(x)<0\}\quad \text {and}\quad \omega _m:=\{x\in W:\,\widetilde{F}_m(x)<0\}, \end{aligned}$$

(6.17)

with boundaries

$$\begin{aligned} \partial \Omega _m= \{x\in W:\,F_m(x)=0\}\quad \text {and}\quad \partial \omega _m= \{x\in W:\,\widetilde{F}_m(x)=0\}. \end{aligned}$$

(6.18)

In particular, since $F_m(x)<F(x)<\widetilde{F}_m(x)$ for all $x\in W$, owing to (6.10) we have

$$\begin{aligned} \omega _m\Subset \Omega \Subset \Omega _m\quad \text {for all } m\in {\mathbb {N}}. \end{aligned}$$

We now proceed to prove the remaining properties of Theorem 1 for the outer sets $\Omega _m$. The proofs for the inner sets $\omega _m$ are analogous.

6.4 $\partial \Omega _m,\partial \omega _m$ are smooth manifolds

Let us show that $\partial \Omega _m$ is a smooth manifold, with local charts $\{\psi ^i_m\}_{i=1}^N$ defined on the same coordinate systems as $\{\phi ^i\}_{i=1}^N$.

We fix a constant $\varepsilon _0\in (0,R_\Omega /4)$, and for all $i=1,\dots ,N$ we set

$$\begin{aligned} F^i(y)=F(x) \quad \text {and}\quad F^i_m(y)=F_m(x)\quad \text {for } y=T^i x, x\in W. \end{aligned}$$

Namely $F^i=F\circ (T^i)^{-1}$ and $F^i_m=F_m\circ (T^i)^{-1}$.

Owing to (6.2) we have

$$\begin{aligned} \begin{aligned} \partial \Omega \cap K^i_0\cap K^j_0=(T^i)^{-1}&G_{\phi ^i}\cap K^j_0=(T^j)^{-1}G_{\phi ^j}\cap K^i_0 \\&\text {and} \\ \Omega \cap K^j_0\cap K^i_0=(T^i)^{-1}&S_{\phi ^i}\cap K^j_0\cap K^i_0=(T^j)^{-1}S_{\phi ^j}\cap K^i_0\cap K^j_0, \end{aligned} \end{aligned}$$

(6.19)

whenever $\partial \Omega \cap K^i_0\cap K^j_0\ne \emptyset$.

This piece of information will allow us to use the transversality property. Specifically, thanks to (6.19) we may apply Propositions 2-3 with functions $\phi =\phi ^j$, $\psi =\phi ^i$, isometry $T=T^i(T^j)^{-1}$, and defining set

$$\begin{aligned} U=U^{j,i}=\Pi \Big ( G_{\phi ^j}\cap T^j K^i_0\Big )\subset B'_{R_\Omega }. \end{aligned}$$

Claim 1. There exists $m_0>0$ such that, for all $i=1,\dots ,N$, for all $m\ge m_0$ and all $x\in \big \{\frac{-3L_\Omega }{m_0}\le F\le \frac{3L_\Omega }{m_0}\big \}\cap K^i_{\varepsilon _0}$, we have

$$\begin{aligned} \frac{\partial F^i_m}{\partial y_n}(y)\ge \frac{1}{2\sqrt{1+L_\Omega ^2}},\quad \text {for all }y=T^i x\in B'_{R_\Omega -\varepsilon _0}\times (-\ell ,\ell ). \end{aligned}$$

(6.20)

Suppose by contradiction this is false; then for every $k\in {\mathbb {N}}$, we may find $m_k\ge k$ and a sequence $x^k\in \big \{-\frac{3L_\Omega }{k}\le F\le \frac{3L_\Omega }{k}\big \}$ such that $y^k=T^i x^k\in B'_{R_\Omega -\varepsilon _0}\times (-\ell ,\ell )$ and

$$\begin{aligned} \frac{\partial F_{m_k}^i}{\partial y_n}(y^k)<\frac{1}{2\sqrt{1+L_\Omega ^2}},\quad \text {for all }k\in {\mathbb {N}}\end{aligned}$$

(6.21)

By compactness, we may extract a subsequence, still labeled as $x^k$, such that $x^k\rightarrow x^0$, and in particular $x^0\in \overline{K^i_0}$ and $F(x^0)=0$, hence $x^0\in \partial \Omega \cap \overline{K^i_0}$ due to (6.10).

Then, by the chain rule we have

$$\begin{aligned} \frac{\partial f^i_m}{\partial y_n}(x)=1\quad \text {and}\quad \frac{\partial f^j_m}{\partial y_n}(x)=\big (\mathcal {R}^j(\mathcal {R}^i)^t\big )_{nn}-\sum _{s=1}^{n-1}\frac{\partial \phi ^{j}_m}{\partial z'_s}\big (z' \big )\,\big (\mathcal {R}^j(\mathcal {R}^i)^t\big )_{sn}, \end{aligned}$$

(6.22)

if $x\in \textrm{supp}\,\xi _j$, where $z'=\Pi \,T^j x$. We now distinguish two cases:

(i) $j\in \{1,\dots ,N\}$ is such that $x^0\not \in \textrm{supp}\,\xi _j$. Then $\textrm{dist}\big (x^0,\textrm{supp}\,\xi _j\big )>0$, hence $x^k\not \in \textrm{supp}\,\xi _j$ for all $k\ge k_0$ large enough.

(ii) $j\in \{1,\dots ,N\}$ is such that $x^0\in \textrm{supp}\,\xi _j$. In this case, it follows that $x^0\in \partial \Omega \cap K^i_0\cap B_{\frac{R_\Omega }{4}}(x^j)$, so that from (6.19) we have $T^j x^0\in G_{\phi ^j}\cap B_{\frac{R_\Omega }{4}}\cap T^j \overline{K^i_{\varepsilon _0}}$. By setting $(z^k)'=\Pi \,T^j x^k$, we thus have

$$\begin{aligned} B'_{\frac{1}{m_k}}\big ((z^k)'\big )\Subset \Pi \Big ( G_{\phi ^j}\cap T^j K^i_0\Big ) , \end{aligned}$$

for all $k\ge k_0$ large enough. Recalling the remarks after (6.19), by applying Proposition 3, and in particular the transversality property (5.11) in (6.22), we infer

$$\begin{aligned} \frac{\partial f^j_{m_k}}{\partial y_n}\big (x^k\big )=\big (\mathcal {R}^j(\mathcal {R}^i)^t\big )_{nn}-\sum _{s=1}^{n-1}\frac{\partial \phi ^{j}_{m_k}}{\partial z'_s}\big ((z^k)' \big )\,\big (\mathcal {R}^j(\mathcal {R}^i)^t\big )_{sn}\ge \frac{1}{\sqrt{1+L^2_\Omega }}, \end{aligned}$$

provided $k\ge k_0$ is large enough.

