Smooth approximation of Lipschitz domains, weak curvatures and isocapacitary estimates

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 $q\geq 1$. The sequences of approximating sets is also characterized by uniform isocapacitary estimates with respect to the initial domain $\Omega$.


Introduction
In this paper we are concerned with inner and outer approximation of bounded Lipschitz domains Ω of the Euclidean space R n , n ≥ 2. Specifically, we construct two sequences of C ∞ -smooth bounded domains {ω m }, {Ω m } such that ω m ⋐ Ω ⋐ Ω m for all m ∈ N, which also satisfy natural covergence properties like, for instance, in the sense of the Lebesgue measure and in the sense of Hausdorff to Ω.
Geometric quantities like a Lipschitz characteristic L Ω = (L Ω , R Ω ) and the diameter d Ω of the domain Ω are comparable to the corresponding ones of its approximating sets ω m , Ω m .Here, the constant R Ω 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 Ω is their Lipschitz constant-see Section 2 for the precise definition of a Lipschitz characteristic of Ω.
Furthermore, the smooth charts locally describing the boundaries ∂ω m , ∂Ω m are defined on the same reference systems as the local charts describing ∂Ω, together with strong convergence in the Sobolev space W 1,p for all p ∈ [1, ∞).
If in addition the local charts describing ∂Ω belong to the Sobolev space W 2,q for some q ∈ [1, ∞), then we also have strong convergence in the W 2,q -sense.In a certain way, this means that the second fundamental forms B ωm and B Ωm of the regularized sets converge in L q to the "weak" curvature B Ω of the initial domain Ω.
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 [19], followed by Massari & Pepe [14] 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 Ω via mollification and convolution, and then define the approximating set Ω m as a suitable superlevel set of the mollified characteristic functions-see for instance [1,Theorem 3.42] or [13,Section 13.2].We point out that Schmidt [20] and Gui, Hu & Li [7] constructed smooth approximating domains strictly contained in Ω under additional assumptions on the finite perimeter domain Ω , whereas an outer approximation via smooth sets is given by Doktor [6] when the domain Ω 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 Ω ε are defined as the ε-superlevel set of the regularized distance function, which in turn is obtained via mollification of the usual signed distance function.Here, the parameter ε can be taken either positive or negative, according to the preferred method of approximation, whether from the inside or outside of Ω.
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.∂Ω ∈ C 1,1 .Namely, we do not recover any quantitative information or convergence property regarding the second fundamental forms B Ωm from the original one B Ω .This is because first-order estimates regarding Ω 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 B Ωm , such pointwise analysis needs to be extended to second-order derivatives, and this calls for the application of the implicit function theorem, for which Ω 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 Ω, like its characteristic function or signed distance, whereas the second fundamental form of hypersurfaces of 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 ∂Ω.Thus, our approach seems more suitable when dealing with weak curvatures, though at the cost of requiring Ω 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 Ω m , but only on their Lipschitz characteristics or other suitable quantities possibly depending on the second fundamental form B Ωm .For instance, various investigations such as [2,3,5,16,17] showed that global regularity of solutions to linear and quasilinear PDEs may depend on a weighted isocapacitary function for subsets ∂Ω, the weight being the norm of the second fundamental form on ∂Ω.
This function, which we denote by K Ω , is defined as and it was first introduced in [5].Above, cap(E, B r (x)) denotes the standard capacity of a compact set E relative to the ball B r (x), i.e.
cap(E, B r (x)) = inf where C 0,1 c (A) is the set of Lipschitz continuous functions with compact support in A. We remark that, in order for K Ω (r) to be well defined, it suffices that ∂Ω is Lipschitz continuous and belongs to W 2,1 , as it can be inferred from inequalities (2.8) below.
Plan of the paper.The rest of the paper is organized as follows: in Section 2, we explain some non-standard notation used throughout the paper, and provide the definitions of L Ω -Lipschitz domain, of W 2,q -domain and of weak curvature.
In Section 3 we state in detail our main results, and we provide a few comments and an outline of their proofs.
In Section 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 Section 5 we introduce the notion of transversality of a unit vector n to a Lipschitz function ϕ, and we show a very interesting fact, i.e. this transversality property is equivalent to the graphicality of ϕ with respect to the coordinate system (y ′ , y n ) having 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 (ϕ).
