A Density Result for Homogeneous Sobolev Spaces on Planar Domains

We show that in a bounded simply connected planar domain Ω the smooth Sobolev functions Wk,∞(Ω) ∩ C∞(Ω) are dense in the homogeneous Sobolev spaces Lk,p(Ω).

obtained a density result assuming Ω to be starshaped or to satisfy an interior segment condition. For the smaller class of functions C ∞ (R 2 ) they also required an extra assumption on the boundary points to be m 2 -limit points. (See also Bishop [2] for a counterexample on a related question.) The result of Koskela-Zhang [14] showing that W 1,∞ (Ω) is dense in W 1,p (Ω) for every bounded simply connected planar domain was generalized to higher dimensions by . They showed that simply connectedness is not sufficient to give such a density result, but Gromov hyperbolicity in the hyperbolic distance is. In this paper we provide another generalization to the Koskela-Zhang result by going to higher order Sobolev spaces. We show that if we restrict attention to the homogenous norm, then being simply connected is sufficient for domains in the plane.
The approach in [13] differs from ours in that there the approximating functions are defined via shifting matters to the disk via the Riemann mapping. Instead, we directly make a Whitney decomposition of the domain and a rough reflection to define our approximating sequence. We achieve this via an elementary use of simply connectedness in the plane. In both of these approaches the values of the function in a suitable compact set are used to define a smooth function in the entire domain which approximates the original function in Sobolev norm. For this we employ similar tools as used by Jones in [8].
In p-Poincaré domains, that is domains Ω where a p-Poincaré inequality Ω |u − u D | p dx ≤ C Ω |∇u| p dx holds, we can bound the integrals of the lower order derivatives by the integrals of the higher order ones and thus we obtain the following corollary to our Theorem 1.1.
For instance Hölder-domains are p-Poincaré domains for p ≥ 2, see Smith-Stegenga [21]. It still remains an open question whether Corollary 1.2 holds if one drops the assumption of being a p-Poincaré domain.
Next we come to the question of density of C ∞ (R 2 ) functions in L k,p (Ω) in our setting of bounded simply connected domains. We have the following corollary which is analogous to [14,Corollary 1.2], where it is shown that Ω being Jordan is sufficient. A small modification of the argument there applies to our situation as well. See the end of Section 4 for the proof.
In Section 2, we collect the necessary ingredients which will be used for defining the approximating sequence; these include a suitable Whitney-type decomposition of a simply connected domain and a local polynomial approximation of Sobolev functions. In Section 3 we describe a partition of the domain using the Whitney-type decomposition of Section 2, which is needed for obtaining a suitable partition of unity. Then in Section 4, we define the approximating sequence and present the necessary estimates for proving Theorem 1.1 and Corollary 1.3.

Preliminaries
For sets A, B ⊂ R 2 we denote the diameter of A by diam(A) and the distance between A and B by dist(A, B). We denote by B(x, r) the open ball with center x ∈ R 2 and radius r > 0 and more generally, by B(A, r) the open r-neighbourhood of a set A ⊂ R 2 . Given a connected set E ⊂ R 2 and points x, y ∈ E, we define the inner distance d E (x, y) between x and y in E to be the infimum of lengths of curves in E joining x to y. (Notice that in general the infimum might have value ∞.) We write the inner distance in E between sets A, B ⊂ E as dist E (A, B).
We will use the following facts in plane topology whose proofs can be found in the book of Newman [17, Chapter VI, Theorem 5.1 and Chapter V, Theorem 11.8].
In the case where Ω is Jordan and γ is homeomorphic to a closed interval, the two connected components of Ω \ γ have boundaries γ ∪ J 1 and γ ∪ J 2 , where J 1 and J 2 are the two connected components of ∂Ω \ γ.