In both cases, we have found that

$$\begin{aligned} \frac{\partial f^j_{m_k}}{\partial y_n}(x^k)\,\xi _j\big (x^k\big )\ge \frac{\xi _j(x^k)}{\sqrt{1+L_\Omega ^2}}\quad \text {for all } j=1,\dots ,N \text { and } k\ge k_0. \end{aligned}$$

(6.23)

Also, owing to (6.15) and (6.8) we have

$$\begin{aligned} \begin{aligned} |f^j_{m_k}(x^k)|\,\Big |\frac{\partial \xi _j(x^k)}{\partial y_n}\Big |&\le |f^j_{m_k}(x^k)-f^j(x^k)|\,|\nabla \xi _j(x^k)|+|f^j(x^k)|\,|\nabla \xi _j(x^k)|\\&\le \frac{1}{m_k}+|f^j(x^k)|\,|\nabla \xi _j(x^k)| \xrightarrow {k\rightarrow \infty } |f^j(x^0)|\,|\nabla \xi _j(x^0)|=0, \end{aligned} \end{aligned}$$

and $|\nabla \xi _0(x^k)|\rightarrow |\nabla \xi _0(x^0)|=0$ since $x^0\in \partial \Omega$. By coupling this piece of information with (6.5), (6.21) and (6.23), we finally obtain

$$\begin{aligned} \begin{aligned} \frac{1}{2\sqrt{1+L_\Omega ^2}} >\frac{\partial F^i_{m_k}}{\partial y_n}(y^k)&=\sum _{j=1}^N\frac{\partial f^j_{m_k}}{\partial y_n}(x^k)\,\xi _j(x^k)+\sum _{j=1}^N f^j_{m_k}(x^k)\,\frac{\partial \xi _j}{\partial y_n}(x^k)-\frac{\partial \xi _0}{\partial y_n}(x^k)\\&\ge \sum _{j=1}^N \frac{\xi _j(x^k)}{\sqrt{1+L_\Omega ^2}}+\sum _{j=1}^N f^j_{m_k}(x^k)\,\frac{\partial \xi _j}{\partial y_n}(x^k)-\frac{\partial \xi _0}{\partial y_n}(x^k)\\&\xrightarrow {k\rightarrow \infty }\sum _{j=1}^N\frac{\xi _j(x^0)}{\sqrt{1+L_\Omega ^2}}=\frac{1}{\sqrt{1+L_\Omega ^2}}, \end{aligned} \end{aligned}$$

which is a contradiction, and thus (6.20) holds true.

Claim 2. There exists $m_0>0$ such that $\forall y'\in B'_{R_\Omega -\varepsilon _0}$, $\forall m\ge m_0$, $\exists y_n\in (-\ell ,\ell )$ with $y=(y',y_n)=T^i x\in T^i W$ satisfying $F_m^i(y)\ge 0$.

Again, assume by contradiction this is false. Then for all $k\in {\mathbb {N}}$, we may find sequences $m_k\ge k$ and $(y^k)'\in B'_{R_\Omega -\varepsilon _0}$ such that

$$\begin{aligned} F^i_{m_k}\big ( (y^k)',y_n)<0\quad \text {for all }y_n\in (-\ell ,\ell ) \text { such that } \big ((y^k)',y_n\big )\in T^i W. \end{aligned}$$

(6.24)

By compactness, we may find a subsequence, still labeled as $(y^k)'$, satisfying $(y^k)'\rightarrow (y^0)'\in \overline{B}'_{R_\Omega -\varepsilon _0}$. Fix $w_n\in (-\ell ,\ell )$ such that $\big ((y^0)',w_n\big )\in T^i W$, and let $\{w^k_n\}_{k\in {\mathbb {N}}}\subset {\mathbb {R}}$ be a sequence satisfying $w^k_n\xrightarrow {k\rightarrow \infty } w_n$. Then $\big ( (y^k)', w_n^k\big )\rightarrow \big ( (y^0)', w_n\big )$, so that $\big ( (y^k)', w_n^k)\in T^i W$ for $k\ge k_0$ large enough being W open, and from (6.24) we have $F^i_{m_k}\big ((y^k)',w_n^k \big )<0$. By using (6.16) and the Lipschitz continuity of F, it is readily shown that

$$\begin{aligned} \lim _{k\rightarrow \infty } F^i_{m_k}\big ( (y^k)',w_n^k)=F^i\big ((y^0)', w_n), \end{aligned}$$

whence $F^i\big ((y^0)', w_n)\le 0$ for all $w_n$ as above, but this contradicts the fact that $F^i\big ( (y^0)',w_n\big )>0$ whenever $w_n>\phi ^i\big ((y^0)'\big )$ due to (6.10), hence Claim 2 is proven.

Now let $y'\in B'_{R_\Omega -\varepsilon _0}$; by (6.16) and since $F^i\big (y',\phi ^i(y')\big )=0$, we have $F^i_m\big (y',\phi ^i(y')\big )<0$. Thus, owing to Claim 2 we may find $y_n$ such that $F^i_m(y',y_n)=0$.

The monotonicity property (6.20) of Claim 1, and the fact that $\partial \Omega _m=\{F_m=0\}\subset \{\frac{L_\Omega }{m}\le F\le \frac{3L_\Omega }{m}\}$ due to (6.16) ensure that such point $y_n$ is unique for all $y'\in B'_{R_\Omega -\varepsilon _0}$. This entails the existence of a function $\psi ^i_m:\,B'_{R_\Omega -\varepsilon _0}\rightarrow {\mathbb {R}}$ such that

$$\begin{aligned} F^i_m\big (y',\psi ^i_m(y')\big )=0 \quad \text {for all }y'\in B'_{R_\Omega -\varepsilon _0}. \end{aligned}$$

(6.25)

Owing to (6.10) and (6.16), we also have that $\psi ^i_m(y')>\phi ^i(y')$ for all $y'\in B'_{R_\Omega -\varepsilon _0}$.

Then, since $F^i_m$ are $C^\infty$-smooth, thanks to (6.20) and (6.25), we may repeat the proof of the implicit function theorem in order to obtain $\psi ^i_m\in C^\infty \big (B'_{R_\Omega -\varepsilon _0}\big )$.

Moreover, via a compactness argument as in Claim 1-2 and (6.1), one can prove that

$$\begin{aligned} \begin{aligned} \bigg \{-\frac{3L_\Omega }{m}\le \,&\, F\le \frac{3 L_\Omega }{m}\bigg \}\subset \bigcup _{i=1}^N B_{\frac{R}{8}}(x^i)\\ \bigg \{-\frac{3L_\Omega }{m}\le \,&\, F\le \frac{3 L_\Omega }{m}\bigg \}\cap \textrm{supp}\,\xi _0=\emptyset , \quad \text {for all } m>m_0, \end{aligned} \end{aligned}$$

(6.26)

so that, in particular, the cylinders $\big \{K^i_{2\varepsilon _0}\big \}_{i=1}^N$ are an open cover of $\partial \Omega _m$, and $\partial \Omega _m\cap \, \textrm{supp}\,\xi _0=\emptyset$ provided $m>m_0$ is large enough

We have thus proven that $\partial \Omega _m$ is a $C^\infty$-smooth manifold for $m>m_0$, with local boundary charts $\{\psi ^i_m\}_{i=1}^N$ defined on the same coordinate cylinders as $\{\phi ^i\}_{i=1}^N$, that is (see Fig. 1 above).

$$\begin{aligned} \begin{aligned}&T^i \partial \Omega _m \cap \big (B'_{R_\Omega -\varepsilon _0}\times (-\ell ,\ell )\big )=\big \{(y', \psi ^i_m(y')):\,y'\in B'_{R_\Omega -\varepsilon _0}\big \}, \\&T^i \Omega _m \cap \big (B'_{R_\Omega -\varepsilon _0}\times (-\ell ,\ell ) \big )=\big \{(y',y_n):\,y'\in B'_{R_\Omega -\varepsilon _0},\,-\ell<y_n<\psi ^i_m(y')\big \}. \end{aligned} \end{aligned}$$

(6.27)

6.5 Approximation properties

First, we show that there exists $m_0>0$ such that

$$\begin{aligned} \Vert \psi ^i_m-\phi ^i\Vert _{L^\infty (B'_{R_\Omega -2\,\varepsilon _0})}\le \frac{6\,L_\Omega \,\sqrt{1+L_\Omega ^2}}{m}\quad \text {for all } m>m_0. \end{aligned}$$

(6.28)

Assume by contradiction this is false; then we may find sequences $m_k\uparrow \infty$ and $(y^k)'\in B'_{R_\Omega -2\varepsilon _0}$ such that

$$\begin{aligned} \psi ^i_{m_k}\big ((y^k)' \big )-\phi ^i\big ((y^k)'\big )> \frac{6\,L_\Omega \,\sqrt{1+L_\Omega ^2}}{m_k} \end{aligned}$$