As a byproduct, we will find an interesting, yet expected result: if ∂Ω ∈ W 2,q , then any Lipschitz function locally describing ∂Ω is of class W 2,q .This means that second-order Sobolev regularity is an intrinsic property of the local charts describing ∂Ω-see Corollary 5.7.
Finally, Section 6 is devoted to the proof of the main Theorem 3.1.

Basic notation and definitions
In this section, we provide the relevant definitions and notation of use throughout the rest of the paper.
, and a function v : U → R, we shall denote by ∇v its d-dimensional gradient, and ∇ 2 v its hessian matrix.We will often use the short-hand notation for its level and sublevel sets {v < 0} := {z ∈ U : v(z) < 0}.
• We denote by W k,p (Ω) the usual Sobolev space of L p (Ω) weakly differentiable functions having weak k-th order derivatives in L p (Ω).
For any α ∈ (0, 1], the spaces C k (Ω) and C k,α (Ω) will denote, respectively, the space of functions with continuous and α-Hölder continuous derivatives up to order k ∈ N.
• Point of R n will be written as x = (x ′ , x n ), with x ′ ∈ R n−1 and x n ∈ R. We write B r (x) to denote the n-dimensional ball of radius r > 0 and centered at x ∈ R n .Also, B ′ r (x ′ ) will denote the (n − 1)-dimensional ball of radius r > 0 and centered at x ′ ∈ 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 ∈ N, and for a given matrix X ∈ M d×d , we shall denote by |X| its Frobenius Norm |X| = 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 dist H (A, B) their Hausdorff distance.• For a given function ϕ : U → R with U ⊂ R n−1 open, we write G ϕ and S ϕ to denote its graph and subgraph in R n , i.e.
and we will write In the following, we specify the definition of Lipschitz domain and of Lipschitz characteristic.
It is easily seen that the above definition coincides with the standard one for uniformly Lipschitz domains-see e.g.[8,Section 2.4].Our definition has the advantage of pointing out L Ω = (L Ω , R Ω ) which appears in the characterization of our approximation sets.
We also remark that, in general, a Lipschitz characteristic L Ω = (L Ω , R Ω ) is not uniquely determined.For instance, if ∂Ω ∈ C 1 , then L Ω may be taken arbitrarily small, provided that R Ω is chosen sufficiently small.The function ϕ in definition 2.1 is typically called local (boundary) chart.By Rademacher's theorem, this function is differentiable for H n−1 -almost every x ′ , with gradient ∇ϕ bounded by L Ω .In particular, this implies that any Lipschitz domain Ω admits a tangent plane on H n−1 -almost every point of its boundary.
Moreover, the local chart ϕ naturally endows ∂Ω of a local parametrization ι ϕ (x ′ ) = x ′ , ϕ(x ′ ) , under which the first fundamental form g = {g ij } n−1 i,j=1 is given by (2.4) where δ ij denotes the Kronecker's delta, and x ′ is a point of differentiability of ϕ.Then, the inverse matrix g −1 = {g ij } n−1 i,j=1 can be explictly computed: (2.5) Since ∂Ω admits a tangent plane H n−1 -almost everywhere, we may want to define a notion of weak second fundamental form, which extends the classical one for C ∞ -smooth domains of R n .For this purpose, we need some additional regularity assumptions on ϕ, and in particular on its second-order derivatives.Definition 2.2 (W 2,q domains and weak curvature).Let q ∈ [1, ∞).We say that a bounded Lipschitz domain Ω is of class W 2,q if the local boundary chart ϕ satisfying (2.2) belongs to the Sobolev space for almost all points x ′ of differentiability of ϕ.Its norm is then given by (2.7) where g −1 is the inverse matrix of g given by (2.5).