A dyadic decomposition.
Although it is standard to consider a Whitney decomposition of a domain in R d (see for instance Whitney [23] or the book of Stein [22, Chapter VI]), we will use a precise construction of such a decomposition. We present this construction below. Here and later on we denote the sidelength of a square Q by l(Q).
For notational convenience we start the Whitney decomposition below from squares with sidelength 2 −1 . Formally, by rescaling, we may consider all bounded domains Ω ⊂ R 2 to have diam(Ω) ≤ 1 in which case no Whitney decomposition would have squares larger than the ones used below regardless of the starting scale.

Definition 2.2 (Whitney decomposition).
Let Ω ⊂ R 2 be a bounded (simply connected) open set. Let Q n be the collection of all closed dyadic squares of sidelength 2 −n . Define a Whitney decomposition asF := n∈NF n where the setsF n are defined recursively as follows. DefineF

Lemma 2.3. A Whitney decomposition given by Definition 2.2 has the following properties.
Although the proof is very elementary, we give it here for completeness.
In order to see (W2), let Q ∈F n . Then all Q ′ ⊂ Ω for all Q ′ ∈ Q n with Q ′ ∩ Q = ∅. Consequently, dist(Q, Ω c ) > 2 −n = l(Q). For the upper bound, suppose distQ, Ω c > 3 In either case, Q / ∈F n giving a contradiction. Property (W3) holds by the recursion in the definition and the fact that the dyadic squares are nested.
Suppose (W4) is not true. Then there exist Q 1 ∈F n and Q 2 ∈F m with n < m − 1 and and so either Q 3 ∈F n+1 or Q 3 ⊂F n . In both cases Q 2 ⊂F n+1 and so Q 2 / ∈F m .

By a chain of dyadic squares
We say that the chain connects Q 1 and Q m .

2.2.
Approximating polynomials. We record here the following two Lemmas from [8] which will be used when estimating the approximation in Section 4. Lemma 2.4 (Lemma 2.1, [8]). Let Q be any square in R 2 and P be a polynomial of degree k defined in R 2 . Let E, F ⊂ Q be such that |E|, |F | > η|Q| where η > 0. Then Given a function u ∈ C ∞ (Ω) and a bounded set E ⊂ Ω, we define (see [8]) the polynomial approximation of u in E , P k (u, E) to be the polynomial of order k − 1 which satisfies Once k is fixed, we denote the polynomial approximation of u in a dyadic square Q as P Q The next lemma is a consequence of Poincaré inequality for Lipschitz domains.