(6.29)

Up to a subsequence, we have $(y^k)'\rightarrow (y^0)'\in \overline{B}'_{R_\Omega -2\varepsilon _0}$, and $\psi ^i_{m_k}\big ((y^k)'\big )\rightarrow \ell _0\in {\mathbb {R}}$. Furthermore, since $\Big ( (y^k)',\psi ^i_m\big ((y^k)'\big )\Big )\in \{F^i_{m_k}=0\}\subset T^i\{\frac{L_\Omega }{m_k} \le F\le \frac{3L_\Omega }{m_k}\}$, we readily infer that $F^i\big ((y^0)',\ell _0 \big )=0$, whence $\ell _0=\phi ^i\big ( (y')^0\big )$ due to (6.10) and (6.2). By continuity we also have $\phi ^i\big ((y^k)'\big )\rightarrow \phi ^i\big ((y^0)'\big )$, which implies that

$$\begin{aligned} \psi ^i_{m_k}\big ((y^k)' \big )-\phi ^i\big ((y^k)'\big )\xrightarrow {k\rightarrow \infty }0. \end{aligned}$$

Then, for all $t\in [0,1]$, we have

$$\begin{aligned} \begin{aligned} \Big |F^i\Big ((y^k)', t\,\psi ^i_{m_k}\big ((y^k)' \big ) +(1-t)&\phi ^i\big ((y^k)'\big ) \Big )-F^i\Big ( (y^k)',\phi ^i\big ((y^k)' \big )\Big )\Big | \\&\le L_{F^i}\,t\,|\psi ^i_{m_k}\big ((y^k)' \big )-\phi ^i\big ((y^k)'\big )|\xrightarrow {k\rightarrow \infty }0, \end{aligned} \end{aligned}$$

where $L_{F^i}$ denotes the Lipschitz constant of $F^i$– recall that $F^i=F\circ (T^i)^{-1}$, and F is Lipschitz continuous. This implies that for all $k\ge k_0$ large enough, the line segment

$$\begin{aligned} \big \{(y^k)'\big \}\times \big [\phi ^i\big ((y^k)' \big ),\psi ^i_{m_k}\big ((y^k)' \big ) \big ]\subset T^i\Big \{ -\frac{3\,L_\Omega }{m_0}\le F\le \frac{3\,L_\Omega }{m_0}\Big \}. \end{aligned}$$

Therefore, by using (6.2), (6.10) (6.16), (6.20) and (6.29), we obtain

$$\begin{aligned} \begin{aligned} \frac{3\,L_\Omega }{m_k}&\ge F^i\Big ((y^k)',\phi ^i\big ((y^k)' \big ) \Big )-F^i_{m_k}\Big ((y^k)',\phi ^i\big ((y^k)' \big ) \Big )=-F^i_{m_k}\Big ((y^k)',\phi ^i\big ((y^k)' \big ) \Big ) \\&=F^i_{m_k}\Big ((y^k)',\psi ^i_{m_k}\big ((y^k)' \big )\Big )-F^i_{m_k}\Big ((y')^k,\phi ^i\big ((y')^k \big ) \Big ) \\&=\Bigg (\int _0^1\frac{\partial F^i_{m_k}}{\partial y_n}\Big ((y^k)',t\,\psi ^i_{m_k}\big ((y^k)'\big )+(1-t)\,\phi ^i\big ( (y^k)'\big ) \Big )\,dt \Bigg )\,\Big [\psi ^i_{m_k}\big ((y^k)'\big )-\phi ^i \big ((y^k)'\big ) \Big ] \\&> \frac{1}{2\sqrt{1+L_\Omega ^2}}\,\frac{6\,L_\Omega \, \sqrt{1+L^2}}{m_k}=\frac{3\,L_\Omega }{m_k},\quad \text {for all } k\ge k_0 \text { large enough,} \end{aligned} \end{aligned}$$

which is a contradiction, hence (6.28) holds true.

Now, recalling that $\{K^j_{2\varepsilon _0}\}_{j=1}^N$ is an open cover of $\partial \Omega$ and $\partial \Omega _m$, from (6.2), (6.27) and (6.28), one can easily obtain that

$$\begin{aligned} \textrm{dist}_\mathcal {H}\big (\partial \Omega _m,\partial \Omega ) \le \frac{6\,L_\Omega \sqrt{1+L_\Omega ^2}}{m}. \end{aligned}$$

This convergence property in the sense of Hausdorff immediately implies that $d_{\Omega _m}\le c(n)\,d_\Omega$, and $\lim _{m\rightarrow \infty }|\Omega _m\setminus \Omega |=0$—see for instance [9, Proposition 2.2.23]—and thus (3.1), (3.2) and (3.3) are proven.

Let us now introduce the transition maps related to the local charts of $\partial \Omega$ and $\partial \Omega _m$.

First of all, note that thanks to (6.27), we have

$$\begin{aligned} \begin{aligned} \partial \Omega _m\cap K^i_{\varepsilon _0}\cap K^j_{\varepsilon _0}=(T^i)^{-1}&G_{\psi ^i_m}\cap K^j_{\varepsilon _0}=(T^j)^{-1}G_{\psi ^j_m}\cap K^i_{\varepsilon _0} \\&\text {and}\\ \Omega _m\cap K^j_{\varepsilon _0} \cap K^i_{\varepsilon _0} =(T^i)^{-1}&S_{\psi ^i_m}\cap K^j_{\varepsilon _0}\cap K^i_{\varepsilon _0} =(T^j)^{-1}S_{\psi ^j_m}\cap K^i_{\varepsilon _0} \cap K^j_{\varepsilon _0}, \end{aligned} \end{aligned}$$

(6.30)

whenever $\partial \Omega _m\cap K^i_{\varepsilon _0}\cap K^j_{\varepsilon _0}\ne \emptyset$.

For all $i\in \{1,\dots ,N\}$, we define the set of indexes

$$\begin{aligned} \mathcal {I}_i:=\big \{ j\in \{1,\dots ,N\}:\,\partial \Omega \cap K^i_{2\varepsilon _0}\cap K^j_{2\varepsilon _0}\ne \emptyset \big \}. \end{aligned}$$

(6.31)

If $j\in \mathcal {I}_i$, then owing to (6.2) there exists $y'\in B'_{R_\Omega -2\varepsilon _0}$ such that $(T^i)^{-1}\big (y',\phi ^i(y')\big )\in \partial \Omega \cap K^j_{2\varepsilon _0}$. Since $\phi ^j$ is $L_\Omega$-Lipschitz continuous and $\phi ^j(0')=0$, we have $|\phi ^j(z')|\le L_\Omega \,|z'|$, so it follows from (6.19), (6.27) and (6.28) that $(T^i)^{-1}\big (y',\psi ^i_m(y')\big )\in \partial \Omega _m\cap K^i_{\varepsilon _0}\cap K^j_{\varepsilon _0}$ for all $m\ge m_0$ large enough.