The reader may verify that identities (2.4)-(2.7)concur with the usual ones when ∂Ω is a smooth hypersurface of R n -see e.g.[11, pp. 246-249].However, these definitions also make sense when ϕ is merely Lipschitz continuous and belongs to the Sobolev space W 2,1 .Indeed, the following inequalities hold true: In order to prove (2.8), we first recall that for all symmetric matrices X, Y , with X definite positive, we have the elementary linear algebra inequalities , where λ min , λ max denote the smallest and largest eigenvalues of X-see e.g.[2, Lemma 3.6] and its proof.Then, owing to (2.5), we observe that the largest and smallest eigenvalues of the matrix g −1 are respectively 1 and (1 + |∇ϕ| 2 ) −1 , and since |∇ϕ| ≤ L Ω we immediately infer (2.8).Inequalities (2.8) also show that (locally) second fundamental form B Ω 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 ∂Ω ∈ W k,q .Similarly, standard definitions follow for domains of class C k and C k,α .

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 3.1.Let Ω ⊂ R n be a bounded, Lipschitz domain, with Lipschitz characteristic the following convergence property hold true the Hausdorff distances safisfy and we may choose their Lipschitz characteristics L Ωm = (L Ωm , R Ωm ) and , for all m ∈ N. (3.4) Moreover, the smooth boundaries ∂ω m , ∂Ω m are described with the help of the same coordinate systems as ∂Ω, i.e. there exist finite number of local boundary charts and {φ i m } N i=1 which describe ∂Ω, ∂Ω m and ∂ω m respectively, such that for each i = 1, . . ., N the functions ψ i m , φ i m ∈ C ∞ are defined on the same reference system as ϕ i , and ), for all i = 1, . . ., N , and any fixed constant for all m ∈ N and r ≤ r 0 (n, L Ω ).
Let us briefly comment on our result.Part (i) of Theorem 3.1 is mostly analogous to [6, Theorem 5.1]; as expected from domains Ω with Lipschitz continuous boundary, the local charts of ∂Ω m , ∂ω m converge to the corresponding local charts of ∂Ω in W 1,p for all p ∈ [1, ∞).In particular, by the classical Morrey-Sobolev's embedding Theorems, this entails an "almost Lipschitz convergence", i.e. the local charts ψ i m and φ i m converge to ϕ i in every Hölder space C 0,α with α ∈ (0, 1).
The main novelty of our result is given in Part (ii), where information about the second fundamental forms B ωm and B Ωm (or equivalently ∇ 2 φ i m and ∇ 2 ψ i m ) is retrieved when ∂Ω is endowed with a weak curvature.For instance, by definition (2.6) and from the results of Theorem 3.1, via a standard covering argument it is easy to show that (3.8) Other than this, we obtain the isocapacitary estimate (3.7),where K Ω (r) and K Ωm , K ωm are the functions defined in (1.1) relative to Ω, Ω m and ω m , respectively.In the proof of (3.7), we will also explicitly write the constant c appearing therein.
Finally, the fixed parameter ε 0 ∈ (0, R Ω /2) appearing in (3.5) and (3.6) is purely technical, and does not affect the validity of the convergence results since the boundaries ∂Ω, ∂Ω m and ∂ω m all share the same coordinate cylinders of the kind Outline of the proof.We fix a covering of ∂Ω, with corresponding partition of unity {ξ i } i and local boundary charts {ϕ i } i , which are L Ω -Lipschitz continuous.
Then we regularize each function ϕ i via convolution, and add (or subtract) a suitable constant, so that we obtain C ∞ -smooth functions {ϕ i m } i such that ϕ i m > ϕ i ( or ϕ i m < ϕ i ).However, in the original reference system, the graphs of these smooth functions G ϕ i m are not "glued" together, and thus their union is not the boundary of a domain, unlike the graphs G ϕ i whose union describes ∂Ω-see Figure 1 below.