Decomposition of the domain
From now on we fix a bounded simply connected domain Ω ⊂ R 2 and a Whitney decom-positionF of Ω given by Definition 2.2. For our purposes we need to choose at each level a nice enough subcollection ofF n , namely we take connected components of the Whitney decomposition (see Figure 1). More precisely we fix Q 0 ∈F 1 and for each n ∈ N let C n be the connected component of the interior ofF n that has int Q 0 as a subset. We define F n,j := {Q ∈F j : int Q ⊂ C n } and using this the families of squares The collection of boundary layer squares in D n is denoted by With this notation we have the following lemma.
Lemma 3.1. The above collections have the properties: Proof. The property (i) is obvious by the definitions of F n,j and D n since C n ⊂ C n+1 .
For (ii) it suffices to prove that for every Q ∈F there exists n ∈ N so that Q ∈ D n . Let Q ∈ F n . Since Ω is connected and open, there exists a path γ in Ω joining Q to Q 0 . By the fact thatF j ⊂ intF j+1 and the property (W1) of the decompositionF we have that Ω = ∪ j∈N intF j . Then by the compactness of γ there exists m ≥ n so that γ ⊂ intF m .
Assume that the claim is false. Then for the two squares Q ∈ Q n that intersect both Q 1 and Q 2 it is true that Q ∈F n and Q ⊂ Q ′ for all Q ′ ∈F n−1 . Let q 1 and q 2 be the centres of the squares Q 1 and Q 2 respectively. Consider a curve γ ′ : [0, 1] → Ω for which γ ′ 0 = q 1 , γ ′ 1 = q 1 and γ ′ ⊂ C n . Such a curve exists by the definition of C n . We may also assume that γ ′ is an injective PSfrag replacements Figure 2. The constructed Jordan curve γ in the proof of Lemma 3.1 ((iii)) has in its interior domain a dyadic square Q that also has to be an element of D n .
curve. Let t 0 := sup{t : and [q 2 , q] respectively. By Jordan curve theorem γ divides R 2 into two components, one of which is precompact (see Figure 2). Denote the precompact component by A.
For small enough ball B around q we have by the definition of γ that B \ γ has exactly two components. Since γ is a Jordan curve one of those components has to contain an interior point of A and thus the whole component lies inside A. On the other hand that component has to intersect with one of the dyadic squares in Q n touching both Q 1 and Q 2 (but being different from Q 1 and Q 2 ). Let Q ∈ Q n be that square. Now for all the neighbouring squaresQ ∈ Q n (except the opposite one) of Q eitherQ ∩ γ([0, 1]) = ∅ implying thatQ ∈ D n orQ is in the precompact component of R 2 \ γ([0, 1]) and thus by simply connectednessQ ⊂ Ω. Since Q 1 ∈ F n , also the opposite square of Q is a subset of Ω. Hence Q ∈F n or Q ⊂ Q ′ ∈ F n−1 which is a contradiction. Thus we have proven (iii).
In order to see (iv), suppose that there exists Q ∈ ∂D n such that Q / ∈ F n . Then Q ∈ F n,i ⊂F i for some i < n. By Property (W4), for all the Q ′ ∈F with Q ′ ∩ Q = ∅ we have Q ′ ∈F j for j ≤ i + 1 ≤ n. Thus, Q ′ ⊂ D n and Q / ∈ ∂D n giving a contradiction. If property (v) fails for some Q ∈ ∂D n , then for every Q ′ ∈F with Q ′ ∩ Q = ∅ we have Q ′ ∈F i for some i ≤ n. Thus, again Q ′ ⊂ D n and Q / ∈ ∂D n giving a contradiction. Finally, we prove property (vi). Since C n is open it suffices to prove that every Jordan curve is loop homotopic to a constant loop. Suppose this is not the case. Then there exists a Jordan curve γ that is not homotopic to a constant loop, and a point x ∈ Ω \ C n that lies inside γ. In particular there exists Q ∈ Q n such that Q ⊂ D n which lies inside γ and for which Q ∩ D n is an edge of a square. Now by similar argument as in (iii) we conclude that Q ∈ D n , which is a contradiction.