Henceforth, for all $j\in \mathcal {I}_i$, (6.19) and (6.30) allow us to define the transition maps ${\mathcal {C}}^{i,j}, {\mathcal {C}}^{i,j}_m$ from $\phi ^i$ to $\phi ^j$ and from $\psi ^i_m$ to $\psi ^j_m$ respectively, i.e.

$$\begin{aligned} \begin{aligned} {\mathcal {C}}^{i,j}y'&=\Pi \,T^j(T^i)^{-1}\big (y',\phi ^i(y')\big ) \\ {\mathcal {C}}^{i,j}_m y'&=\Pi \,T^j(T^i)^{-1}\big (y',\psi ^i_m(y')\big ), \end{aligned} \end{aligned}$$

(6.32)

which are defined on the open sets

$$\begin{aligned} \begin{aligned} U^{i,j} =\Pi \,\Big (G_{\phi ^i}\cap T^i\,K^j_0\Big )\quad \text {and}\quad U^{i,j}_m =\Pi \,\Big (G_{\psi ^i_m}\cap T^i\,K^j_{\varepsilon _0}\Big ). \end{aligned} \end{aligned}$$

In particular, by their definitions and the arguments of Sect. 5, we may write

$$\begin{aligned} \begin{aligned} x=(T^i)^{-1}\big (y',\phi ^i(y')\big )=(T^j)^{-1} \big ({\mathcal {C}}^{i,j}y',\phi ^j({\mathcal {C}}^{i,j}y')\big )\quad&\text {for } x\in \partial \Omega \cap K^i_0\cap K^j_0\\ x^m=(T^i)^{-1}\big (y',\psi ^i_m(y')\big )=(T^j)^{-1} \big ({\mathcal {C}}_m^{i,j}y',\psi _m^j({\mathcal {C}}^{i,j}_my')\big )\quad&\text {for } x^m\in \partial \Omega _m\cap K^i_{\varepsilon _0}\cap K^j_{\varepsilon _0}. \end{aligned} \end{aligned}$$

(6.33)

and their inverse functions are $({\mathcal {C}}^{i,j})^{-1}={\mathcal {C}}^{j,i}$ and $({\mathcal {C}}^{i,j}_m)^{-1}={\mathcal {C}}^{j,i}_m$. Observe also that ${\mathcal {C}}^{i,i}={\mathcal {C}}^{i,i}_m=\textrm{Id}$.

Furthermore, since $\textrm{supp}\,\xi _j\Subset B_{R_\Omega /4}(x^j)\Subset K^j_{2\varepsilon _0}$, it follows from the definition of $\mathcal {I}_i$ and (6.28) that

$$\begin{aligned} \nabla ^k\xi _j\big ( (T^i)^{-1}(y',\phi ^i(y'))\big )=\nabla ^k\xi _j\big ( (T^i)^{-1}(y',\psi ^i_m(y'))\big )=0\quad \text {if }j\not \in \mathcal {I}_i, \end{aligned}$$

(6.34)

for all $k\in {\mathbb {N}}$, for all $y'\in B'_{R_\Omega -\varepsilon _0}$, and all $m\ge m_0$.

We now claim that for all $j\in \mathcal {I}_i$, there exists an open set $V^{i,j}\subset B'_{R_\Omega -2\varepsilon _0}$ for which we have

$$\begin{aligned} \xi _j\big ( (T^i)^{-1}(y',\phi ^i(y'))\big )=\xi _j \big ( (T^i)^{-1}(y',\psi ^i_m(y'))\big )=0\quad \text {if } y'\not \in V^{i,j}, \end{aligned}$$

(6.35)

and such that $V^{i,j}\subset U^{i,j}\cap U^{i,j}_m$ for all $m>m_0$. This in particular implies that both ${\mathcal {C}}^{i,j}$ and ${\mathcal {C}}^{i,j}_m$ are defined on $V^{i,j}$.

To this end, let

$$\begin{aligned} V^{i,j}:=\Pi \Big ( G_{\phi ^i}\cap T^i K^j_{2\varepsilon _0}\Big )\cap B'_{R_\Omega -2\varepsilon _0}. \end{aligned}$$

Then, owing to (6.28) it is immediate to verify that

$$\begin{aligned} B'_{R_\Omega -2\varepsilon _0}\cap \bigg ( \Pi \Big ( G_{\phi ^i}\cap T^i B_{R_\Omega /4}(x^j)\Big )\cup \Pi \Big ( G_{\psi _m^i}\cap T^i B_{R_\Omega /4}(x^j)\Big )\bigg )\Subset V^{i,j}, \end{aligned}$$

(6.36)

whenever $m>m_0$ is large enough, and thus (6.35) is satisfied by our choice of set $V^{i,j}$.

Clearly $V^{i,j}\subset U^{i,j}$, so we are left to verify that $V^{i,j}\subset U^{i,j}_m$. To this end, let $y'\in V^{i,j}$; then by (6.30) and (6.33) we may write

$$\begin{aligned} T^j(T^i)^{-1}(y',\phi ^i(y'))=({\mathcal {C}}^{i,j}y',\phi ^j({\mathcal {C}}^{i,j}y'))\in B'_{R_\Omega -2\varepsilon _0}\\ \times \big (-L_\Omega (R_\Omega -2\varepsilon _0),L_\Omega (R_\Omega -2\varepsilon _0)\big ), \end{aligned}$$

where in the latter inclusion we made use of the inequality $|\phi ^j(z')|\le L_\Omega \,|z'|$. Therefore, thanks to (6.28), for $m>m_0$ we have $(T^i)^{-1}(y',\psi ^i_m(y'))\in \partial \Omega _m\cap K^i_{\varepsilon _0}\cap K^j_{2\varepsilon _0}$, hence $y'\in U^{i,j}_m$ by (6.30) and the definition of $U^{i,j}_m$, so the claim is proven.

We now prove that

$$\begin{aligned} \bigcup _{j\in \mathcal {I}_i}V^{i,j}=B'_{R_\Omega -2\varepsilon _0}. \end{aligned}$$

(6.37)

Since $\{K^j_{2\varepsilon _0}\}_{j=1}^N$ is a cover of $\partial \Omega$, from (6.2) and by the definition of $\mathcal {I}_i$ (6.31), we have that $\{T^i K^j_{2\varepsilon _0}\}_{j\in \mathcal {I}_i}$ is an open cover of $G_{\phi ^i}\cap K^i_{2\varepsilon _0}$. We now exploit that the projection map $\Pi$ is a homeomorphism from $G_{\phi ^i}$ (with the induced topology) to $B'_{R_\Omega }$. More precisely, we have that

$$\begin{aligned} \Pi :\,G_{\phi ^i}\cap K^i_{2\varepsilon _0}\rightarrow B'_{R_{\Omega -2\varepsilon _0}} \end{aligned}$$

is a homeomorphism by definition of $K^i_{2\varepsilon _0}$ and (6.2). From these two observations, it follows that

$$\begin{aligned} \begin{aligned} \bigcup _{j\in \mathcal {I}_i} V^{i,j}&=\bigcup _{j\in \mathcal {I}_i} \Pi \Big ( G_{\phi ^i}\cap T^i K^j_{2\varepsilon _0}\Big )\cap B'_{R_\Omega -2\varepsilon _0}=\Pi \Bigg (\bigcup _{j\in \mathcal {I}_i} \big (G_{\phi ^i}\cap K^i_{2\varepsilon _0}\cap T^i K^j_{2\varepsilon _0}\big )\Bigg )\\&=\Pi \Big (G_{\phi ^i}\cap K^i_{2\varepsilon _0}\Big )=B'_{R_{\Omega -2\varepsilon _0}}, \end{aligned} \end{aligned}$$

that is (6.37).

Then, owing to (6.28) and by proceeding as in the derivation of (5.18), we obtain

$$\begin{aligned} \Vert {\mathcal {C}}^{i,j}_m-{\mathcal {C}}^{i,j}\Vert _{L^\infty (V^{i,j})}\le \frac{6\,L_\Omega \sqrt{1+L_\Omega ^2}}{m}\quad \text {for all } m>m_0. \end{aligned}$$

(6.38)

Our next goal is to obtain estimates on $\nabla \psi ^i_m$. To this end, we differentiate equation $F_m^i(y',\psi ^i_m(y'))=0$ with respect to $y'_k$, for $k=1,\dots ,n-1$, and recalling (6.34) we find

$$\begin{aligned} \frac{\partial \psi ^i_m }{\partial y'_k}(y')=-\bigg (\frac{\partial F_m^i(y',\psi ^i_m(y'))}{\partial y_n}\bigg )^{-1}\sum _{j\in \mathcal {I}_i}\Bigg \{ \frac{\partial f^j_m(x^m)}{\partial y_k'}\,\xi _j(x^m)+f_m^j(x^m)\,\frac{\partial \xi _j(x^m)}{\partial y_k'}\Bigg \},\nonumber \\ \end{aligned}$$

(6.39)

where $x^m=(T^i)^{-1}\big (y',\psi ^i_m(y')\big )$, $y'\in B'_{R_\Omega -2\varepsilon _0}$.