To overcome this problem, we define a suitable C ∞ -smooth function F m , built upon {ϕ i m } i and {ξ i } i -see equation (6.14) below-and define the regularized set Ω m as the sublevel set {F m < 0}, so that ∂Ω m = {F m = 0}, and by construction we will have In order to show that ∂Ω m is a smooth manifold, we prove that the gradient of F m along the directions of graphicality of ϕ i is greater than a positive constant depending on L Ω -see estimate (6.20).This property of F m will be proven by exploiting the so-called transversality condition of ϕ i , which is inherited via convolution by ϕ i m as well.Therefore, F m is strictly monotone along these directions, which entails that its zero-level set ∂Ω m is a smooth manifold with local boundary charts ψ i m defined on the same reference system as ϕ i .Thanks to the properties of convolution, we show that F m converge to the boundary defining function F of Ω built upon {ϕ i } i and {ξ i } i -see equations (6.9) and (6.10)-and thus ψ i m converge uniformly to ϕ i .Then, as in the proof of the implicit function theorem, we differentiate the identity F m y ′ , ψ i m (y ′ ) = 0, so that we may express the gradient ∇ψ i m (and its Hessian ∇ 2 ψ i m ) in terms of {ϕ j m , ∇ϕ j m } j (and {∇ 2 ϕ 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 |∇ 2 ψ i m | obtained in the previous steps, as to evaluate weighted Poincaré type quotients of the kind in terms of the corresponding quotient with weight |B Ω |, and then (3.7) will follow from the celebrated isocapacitary equivalency Theorem of Maz'ya [15], [18,Theorem 2.4.1].
In red: the graphs of the regularized local charts (up to isometry).
Our next and final result shows the flexibility of our approximation method, which takes into account even higher regularity of the domain Ω.Theorem 3.2.Under the same notations as Theorem 3.1, we have that ).The proof of Theorem 3.2 can be easily carried out by extending the proof and estimates of Theorem 3.1 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.

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 4.1.Let U ⊂ R n−1 be a bounded domain, K > 0 be a constant, and {Ψ m } m∈N be a family of bi-Lipschitz maps on U such that and there exists a bi-Lipschitz map Ψ : By Lebesgue differentiation theorem and since Ψ is a bi-Lipschitz map, we have that U ϕ is a subset of U with full measure.Also, thanks to (4.2) and the fact that Ψ U ⋐ O, we have that ϕ and M m (ϕ) are well defined on a neighbourhood of Ψ m (U ) for m > m 0 large enough.Then, for all x ′ ∈ U ϕ we have Above we used the fact that Ψ(x) is a Lebesgue point of ϕ, and as a consequence of (4.2).Now fix ε > 0, and take a function By applying Jensen inequality, the change of variables where we also used estimates (4.1) and (4.4).Then, by using (4.2) and (4.5), it is immediate to verify that and finally, via a change of variables y ′ = Ψ(x ′ ), and (4.4) we get Henceforth, by plugging the last three estimates into (4.6),we find and thus (4.3) follows by the arbitrariness of ε. □ We close this section recalling a variant of Lebesgue dominated convergence Theorem which will be useful later on.
. on E, and ´E g q k dx → ´E g q dx.Then f ∈ L q (E), and

Transversality and graphicality
Throughout this section, we shall consider an isometry T of R n , such that (5.1) where R = R ij n i,j=1 is an orthogonal matrix of R n , and where e n denotes the n-th canonical vector of R n , i.e. e n = (0, . . ., 0, 1), R t is the transpose matrix of R, and S n−1 is the unit sphere on R n .
Here we introduce the geometric notion of transversality, which was already used in [9] in a wider sense.The definition given here suffices to our purposes.Definition 5.1 (Transversality).Let ϕ : U → R be a Lipschitz continuous function on U ⊂ R n−1 open.We say that a unit vector n ∈ S n−1 is transversal to ϕ if there exists κ > 0 such that n where ν denotes the outward normal to G ϕ with respect to the subgraph S ϕ .
The next proposition shows a very interesting feature: the transversality of n ∈ S n−1 to a Lipschitz function ϕ is equivalent to the graphicality (and subgraphicality) of ϕ with respect to any reference system having e n = n, that is after performing a rotation of the axes through R, the graph and subgraph of ϕ are mapped onto the graph and subgraph of another function ψ-see identities (5.2) below.
then we have the transversality condition Let us comment on this result.Part (i) states that if G ϕ and S ϕ are, respectively, the graph and subgraph of an L-Lipschitz function ψ with respect to the reference system z = (z ′ , z n ) having 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 ϕ with respect to the coordinate system z = (z ′ , z n ), and it also provides a Lipschitz estimate to ψ.