The next lemma shows that we can connect the boundary of D n to the boundary of Ω with a short curve in the complement of D n . Lemma 3.2. For each point x ∈ ∂D n , there exists an injective curve γ : Proof. Let Q ∈ ∂D n be such that x ∈ Q ∩ ∂D n . By Lemma 3.1 (v) we have that there exists a square Q ′ ∈ Q n touching Q at x so that Q ′ / ∈F n and Q ′ ⊂Q for everyQ ∈F j , j < n. Thus, there exists a neighbouring square Q ′′ ∈ Q n of Q ′ and a point y ∈ ∂Ω ∩ Q ′′ . Let γ 1 be a curve corresponding to a line segment connecting x to a point z ∈ Q ′ ∩ Q ′′ and let γ 2 be a curve corresponding to a line segment connecting z to y. Moreover, let t 0 := inf{t : Observe that by Lemma 3.1 (iv) we have ∂D n = Q∈∂Dn (Q∩∂D n ). Thus, by Lemma 3.1 (iii) we have that ∂D n is locally homeomorphic to the real line. Since by Lemma 3.1 (vi) C n is simply connected, we have that ∂D n = ∂C n is connected. Hence, ∂D n is a Jordan curve. Thus, we may write where I i = [y i , y i+1 ] is an edge of a square in F n with vertices y i and y i+1 , and y 1 = y Ln+1 . For the rest of the paper we fix a constant M > (4 √ 2 + 2). However, the following lemma is true for any M > 0 and with C depending on M. Lemma 3.3. There exists C ∈ N so that for any n ∈ N and x, y ∈ ∂D n with d ∂Dn (x, y) ≥ 2 −n C, and for any γ in Ω\int D n connecting x to y we have that γ ∩(Ω \ B(x, M2 −n )) = ∅. In particular, L(γ) ≥ M2 −n .
Proof. By taking a slightly larger C, namely C + 2, we may assume that x = y i and y = y j for some i and j, where y i , y j are two endpoints of intervals from the collection {I i } forming the boundary as noted above. Moreover, by symmetry we may assume that i < j and j − i ≤ n + 1 − j. Since each I i is a side for two squares in Q n , by taking C large enough, we obtain where the union is taken over all Q ∈ Q n having I m as one of it sides for some i < m ≤ j−1. Therefore, one of the intervals I m 1 , for i < m 1 ≤ j −1, has to intersect with the complement of the ball B(x, 2M2 −n ). Let Q ′ 1 ∈ ∂D n be the boundary square corresponding to that interval and let q 1 ∈ I m 1 \ B(x, 2M2 −n ). By symmetry, there also exists Q ′ 2 ∈ ∂D n whose side is some I m 2 with m 2 / ∈ {i+1, i+1, . . . , j −1} such that there is q 2 ∈ I m 2 \B(x, 2M2 −n ). Suppose now that there exists a curve γ in Ω\int D n joining x to y with γ ⊂ B(x, M2 −n ). We may assume that γ is injective, and by compactness that γ(t) ∈ Ω \ D n for every t ∈ (0, 1). Then, for i = 1, 2 we have that B(Q ′ i , 2 √ 2l(Q ′ i )) ⊂ B(q, M2 −n ) and hence Figure 3. In the proof of Lemma 3.3 we assume towards a contradiction that x and y can be connected by a short curve γ in Ω \ D n . This will imply that one more square in Q n (here Q ′′ 1 ) will be a subset of D n .