For all $l=1,\dots ,n$, by using the chain rule and recalling the definition of ${\mathcal {C}}^{i,j}_m$, we find

$$\begin{aligned} \begin{aligned}&\frac{\partial f^i_m}{\partial y'_l}(x^m)=-\frac{\partial \phi _m^i}{\partial y'_l}(y')\quad \text {and}\quad \frac{\partial f^i_m}{\partial y_n}(x^m)=1 \\&\frac{\partial f^j_m}{\partial y_l}(x^m)=\big (\mathcal {R}^j(\mathcal {R}^i)^t\big )_{nl}-\sum _{r=1}^{n-1}\frac{\partial \phi ^j_m}{\partial z'_r}({\mathcal {C}}^{i,j}_m y')\,\big (\mathcal {R}^j(\mathcal {R}^i)^t\big )_{rl}, \end{aligned} \end{aligned}$$

(6.40)

for all $j\in \mathcal {I}_i$ such that $x^m\in \textrm{supp}\,\xi _j$. Since $\phi ^j_m$ are $L_\Omega$-Lipschitz continuous, from (6.40) it follows that

$$\begin{aligned} \sum _{l=1}^n \Big |\frac{ \partial f^j_m(x^m)}{\partial y_l}\Big |\le c(n)(1+L_\Omega ),\quad \text {for all }j\in \mathcal {I}_i. \end{aligned}$$

(6.41)

Moreover, from (6.15), (6.28) and (6.8), we find that $f^j_m(x^m)\,|\nabla \xi _j(x^m)|\xrightarrow {m\rightarrow \infty } f^j(x^0)\,|\nabla \xi _j(x^0)|=0$, where $x^0=(T^i)^{-1}\big (y',\phi ^i(y')\big )\in \partial \Omega$.

By making use of this piece of information, (6.41) and (6.20), from (6.39) we finally obtain the gradient estimate

$$\begin{aligned} |\nabla \psi ^i_m(y')|\le c(n)\big (1+L_\Omega ^2\big ),\quad \text {for all } y'\in B'_{R_\Omega -2\varepsilon _0}, \end{aligned}$$

(6.42)

for all $i=1,\dots ,N$ and $m>m_0$ large enough. In particular, owing to (6.28), (6.27) and (6.42), it is readily seen that $\Omega _m$ are $\mathcal {L}_{\Omega _m}$-Lipschitz domains, with

$$\begin{aligned} L_{\Omega _m}\le c(n)\big (1+L_\Omega ^2\big )\quad \text {and}\quad R_{\Omega _m}\ge \frac{R_\Omega }{c(n)\,\big (1+L_\Omega ^2\big )}, \end{aligned}$$

and (3.4) is proven.

Next, the definition of ${\mathcal {C}}^{i,j}$ and ${\mathcal {C}}^{i,j}_m$, (6.42) and the $L_\Omega$-Lipschitz continuity of $\phi ^i$ imply

$$\begin{aligned} \sup _{i=1,\dots ,N}\sup _{j\in \mathcal {I}_i}\Big \{\Vert \nabla {\mathcal {C}}^{i,j}\Vert _{L^\infty }+\Vert \nabla {\mathcal {C}}^{i,j}_m\Vert _{L^\infty }\Big \}\le c(n)(1+L_\Omega ^2)\quad \text {for all }m>m_0, \end{aligned}$$

(6.43)

and in particular ${\mathcal {C}}^{i,j}$ and ${\mathcal {C}}^{i,j}_m$ are uniformly bi-Lipschitz transformations.

Hence, thanks to (6.38) and (6.43), we are in the position to apply Proposition 1 and get

$$\begin{aligned} \frac{\partial \phi ^j_m}{\partial z'_r}({\mathcal {C}}^{i,j}_m y')\xrightarrow {m\rightarrow \infty } \frac{\partial \phi ^j}{\partial z'_r}({\mathcal {C}}^{i,j}y')\quad \text {for } \mathcal {H}^{n-1} \text {-a.e. }y'\in V^{i,j}. \end{aligned}$$

(6.44)

From this, (6.20), (6.35), (6.37), (6.40) and identity (6.39) we find

$$\begin{aligned} \nabla \psi _m^i(y')\xrightarrow {m\rightarrow \infty } G(y')\quad \text {for } \mathcal {H}^{n-1}\text {-a.e. }y'\in B'_{R_\Omega -2\varepsilon _0}, \end{aligned}$$

where G is a bounded vector valued function which can be explictly written. From (6.42) and on applying dominated convergence theorem, we get that $\nabla \psi ^i_m\xrightarrow {m\rightarrow \infty } G$ in $L^p(B'_{R-2\varepsilon _0})$ for all $p\in [1,\infty )$. On the other hand, (6.28) and the uniqueness of the distributional limit imply that $G=\nabla \phi ^i$, hence (3.5) is proven.

6.6 Curvature convergence

Assume now that $\partial \Omega \in W^{2,q}$ for some $q\in [1,\infty )$. Then the local charts $\phi ^i\in W^{2,q}(B'_{R_\Omega })$.

Let $y'\in B'_{R_{\Omega -2\varepsilon _0}}$, and differentiate twice the identity $F_m^i(y',\psi ^i_m(y'))=0$ with respect to $y'_k\,y'_l$ for $k,l=1,\dots n-1$, as to find

$$\begin{aligned} \begin{aligned}&\frac{\partial ^2 \psi ^i_m}{\partial y_k'\partial y_l'}(y')=-\bigg (\frac{\partial F^i_m(y',\psi ^i_m(y'))}{\partial y_n}\bigg )^{-1}\\ \Bigg \{&\frac{\partial ^2 F^i_m (y',\psi ^i_m(y'))}{\partial y_k'\partial y_l'} +\frac{\partial ^2 F_m^i (y',\psi ^i_m(y'))}{\partial y_l'\partial y_n}\,\frac{\partial \psi ^i_m}{\partial y_k'}(y')+ \\&+\frac{\partial ^2 F_m^i (y',\psi ^i_m(y'))}{\partial y_k'\partial y_n}\,\frac{\partial \psi ^i_m}{\partial y_l'}(y') + \\&+\frac{\partial ^2 F_m^i (y',\psi ^i_m(y'))}{\partial y_n\partial y_n}\,\frac{\partial \psi ^i_m}{\partial y_k'}(y')\,\frac{\partial \psi ^i_m}{\partial y_l'}(y')\Bigg \}. \end{aligned} \end{aligned}$$

(6.45)

By differentiating twice $F^i_m=F_m\circ (T^i)^{-1}$, and recalling Definition (6.14), for all $l,r=1,\dots n$ we get

$$\begin{aligned} \begin{aligned} \frac{\partial ^2 F^i_m}{\partial y_r\partial y_l}(y',\psi ^i_m(y'))=\sum _{j\in \mathcal {I}_i}&\Bigg \{\frac{\partial ^2 f^j_m }{\partial y_r\partial y_l}(x^m)\,\xi _j(x^m)+\frac{\partial f^j_m}{\partial y_r}(x^m)\,\frac{\partial \xi _j}{\partial y_l}(x^m) \\&+\frac{\partial f^j_m}{\partial y_l}(x^m)\,\frac{\partial \xi _j}{\partial y_r} (x^m)+f^j_m(x^m)\,\frac{\partial ^2\xi _j}{\partial y_r\partial y_l}(x^m)\Bigg \}, \end{aligned} \end{aligned}$$

(6.46)

where $x^m=(T^i)^{-1}(y',\psi ^i_m(y'))$. The above summation is only over the set of indices $\mathcal {I}_i$, since $\nabla ^k \xi _j(x^m)=0$ owing to (6.35).