Before starting the proof, we need to introduce the so-called transition map C from ϕ to ψ.Under the same notation as Proposition 5.2, the transition map C : U → V is defined as Here Π : R n → R n−1 is the projection map Π(x ′ , x n ) = x ′ .Observe that, when identities (5.2) hold true, by the very definition of C we have the equation In particular, this implies that C is a bijection, with inverse function C −1 : V → U given by Proof of Proposition 5.2.(i) By Rademacher's theorem, the normal vector ν to G ϕ outward with respect to S ϕ is well defined H n−1 -almost everywhere, and thanks to (5.2) and the definition of C, we may write Therefore, since Rn = e n and |∇ψ| ≤ L, from (5.4) we infer Now let f : T (U × R) → R be the function defined as f (z) = f (x) for z = T x.Recalling Rn = e n , via the chain rule we compute Thus, from expression (5.4) of ν(x ′ ) and estimate (5.3), we obtain Therefore, owing to (5.8) and the implicit function theorem, we immediately infer the existence of a function Thereby, (5.2) follows from the very definition of f and (5.6).Finally, by using (5.5) we infer that |∇ψ(Cx ′ )| ≤ L for all x ′ ∈ U , whence ∥∇ψ∥ L ∞ (V ) ≤ L since the transition map C is a bijection.□ Remark 5.3.We point out that inequality (5.8), when evaluated at points z = T x ′ , ϕ(x ′ ) , holds true if ϕ and ψ are merely Lipschitz continuous and satisfy (5.2).Indeed, since 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 (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 (ϕ).This is the content of the following proposition Proposition 5.4.Let U, V ⊂ R n−1 be open bounded , let T be an isometry of the form (5.1), and n = R t e n .Let ϕ : U → R and ψ : V → R be L-Lipschitz functions satisfying (5.2).If we set and for some sequence {c m } m∈N ⊂ R we define (5.10) In addition, we have the transversality condition

and
(5.12) where ν m is the outward unit normal to G ϕm with respect to the subgraph S ϕm .
Next, from the L-Lipschitz continuity of ϕ, we have for all x ′ , y ′ ∈ U m , hence ϕ m is L-Lipschitz continuous as well.From this and (5.11), we get that is (5.12).Next, since ρ m is radially symmetric and ϕ is L-Lipschitz continuous, for all x ′ ∈ U m we get and thus (5.10) follows.□ Since we have proven that the regularized function M m (ϕ) satisfies the transversality condition, Part (ii) of Proposition 5.2 entails its "graphicality" with respect to the coordinate system having n = e n .Proposition 5.5.Under the same assumptions of Proposition 5.4, there exist V m ⊂ R n−1 open bounded such that (5.13) dist and if C m is the transition map of ϕ m , we have that (5.17) Proof.From the results of Part (ii) of Proposition 5.2 and (5.12), there exist V m ⊂ R n−1 open bounded, and a function ψ m ∈ C ∞ (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 ϕ m is the function , and for all x ′ ∈ U m we have so that from (5.10) we infer for all x ′ ∈ U m .In particular (5.18) The first inequality in (5.18) On the other hand, by definition of U m , for any x ′ ∈ U we may find Since Π and T are 1-Lipschitz continuous, and ϕ is L-Lipschitz continuous, it follows that since C(U ) = V .Hence, by using the triangle inequality we get that is (5.13).
Next, on assuming that V m ∩ V ̸ = ∅, and C m being a bijection between U m and V m , we may take a point y ′ ∈ V m ∩ V such that y ′ = C m x ′ for some x ′ ∈ U m From (5.18) we find By using these two estimates and the L-Lipschitz continuity of ψ, we obtain 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).□ The next proposition shows that if ϕ ∈ W 2,q , then ψ ∈ W 2,q as well.Namely, graphicality preserves Sobolev second-order regularity for Lipschitz functions.Proposition 5.6.Under the same assumptions of Propositions 5.4-5.5, if in addition ϕ ∈ W 2,q loc (U ) for some q ∈ [1, ∞], then ψ ∈ W 2,q loc (V ).Proof.In the following proof, we will make use of Propositions 5.4-5.5 with c m ≡ 0.