PSfrag replacements
i which is not a subset of D n , see Figure 3. We claim that either Q ′′ 1 or Q ′′ 2 lies inside the Jordan curve γ ′ obtained by concatenating the curve γ and the part of the boundary, denoted by γ ′′ , obtained from the intervals {I h } j−1 h=i , or by concatenating γ and ∂D n \ γ ′′ . This can be seen in the following way. Consider Ω h − → R 2 ֒→ S 2 , where h is a homeomorphism and the inclusion R 2 ֒→ S 2 is the inverse of the stereographic projection. Under this composite map S 2 \D n is a simply connected domain. Hence, by Lemma 2.1 (S 2 \D n )\γ has exactly two components whose boundaries are the two connected components of ∂D n \ γ together with γ. Thus, (Ω \ D n ) \ γ = (S 2 \ D n ) \ γ has exactly two components. Since ∂Q ′′ 1 ∩ ∂D n and ∂Q ′′ 2 ∩ ∂D n are in two different connected components of ∂D n \ γ, we conclude that Q ′′ 1 and Q ′′ 2 are in different components of (Ω \ D n ) \ γ. We denote the Q ′′ i that lies inside the Jordan curve by Q ′′ . Since Q ′′ ⊂ B(Q ′ , √ 2l(Q ′ )), we have that every neighbouring square of Q ′′ either lies inside γ ′ or is an element of ∂D n . In particular, by the simply connectedness of Ω they all are subsets of Ω. Hence, Q ′′ ⊂ D n which is a contradiction. Thus, we have proven that γ ∩ (Ω \ B(x, M2 −n )) = ∅.
Let us now partition Ω \ D n in the following way. Recall (3.1). Notice that for large enough n we have that L n ≥ 2C. Define x 1 := y 1 and then x m := y (m−1)C until L n +1−(m− Figure 4. Here the domain Ω is decomposed into the core part D 3 and eight boundary partsH i . A neighbourhood H 8 ofH 8 is also illustrated. 1)C < 2C. Notice that for every i = j we have d ∂Dn (x i , x j ) ≥ 2 −n C. We now partition the set Ω \ D n up to Lebesgue measure zero into connected sets {H j } m j=1 whereH j is the open set bounded by γ j , γ j+1 given by Lemma 3.2 for points x j and x j+1 , and J j := C(j+1) i=Cj I i (with interior in Ω \ D n ). This partition is well defined by Lemma 2.1. Notice that since L(γ i ) ≤ M for all i, we have that γ i ∩γ j = ∅ for all i = j. Let us define H j as the connected component containingH j of the set Ω ∩ H j ∪ B R 2 (γ j ∪ γ j+1 ∪ J j , δ) , where δ = 2 −n−3 . See Figure 4 for an illustration of the decomposition. Although the decomposition depends on n, for simplicity we do not display the dependence in the notation. A crucial property of our decomposition is the following lemma. Proof. Trivially γ i+1 ∈ H i ∩ H i+1 . Thus, we only need to show that H j ∩ H i = ∅ implies |i − j| ≤ 1. We may assume that i = j. Let x ∈ H i ∩ H j .
Suppose first that x ∈H i . Then, by (path) connectedness of H j there exists a path γ in H j from x toH j . Let Then, γ(t 0 ) / ∈H i but γ(t) ∈ H i ∩H j . Thus it suffices to consider the case when x / ∈H i ∪H j . Suppose now that x ∈ D n . Since δ < 2 −n 2 , we have that x ∈ Q for some Q ∈ ∂D n . Then, there are neighbouring squares Q i , Q j ∈ Q n of Q for which Q i ∩H i = ∅ and Q j ∩H j = ∅.
Since δ is small, we may choose the Q i , Q j so that Q i ∩ Q j = ∅. If Q i = Q j or if Q i and Q j have a common edge, then there is a curve γ ′ in Q i ∪ Q j fromH i toH j with L(γ ′ ) < 2δ. If Q i ∩ Q j is a singleton, then by Lemma 3.1 (iii) the neighbouring square Q ′ = Q of both Q i and Q j lies in Ω \ int D n . Indeed, if this were not the case, then Q ′ , Q ∈ F n and Q ′ ∩ Q is a singleton, implying that Q i ∈ D n or Q j ∈ D n . Thus, there exists a curve γ ′ in Ω \ D n joining Q i and Q j with L(γ ′ ) < 4δ.
Combining these observations with the analogous ones for Q j , we have that J i and J j can be connected by a curve in Ω \ D k with length less than 4δ + 4 √ 2 · 2 −n < 2 −n M. Hence, we have by Lemma 3.3 that dist ∂Dn (J i , J j ) ≤ C. Thus, |i − j| ≤ 1 in cyclical manner.
We are left with the case where x ∈ Ω \ (D n ∪H i ∪H j ). By definition we have that B (D n , 2δ) ⊂ Ω. Thus, if dist(x, J i ) < δ, we may join x to J i by a curve in Ω \ int D n with length less than δ. If dist(x, J i ) ≥ δ, then x ∈ B(γ m , δ), where m ∈ {i, i + 1}. By path connectedness of H i there is a curve γ in H i connecting x to γ i ∪ γ i+1 ∪ J i . We want to prove that x can be joined to γ m in δ-neighbourhood of γ m . If (a subcurve of) γ is not such a curve, then we may define Then, γ| [0,t 0 ] ⊂ B(γ m , δ). Therefore, there exists a point y ∈ γ m with d(γ(t 0 ), y) < δ. In particular, the line segment [γ(t 0 ), y] lies in (Ω \ D n ) ∩ B(γ m , δ) and thus we have proven that there exists a curve γ ′ in (Ω \ D n ) ∩ B(γ m , δ) connecting x to γ m . By the definition of γ m we have that γ ′ ⊂ B(γ m (0), 2 √ 2·2 −n +δ). By the same argument for j we conclude that J i and J j can actually be connected by a curve γ in (Ω \ int D n ) ∩ B(γ(0), 4 √ 2 · 2 −n + 2δ). Hence, by Lemma 3.3 dist ∂Dn (J i , J j ) < C, and thus |i − j| ≤ 1 in cyclical manner.