We also have

$$\begin{aligned} \frac{\partial ^2 f^j_m}{\partial y_r\partial y_l}(x^m)=-\sum _{s,t=1}^{n-1}\frac{\partial ^2 \phi ^j_m}{\partial z'_s\partial z'_t}({\mathcal {C}}^{i,j}_m y')\,\big (\mathcal {R}^j(\mathcal {R}^i)^t\big )_{sr}\big (\mathcal {R}^j(\mathcal {R}^i)^t\big )_{tl} \end{aligned}$$

(6.47)

for all $j\in \mathcal {I}_i$ such that $x^m\in \textrm{supp}\,\xi _j$.

Thanks to (6.15), (6.28) and (6.8), we readily find that $f^j_m(x^m)\,|\nabla \xi _j(x^m)|\rightarrow 0$ and $f^j_m(x^m)\,|\nabla ^2\xi _j(x^m)|\rightarrow 0$. From this, and by using (6.6), (6.20), (6.41), (6.42) and (6.45)-(6.47), we obtain

$$\begin{aligned} |\nabla ^2 \psi ^i_m(y')|\le c(n)(1+L_\Omega ^5)\,\sum _{j\in \mathcal {I}_i}\bigg \{ |\nabla ^2 \phi ^j_m|({\mathcal {C}}^{i,j}_m y')\,\xi _j\big ((T^i)^{-1}(y',\psi ^i_m(y')\big )+\frac{(1+L_\Omega )}{R_\Omega }\bigg \},\nonumber \\ \end{aligned}$$

(6.48)

for all $y'\in B'_{R_\Omega -2\varepsilon _0}$, provided $m>m_0$ is large enough.

Then again, thanks to (6.38) and(6.43), we may apply Proposition 1 and infer

$$\begin{aligned} \nabla ^2 \phi ^j_m({\mathcal {C}}^{i,j}_m y')\rightarrow \nabla ^2 \phi ^j({\mathcal {C}}^{i,j}y')\quad \text {for } \mathcal {H}^{n-1}\text {-a.e. }y'\in V^{i,j}\text { and in } L^q(V^{i,j}). \end{aligned}$$

(6.49)

Finally, recalling (6.35) and (6.37), we may exploit the properties (6.20), (6.28), (6.40), (6.44), (6.45)-(6.49) in order to apply dominated convergence Theorem 3 with dominating functions

$$\begin{aligned} \begin{aligned}&g_m =c(n, L_\Omega , R_\Omega )\,\sum _{j\in \mathcal {I}_i}\bigg \{ |\nabla ^2 \phi ^j_m|({\mathcal {C}}^{i,j}_m y')\,\xi _j\big ((T^i)^{-1}(y',\psi ^i_m(y')\big )+1\bigg \}\\&g =c(n, L_\Omega , R_\Omega )\,\sum _{j\in \mathcal {I}_i}\bigg \{ |\nabla ^2 \phi ^j|({\mathcal {C}}^{i,j} y')\,\xi _j\big ((T^i)^{-1}(y',\phi ^i(y')\big )+1\bigg \} \end{aligned} \end{aligned}$$

This entails

$$\begin{aligned} \nabla ^2 \psi _m^i\rightarrow M,\quad \mathcal {H}^{n-1}\text {-a.e. on } B'_{R_\Omega -2\varepsilon _0} \text { and in } L^q(B'_{R_\Omega -2\varepsilon _0}), \end{aligned}$$

(6.50)

for some matrix valued function M, which can be explictly written in terms of $\phi ^j,\nabla \phi ^j,\nabla ^2 \phi ^j$ and $\xi _j$. On the other hand, (6.28) and the uniqueness of the distributional limit imply that $M=\nabla ^2 \phi ^i$, hence (3.6) is proven.

6.7 Proof of the isocapacitary estimate (3.7)

In the following subsection, we will denote by $\widetilde{M}_m(h)$ the convolution of a function $h\in L^1_{loc}({\mathbb {R}}^n)$ with respect to the first $(n-1)$-variables, i.e.

$$\begin{aligned} \widetilde{M}_m(h)(z',z_n)=\int _{{\mathbb {R}}^{n-1}}h(x',z_n)\,\rho _m(z'-x')\,dx'. \end{aligned}$$

We then have the following elementary lemma, which will be useful later.

Lemma 1

Let $v\in C^{\infty }_c({\mathbb {R}}^n)$. Then, if we set

$$\begin{aligned} \widetilde{v}_m:=\sqrt{\widetilde{M}_m(v^2)}, \end{aligned}$$

we have that $\widetilde{v}_m$ is Lipschitz continuous on ${\mathbb {R}}^n$, and

$$\begin{aligned} |\nabla \widetilde{v}_m|\le c(n)\,\sqrt{\widetilde{M}_m\big (|\nabla v|^2\big )}\quad \text {a.e. on }{\mathbb {R}}^n. \end{aligned}$$

(6.51)

Proof

By Hölder’s inequality, for $k=1,\dots ,n$ we have

$$\begin{aligned} \bigg |\frac{\partial \widetilde{M}_m(v^2)}{\partial x_k}\bigg |=\bigg |\widetilde{M}_m\bigg (\frac{\partial v^2}{\partial x_k}\bigg )\bigg |=2\,\bigg |\widetilde{M}_m\bigg (v\,\frac{\partial v}{\partial x_k}\bigg )\bigg |\le 2\,\sqrt{\widetilde{M}_m(v^2)}\,\sqrt{\widetilde{M}_m\bigg (\bigg |\frac{\partial v}{\partial x_k}\bigg |^2\bigg )}. \end{aligned}$$

Therefore, on setting $\widetilde{v}_{\varepsilon ,m}:=\sqrt{\varepsilon ^2+\widetilde{M}_m(v^2)}$, for all $\varepsilon \in (0,1)$ we have that

$$\begin{aligned} |\nabla \widetilde{v}_{\varepsilon ,m}|=\frac{\big |\nabla \widetilde{M}_m(v^2)\big |}{2\sqrt{\varepsilon ^2+\widetilde{M}_m(v^2)}}\le c(n)\,\frac{\sqrt{\widetilde{M}_m(v^2)}\,\sqrt{\widetilde{M}_m(|\nabla v|^2)}}{\sqrt{\varepsilon ^2+\widetilde{M}_m(v^2)}}\le c(n)\,\sqrt{\widetilde{M}_m\big (|\nabla v|^2\big )}.\nonumber \\ \end{aligned}$$

(6.52)

Thus, the sequence $\{\widetilde{v}_{\varepsilon ,m}\}_{\varepsilon \in (0,1)}$ is uniformly bounded in $C^{0,1}_c({\mathbb {R}}^n)$, and since $\widetilde{v}_{\varepsilon ,m}\xrightarrow {\varepsilon \rightarrow 0^+} \widetilde{v}_m$ on ${\mathbb {R}}^n$, we deduce that $\widetilde{v}_m\in C^{0,1}_c({\mathbb {R}}^n)$ by weak-$*$ compactness, and the thesis follows by letting $\varepsilon \rightarrow 0$ in (6.52) and by Rademacher’s Theorem. $\square$