Fix U 0 ⋐ U open, and set V 0 = C(U 0 ).Since dist H (V m , V ) → 0 due to (5.13), from [8, Proposition 2.2.17] we may find m 0 > 0 large enough such that and set f m (y) ≡ f m (x) for y = T x.Then owing to (5.15), we have that f m y ′ , ψ m (y ′ ) = 0 for all y ′ ∈ V m .By differentiating this expression, we obtain (5.19) and from the chain rule, equation n = R t e n , the definition of C −1 m and (5.11), we have Moreover, thanks to (5.14) and the L-Lipschitz continuity of M m (ϕ), the maps C m are uniformly bi-Lipschitz, i.e.
Thanks to this piece of information and (5.17), we may apply Proposition 4.1 and get e. y ′ ∈ V 0 By combining (5.19)-(5.21),and by using dominated convergence theorem, we find that ∇ψ m converges in L p (V 0 ) to some vector-valued function G for all p ∈ [1, ∞).It then follows from (5.16) and the uniqueness of the distributional limit that G = ∇ψ, hence (5.22) ∇ψ m → ∇ψ H n−1 -a.e. in V 0 and in L p (V 0 ).
Next, we differentiate twice identity f m y ′ , ψ m (y ′ ) = 0, and for k, r = 1, . . ., n − 1 we obtain while from the chain rule and the properties of C m , we obtain (5.24) Then, another application of Proposition 4.1 entails that From this, (5.20), (5.22)-(5.24)and by using dominated convegence Theorem 4.2, we find that ∇ 2 ψ m converges in L q (V 0 ) to some matrix valued function H. Whence H = ∇ 2 ψ due to the uniqueness of the distributional limit, and the proof in the Case q ∈ [1, ∞) is complete due to the arbitrariness of U 0 .
In the Case q = ∞, from (5.20), (5.23) and (5.24) we infer that {ψ m } m is a sequence uniformly bounded in W 2,∞ (V 0 ) with respect to m.Therefore, up to a subsequence, we have that ψ m weakly- * converge in W 2,∞ (V 0 ) to ψ, thus completing the proof.□ At last, we close this section with the following intrinsic property of W 2,q domains.Corollary 5.7.Let Ω be a bounded Lipschitz domains such that ∂Ω ∈ W 2,q for some q ∈ [1, ∞].Then any Lipschitz local chart ψ of ∂Ω is of class W 2,q .
Proof.From Definition 2.2, there exists a Lipschitz local chart ϕ ∈ W 2,q and an isometry T such that (5.2) holds.The thesis then follows from Proposition 5.6.□ As a final remark, let us mention that both Proposition 5.6 and Corollary 5.7 can be easily extended to the W k,q Case.

Proof of Theorem 3.1
This section is devoted to the proof of Theorem 3.1, 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 ∂Ω.By Definition 2.1, for any x 0 ∈ ∂Ω, we may find an L Ω -Lipschitz function ϕ x 0 : B ′ R Ω → R, and an isometry T x 0 of R n such that T x 0 x 0 = 0, and 1 .By compactness, we may find a finite sequence of points as well as L Ω -Lipschitz functions ϕ i and isometries T i satisfying We denote by R i the orthogonal matrix of T i , i.e.T i can be written as Notice also that the cardinality N of this covering of ∂Ω may be chosen satisfying We then set Ω t := {x ∈ Ω : dist(x, ∂Ω) > t} , so that by (6.1) we have (6.4) Starting from this point, we construct a suitable partition of unity: let and where ρt is the standard, radially symmetric convolution kernel on R n , and χ A denotes the indicator function of a set A.

6.3.
Regularization and definition of the smooth approximating sets ω m , Ω m .For i = 1, . . ., N , we can define the smooth functions L Ω m and (6.11) From the results of Proposition 5.4, we deduce that ϕ i m , ϕ i m ∈ C ∞ are L Ω -Lipschitz functions, and for all y ′ ∈ B ′ R Ω −1/m and i = 1, . . ., N .Taking inspiration from (6.8) and (6.10), we are led to define the functions and functions F m , F m : W → R defined as where the products f j m (x) ξ j (x) and f j m (x) ξ j (x) have to be interpreted equal to zero when x ̸ ∈ supp ξ j .
Clearly, F m and F m are C ∞ -smooth functions on W , and since (6.15) for all x ∈ K j 1/m thanks to (6.12), we then have The approximating open sets Ω m , ω m are thus defined as follows (6.17In particular, since F m (x) < F (x) < F m (x) for all x ∈ W , owing to (6.10) we have We now proceed to prove the remaining properties of Theorem 3.1 for the outer sets Ω m .The proofs for the inner sets ω m are analogous.6.4.∂Ω m , ∂ω m are smooth manifolds .Let us show that ∂Ω m is a smooth manifold, with local charts {ψ i m } N i=1 defined on the same coordinate systems as {ϕ i } N i=1 .We fix a constant ε 0 ∈ (0, R Ω /4), and for all i = 1, . . ., N we set F i (y) = F (x) and F i m (y) = F m (x) for y = T i x, x ∈ W . Owing to (6.2) we have whenever ∂Ω ∩ K i 0 ∩ K j 0 ̸ = ∅.This piece of information will allow us to use the transversality property.Specifically, thanks to (6.19) we may apply Propositions 5.2-5.4 with functions ϕ = ϕ j , ψ = ϕ i , isometry T = T j (T i ) −1 , and defining set Claim 1.There exists m 0 > 0 such that, for all i = 1, . . ., N , for all m ≥ m 0 and all Suppose by contradiction this is false; then for every k ∈ N, we may find m k ≥ k and a sequence , for all k ∈ N By compactness, we may extract a subsequence, still labeled as x k , such that x k → x 0 , and in particular x 0 ∈ K i 0 and F (x 0 ) = 0, hence x 0 ∈ ∂Ω ∩ K i 0 due to (6.10).Then, by the chain rule we have (6.22) if x ∈ supp ξ j , where z ′ = Π T j x.We now distinguish two cases: (i) j ∈ {1, . . ., N } is such that x 0 ̸ ∈ supp ξ j .Then dist x 0 , supp ξ j > 0, hence x k ̸ ∈ supp ξ j for all k ≥ k 0 large enough.
Also, owing to (6.15) and (6.8) we have By coupling this piece of information with (6.5), (6.21) and (6.23), we finally obtain which is a contradiction, and thus (6.20) holds true.
Claim 2. There exists m 0 > 0 such that ∀y Again, assume by contradiction this is false.Then for all k ∈ N, we may find sequences By compactness, we may find a subsequence, still labeled as being W open, and from (6.24) we have F i m k (y k ) ′ , w k n < 0. By using (6.16) and the Lipschitz continuity of F , it is readily shown that whence F i (y 0 ) ′ , w n ) ≤ 0 for all w n as above, but this contradicts the fact that F i (y 0 ) ′ , w n > 0 whenever w n > ϕ i (y 0 ) ′ due to (6.10), hence Claim 2 is proven.Now let y ′ ∈ B ′ R Ω −ε 0 ; by (6.16) and since F i y ′ , ϕ i (y ′ ) = 0, we have F i m y ′ , ϕ i (y ′ ) < 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 (6.16) ensure that such point y n is unique for all y ′ ∈ B ′ R Ω −ε 0 .This entails the existence of a function R Ω −ε 0 .Furthemore, owing to (6.10) and (6.16), we have that ψ i m (y ′ ) > ϕ i (y ′ ) for all y ′ ∈ B ′ R Ω −ε 0 , and from the implicit function theorem we also infer that Moreover, via a compactness argument as in Claim 1-2 and (6.1), one can prove that so that, in particular, the cylinders K i 2ε 0

N
i=1 are an open cover of ∂Ω m , and ∂Ω m ∩ supp ξ 0 = ∅ provided m > m 0 is large enough.