Now let $x^0_m\in \partial \Omega _m$; then owing to (6.26) and (6.16), there exists $i\in \{1,\dots ,N\}$ such that $x^0_m\in B_{R_\Omega /8}(x^i)$. Therefore, we may write $x^0_m=(T^i)^{-1}\Big ((y^0)',\psi ^i_m\big ((y^0)'\big )\Big )$ for some $(y^0)'\in B'_{R_\Omega /8}$, and we also set $x^0:=(T^i)^{-1}\Big ((y^0)',\phi ^i\big ((y^0)'\big )\Big )\in \partial \Omega$. Let

$$\begin{aligned} r_0:=\frac{R_\Omega }{C(n)\,\big (1+L_\Omega ^2 \big )}, \end{aligned}$$

for some fixed constant $C(n)>1$ large enough, and consider $r\le r_0$, and $v\in C^\infty _c\big (B_r(x^0_m)\big )$. Then, since $B_r(x^0_m)\Subset B_{R_\Omega /4}(x^i)\Subset K^i_{2\varepsilon _0}$, we have

$$\begin{aligned} \int _{\partial \Omega _m}v^2\,|\mathcal {B}_{\Omega _m}|\,d\mathcal {H}^{n-1}=\int _{B'_{R_\Omega /4}}v^2 \Big ((T^i)^{-1}\big (y',\psi ^i_m(y')\big )\Big )|\mathcal {B}_{\Omega _m}(y')|\, \sqrt{1+|\nabla \psi ^i_m(y')|^2}\,dy'. \end{aligned}$$

Consider the new set of indices

$$\begin{aligned} \mathbb {J}^{x^0_m}_r:=\big \{j\in \mathcal {I}_i:\,B_r(x^0_m)\cap \textrm{supp}\,\xi _j\ne \emptyset \big \}. \end{aligned}$$

Owing to (2.10), (6.33), (6.35), (6.42) and the Hessian estimate (6.48), we obtain

$$\begin{aligned} \begin{aligned}&\int _{\partial \Omega _m}v^2\,|\mathcal {B}_{\Omega _m}|\,d\mathcal {H}^{n-1} \le \sqrt{1+L_\Omega ^2}\, \int _{B'_{R_\Omega /4}}v^2\Big ((T^i)^{-1}\big (y',\psi ^i_m(y')\big )\Big )\,|\nabla ^2 \psi ^i_m(y')|\,dy' \\&\le c(n)\,(1+L_\Omega ^6)\,\sum _{j\in \mathbb {J}_r^{x^0_m}} \int _{V^{i,j}}\Bigg \{v^2\Big ((T^j)^{-1}\big ({\mathcal {C}}^{i,j}_m y',\psi ^j_m({\mathcal {C}}^{i,j}_m y')\big )\Big )\times \\&\times \xi _j\Big ((T^j)^{-1}\big ({\mathcal {C}}^{i,j}_m y',\psi ^j_m({\mathcal {C}}^{i,j}_m y')\big )\Big )\,M_m\big (|\nabla ^2 \phi ^j|\big )({\mathcal {C}}^{i,j}_m y')\Bigg \}\,dy' \\&\hspace{1cm}+c(n)\,\frac{(1+L^7_\Omega )}{R_\Omega }\, |\mathbb {J}_r^{x^0_m}|\int _{B'_{R/4}}v^2\Big ((T^i)^{-1}\big ( y',\psi ^i_m( y')\big )\Big )dy'. \end{aligned} \end{aligned}$$

(6.53)

By using $|\mathbb {J}_r^{x^0_m}|\le N$, (6.3), (3.4) and the results of [2, Corollary 6.6], we get

(6.54)

On the other hand, via the change of variables $z'={\mathcal {C}}^{i,j}_m y'$, by making use of (6.43), (6.36), and observing that $B_r(x^0_m)\Subset K^i_{2\varepsilon _0}\cap K^j_{2\varepsilon _0}$ for all $j\in \mathbb {J}^{x^0_m}_r$, $x^0_m\in \partial \Omega _m$ and $r\le r_0$, we find

$$\begin{aligned} \begin{aligned}&\int _{V^{i,j}}\Bigg \{v^2\Big ((T^j)^{-1}\big ({\mathcal {C}}^{i,j}_m y',\psi ^j_m({\mathcal {C}}^{i,j}_m y')\big )\Big )\,\xi _j\Big ((T^j)^{-1}\big ({\mathcal {C}}^{i,j}_m y',\psi ^j_m({\mathcal {C}}^{i,j}_m y')\big )\Big )\,M_m\big (|\nabla ^2 \phi ^j|\big )({\mathcal {C}}^{i,j}_m y')\Bigg \}\,dy' \\&\le c(n)(1+L_\Omega ^{(n-1)})\,\int _{W^{i,j}} w^2_{j,m}(z',0)\,M_m\big (|\nabla ^2\phi ^j|\big )(z')\,dz', \end{aligned}\nonumber \\ \end{aligned}$$

(6.55)

for some open set $W^{i,j}\Subset {\mathcal {C}}^{i,j}(U^{i,j})$, where we also set

$$\begin{aligned} w_{j,m}(z',z_n):=v\Big ((T^j)^{-1}\big (z',z_n+\psi ^j_m( z')\big )\Big ). \end{aligned}$$

Since $v\in C^\infty _c(B_r(x^0_m))$ and $x^0_m=(T^j)^{-1}\Big ( {\mathcal {C}}^{i,j}_m\big ((y^0)'\big ),\psi ^j_m\big ((y^0)'\big )\Big )$ for all $j\in \mathbb {J}_r^{x^0_m}$, by using (6.42) it is readily seen that

$$\begin{aligned} w_{j,m}\in C^\infty _c\bigg (B_{c(n)(1+L_\Omega ^2)\,r}\Big ({\mathcal {C}}^{i,j}_m\big ((y^0)'\big ),0 \Big )\bigg ), \end{aligned}$$

and from the chain rule we find

$$\begin{aligned} |\nabla w_{j,m}(z',z_n)|\le c(n)(1+L_\Omega ^2)\,\Big |\nabla v\Big ((T^{j})^{-1}\big (z',z_n+\psi _m^j(z')\big )\Big )\Big | \end{aligned}$$

(6.56)

Next, by using Fubini-Tonelli’s Theorem we obtain

$$\begin{aligned} \begin{aligned}&\int _{W^{i,j}} w^2_{j,m}(z',0)\,M_m \big (|\nabla ^2\phi ^j|\big )(z')\,dz' =\int _{W^{i,j}}w_{j,m}^2 (z',0)\int _{B'_{1/m}(z')}|\nabla ^2 \phi ^j(\tilde{z}')|\,\rho _{m}(z'-\tilde{z}')\,d\tilde{z}'\,dz' \\&\le \int \limits _{W^{i,j}+B'_{1/m}}|\nabla ^2\phi ^j(\tilde{z}')| \Big (\int _{B'_{1/m(\tilde{z}')}}w^2_{j,m}(z',0)\,\rho _m(\tilde{z}'-z')\,dz'\Big )\,d\tilde{z}'. \end{aligned} \end{aligned}$$

We have thus found that

$$\begin{aligned} \int _{W^{i,j}} w^2_{j,m}(z',0)\,M_m\big (|\nabla ^2\phi ^j|\big )(z')\,dz'\le \int _{\widetilde{W}^{i,j}}\widetilde{M}_m(w^2_{j,m})(z',0)\,|\nabla ^2\phi ^j(z')|\,dz',\nonumber \\ \end{aligned}$$

(6.57)

for some open set $\widetilde{W}^{i,j}\Subset {\mathcal {C}}^{i,j}(U^{i,j})$, provided $m>m_0$ is large enough.

Thanks to Lemma 1 and inequality (6.38), we easily infer

$$\begin{aligned} \sqrt{\widetilde{M}_m(w^2_{j,m})}\in C^{0,1}_c\bigg (B_{c(n)(1+L_\Omega ^2)(r+\frac{1}{m})}\Big ( {\mathcal {C}}^{i,j}\big ((y^0)'\big ),0\Big )\bigg ), \end{aligned}$$

and

$$\begin{aligned} \Big |\nabla \sqrt{\widetilde{M}_m(w^2_{j,m})}\Big |\le c(n)\,\sqrt{\widetilde{M}_m\big (|\nabla w_{j,m}|^2 \big )}\quad \text {a.e. on }{\mathbb {R}}^n. \end{aligned}$$

(6.58)

Finally, set

$$\begin{aligned} \tilde{h}_{j,m}(x',x_n):=\sqrt{\widetilde{M}_m(w^2_{j,m})}\Big (T^j\big (x',x_n-\phi ^j(x')\big )\Big ) \end{aligned}$$

so that $\tilde{h}_{j,m}$ is Lipschitz continuous on ${\mathbb {R}}^n$. Moreover, thanks to (6.28), for all $j\in \mathbb {J}_{r}^{x^0_m}$, we have that

$$\begin{aligned} B_{c(n)(1+L_\Omega ^3)(r+\frac{1}{m})}(x^0)\Subset K^i_{2\varepsilon _0}\cap K^j_{2\varepsilon _0} \end{aligned}$$

for all $m>m_0$ sufficiently large and all $r\le r_0$, and thus we may write $x^0=(T^j)^{-1}\Big ({\mathcal {C}}^{i,j}\big ((y^0)'\big ),\phi ^j((y^0)') \Big )$ due to (6.33). Recalling that $\phi ^j$ is $L_\Omega$-Lipschitz continous, it follows that

$$\begin{aligned} \tilde{h}_{j,m}\in C^{0,1}_c\Big (B_{c(n)(1+L_\Omega ^3)(r+\frac{1}{m})}(x^0) \Big ), \end{aligned}$$

and from the chain rule

$$\begin{aligned} \big |\nabla \tilde{h}_{j,m}(x',x_n)\big |\le c(n)(1+L_\Omega )\, \Big |\nabla \sqrt{\widetilde{M}_m(w^2_{j,m})}(x',x_n-\phi ^j(x'))\Big |\quad \text {for a.e. }x.\nonumber \\ \end{aligned}$$

(6.59)

Owing to (2.10) and the definition of $\widetilde{h}_{j,m}$, we have

$$\begin{aligned} \begin{aligned}&\int _{\widetilde{W}^{i,j}}\widetilde{M}_m(w^2_{j,m})(z',0)\,|\nabla ^2\phi ^j(z')|\,dz' = \int _{\widetilde{W}^{i,j}}\tilde{h}_{j,m}^2\big ((T^j)^{-1}(z',\phi ^j(z'))\big )\,|\nabla ^2\phi ^j(z')|\,dz' \\&\hspace{0.5cm} \le c(n)(1+L_\Omega ^3)\int _{\widetilde{W}^{i,j}}\tilde{h}_{j,m}^2\big ((T^j)^{-1}(z',\phi ^j(z'))\big ) \,\big |\mathcal {B}_\Omega (z')\big |\sqrt{1+|\nabla \phi ^j(z')|^2}\,dz' \\&\hspace{0.5cm} =c(n)(1+L_\Omega ^3)\,\int _{\partial \Omega } \tilde{h}_{j,m}^2\big |\mathcal {B}_\Omega \big |\,d\mathcal {H}^{n-1} \\&\hspace{0.5cm} \le c(n)(1+L_\Omega ^3)\,\bigg (\sup \,\frac{\int _{\partial \Omega } h^2\,\big |\mathcal {B}_\Omega \big |\,d\mathcal {H}^{n-1}}{\int _{{\mathbb {R}}^n}|\nabla h|^2\,dx}\bigg )\,\int _{{\mathbb {R}}^n}|\nabla \tilde{h}_{j,m}|^2\,dx, \end{aligned} \end{aligned}$$

(6.60)

where the supremum above is taken over all functions $h\in C^{0,1}_c\Big (B_{c(n)(1+L_\Omega ^3)(r+\frac{1}{\,}m)}(x^0) \Big )$.

Henceforth, by coupling (6.3) and estimates (6.53)-(6.60), for all $v\in C^\infty _c\big (B_r(x^0_m)\big )$ we obtain

$$\begin{aligned} \begin{aligned}&\int _{\partial \Omega _m}v^2\,\big |\mathcal {B}_{\Omega _m}\big |\,d\mathcal {H}^{n-1}\le c(n)\,(1+L_\Omega ^{n+4})\,\bigg (\sup \frac{\int _{\partial \Omega } h^2\,\big |\mathcal {B}_\Omega \big |\,d\mathcal {H}^{n-1}}{\int _{{\mathbb {R}}^n}|\nabla h|^2\,dx}\bigg )\sum _{j\in \mathbb {J}_r^{x^0_m}}\int _{{\mathbb {R}}^n}\widetilde{M}_m\big ( |\nabla w_{j,m}|^2\big )\,dx \\&+\tilde{c}\,\int _{{\mathbb {R}}^n}|\nabla v|^2 dx \\&\le c(n)\,(1+L_\Omega ^{n+4})\,\bigg (\sup \frac{\int _{\partial \Omega } h^2\,\big |\mathcal {B}_\Omega \big |\,d\mathcal {H}^{n-1}}{\int _{{\mathbb {R}}^n}|\nabla h|^2\,dx}\bigg )\sum _{j\in \mathbb {J}_r^{x^0_m}}\int _{{\mathbb {R}}^n} |\nabla w_{j,m}|^2\,dx+\tilde{c}\,\int _{{\mathbb {R}}^n}|\nabla v|^2 dx \\&\le c(n)\,(1+L_\Omega ^{n+8})\,N\,\bigg (\sup \frac{\int _{\partial \Omega } h^2\,\big |\mathcal {B}_\Omega \big |\,d\mathcal {H}^{n-1}}{\int _{{\mathbb {R}}^n}|\nabla h|^2\,dx}\bigg )\,\int _{{\mathbb {R}}^n} |\nabla v|^2\,dx+\tilde{c}\,\int _{{\mathbb {R}}^n}|\nabla v|^2 dx \\&\le c'(n)\,(1+L_\Omega ^{n+8})\frac{d_\Omega ^n}{R_\Omega ^n}\,\bigg (\sup \frac{\int _{\partial \Omega } h^2\,\big |\mathcal {B}_\Omega \big |\,d\mathcal {H}^{n-1}}{\int _{{\mathbb {R}}^n}|\nabla h|^2\,dx}\bigg )\,\int _{{\mathbb {R}}^n}|\nabla v|^2 dx+\tilde{c}\,\int _{{\mathbb {R}}^n}|\nabla v|^2 dx, \end{aligned} \end{aligned}$$

where in the second inequality we made use of Fubini-Tonelli’s Theorem, the supremum above is taken over all $h\in C^{0,1}_c\Big (B_{c(n)(1+L_\Omega ^3)(r+\frac{1}{\,}m)}(x^0) \Big )$, and we set

(6.61)

Therefore, for all $x^0_m\in \partial \Omega _m$, $r\le r_0$, we have found

$$\begin{aligned} \begin{aligned} \sup _{v\in C^{\infty }_c(B_r(x^0_m))}&\frac{\int _{\partial \Omega _m} v^2\,\big |\mathcal {B}_{\Omega _m}\big |\, d\mathcal {H}^{n-1}}{\int _{{\mathbb {R}}^n}|\nabla v|^2\,dx} \\&\le \frac{c(n)\,(1+L_\Omega ^{n+8})\,d_\Omega ^n}{R_\Omega ^{n}}\,\Bigg (\displaystyle \sup _{ \begin{array}{c}{x^0\in \partial \Omega }\\ v\in C^{0,1}_c\big (B_{c(n)(1+L_\Omega ^3)(r+1/m)}(x^0)\big ) \end{array} } \frac{\int _{\partial \Omega } v^2\,\big |\mathcal {B}_{\Omega }\big |\,d\mathcal {H}^{n-1}}{\int _{{\mathbb {R}}^n}|\nabla v|^2\,dx}\Bigg )+\tilde{c}. \end{aligned} \end{aligned}$$

From this, (6.61) and the isocapacitary equivalence [19, Theorem 2.4.1], we finally obtain the desired estimate

(6.62)

for all $r\le r_0$ and $m>m_0$, and the proof is complete.