We have thus proven that ∂Ω m is a C ∞ -smooth manifold for m > m 0 , with local boundary charts {ψ i m } N i=1 defined on the same coordinate cylinders as . Approximation properties.First, we show that there exists m 0 > 0 such that (6.27) Assume by contradiction this is false; then we may find sequences m k ↑ ∞ and (y k ) ′ ∈ B ′ R Ω −2ε 0 such that (6.28) Up to a subsequence, we have ( we readily infer that F i (y 0 ) ′ , ℓ 0 = 0, whence ℓ 0 = ϕ i (y ′ ) 0 due to (6.10) and (6.2).By continuity we also have Then, for all t ∈ [0, 1], we have where L F denotes the Lipschitz constant of F .This implies that for all k ≥ k 0 large enough, the line segment Therefore, by using (6.2), (6.10) (6.16), (6.20) and (6.28), we obtain Let us now introduce the transition maps related to the local charts of and ∂Ω m .First of all, note that thanks to (6.26), we have and whenever ∂Ω m ∩ K i ε 0 ∩ K j ε 0 ̸ = ∅.For all i ∈ {1, . . ., N }, we define the set of indexes If j ∈ I i , then owing to (6.2) there exists , so it follows from (6.19), (6.26) and (6.27) that (T Henceforth, for all j ∈ I i , (6. 19) and (6.29) allow us to define the transition maps C i,j , C i,j m from ϕ i to ϕ j and from ψ i m to ψ j m respectively, i.e.
which are defined on the open sets In particular, by their definitions and the arguments of Section 5, we may write and their inverse functions are (C i,j ) −1 = C j,i and (C i,j m ) −1 = C j,i m .Observe also that C i,i = C i,i m = Id.
Furthermore, since supp ξ j ⋐ B R Ω /4 (x j ) ⋐ K j 2ε 0 , it follows from the definition of I i and (6.27) that (6.32) R Ω −ε 0 , and all m ≥ m 0 .We now claim that for all j ∈ I i , there exists an open set V i,j ⊂ B ′ R Ω −2ε 0 for which we have (6.33) , and such that V i,j ⊂ U i,j ∩ U i,j m for all m > m 0 .This in particular implies that both C i,j and C i,j m are defined on V i,j .To this end, let Then, owing to (6.27) it is immediate to verify that (6.34) whenever m > m 0 is large enough, and thus (6.33) is satisfied by our choice of set V i,j .Clearly V i,j ⊂ U i,j , so we are left to verify that V i,j ⊂ U i,j m .To this end, let y ′ ∈ V i,j ; then by (6.29) and (6.31) we may write , where in the latter inclusion we made use of the inequality |ϕ j (z ′ )| ≤ L Ω |z ′ |.Therefore, thanks to (6.27), for m > m 0 we have (T m by (6.29) and the definition of U i,j m , so the claim is proven.We also remark that (6.35) , and the projection map Π is a homeomorphism from G ϕ i (with the induced topology) to B ′ R Ω .Moreover, owing to (6.27) and by proceeding as in the derivation of (5.18), we obtain Our next goal is to obtain estimates on ∇ψ i m .To this end, we differentiate equation F i m (y ′ , ψ i m (y ′ )) = 0 with respect to y ′ k , for k = 1, . . ., n − 1, and recalling (6.32) we find (6.37) where x m = (T i ) −1 y ′ , ψ i m (y ′ ) , y ′ ∈ B ′ R Ω −2ε 0 .For all l = 1, . . ., n, by using the chain rule and recalling the definition of C i,j m , we find for all j ∈ I i such that x m ∈ supp ξ j .Since ϕ j m are L Ω -Lipschitz continuous, from (6.38) it follows that (6.39) , for all j ∈ I i .
We then have the following elementary lemma, which will be useful later.
Owing to (2.8) and the definition of h j,m , we have Therefore, for all x 0 m ∈ ∂Ω m , r ≤ r 0 , we have found From this, (6.58) and the isocapacitary equivalence [18, Theorem 2.4.1],we finally obtain the desired estimate (6.59) for all r ≤ r 0 and m > m 0 , and the proof is complete.
The author has been partially supported by the "Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni" (GNAMPA) of the "Istituto Nazionale di Alta Matematica" (INdAM, Italy).

Proposition 5 . 2 .
Let U ⊂ R n−1 be open, ϕ : U → R be a Lipschitz function, let T be an isometry of the form (5.1), and let n = R t e n .(i) If there exists an L-Lipschitz function ψ for some k ∈ N and (5.3) holds, then there exist V ⊂ R n−1 open, and a function ψ ∈ C k (V ) such that ∥∇ψ∥ L ∞ (V ) ≤ L and (5.2) holds true.

6. 2 .
Boundary defining function.Starting from the partition of unity {ξ i } N i=0 , and the local charts {ϕ i } N i=1 , we can construct the boundary defining function of ∂Ω as in [10, Proposition 5.43].