Approximation
In this section we finish the proof of Theorem 1.1 by making a partition of unity using the decomposition of Ω constructed in Section 3 and by approximating a given function by polynomials in this decomposition. Recall that our aim is to show that for any u ∈ L k,p (Ω) and ǫ > 0 there exists a function u ǫ ∈ W k,∞ (Ω) ∩ C ∞ (Ω) with ∇ k u − ∇ k u ǫ L p (Ω) ǫ. By noting that L k,p (Ω) ∩ C ∞ (Ω) is dense in L k,p (Ω) we may assume that function u ∈ L k,p (Ω) ∩ C ∞ (Ω). From now on, let u and ǫ > 0 be fixed.
Using the notation from Section 3, we write the domain Ω as the union of the core part D n and the boundary regions {H i } l i=1 . For eachH i we let I i be the collection of squares Q in ∂D n such that Q ∩H i = ∅, which are bounded in number independently of n. We need to decide what polynomial to attach to each set H i . For this purpose, for each 1 ≤ i ≤ l we assign a square Q i ∈ I i . We call Q i the associated square of H i .
Given Q ∈ I i we set P Q := i+1 j=i−1 {Q ′ ∈ I j }, which is a collection of squares from a suitable neighbourhood of Q.
Recall the approximating polynomials P Q introduced in Section 2.2. We abbreviate P i = P Q i for the associated squares Q i .
We make a smooth partition of unity by using a Euclidean mollification. (Compare to [14] where the inner distance in Ω was used for the mollification.) Let ρ r denote a standard Euclidean mollifier supported in B(0, r). We start with a collection of functions {ψ i } l i=0 , whereψ 0 = χ Dn * ρ 2 −n−5 andψ i = χH i * ρ 2 −n−5 | H i for i ≥ 1. Using this we obtain a partition of unity {ψ i } l i=0 by setting ψ i =ψ i / l j=0ψ j . Now the partition of unity {ψ i } l i=0 satisfies the following. (1) The function ψ 0 is supported in B(D n , 2 −n 10 ).
We will fix n later such that for the function u ǫ defined as First of all, we consider only n large enough so that (4.1) Now, we need to show that n can actually be chosen large enough so that also So, we compute for Q ∈ I i and |α| = k where A 1 and A 2 are the first and second terms on the right hand side of the inequality and we used that for β < α, j ∇ α−β ψ j = 0 and order of P i is at most k − 1. We first estimate A 1 as where in the third inequality we used that Q i (associated square ofH i ) and Q may be joined by a chain of bounded number of squares from I i by our construction, and therefore we may apply Lemma 2.5. Similarly we estimate A 2 as where again in the second inequality we used that if ψ j (x) = 0 for x ∈ Q ∈ I i then by our construction Q j and Q can be joined by a chain of bounded number of squares as j is either i − 1, i or i + 1 (cyclically); and therefore we can apply Lemma 2.5. For Q ∈F \ D n such that Q ∩ spt(ψ 0 ) = ∅, we assign to Q a square Q ′ ∈ I i , such that Q ∩ Q ′ = ∅. Note that such a square Q ′ exists by our construction. Then Q and Q ′ can be joined by a chain of bounded (by an absolute constant) number of squares from D n+1 . We choose such a chain for Q and denote it by B Q . We also set J n := {Q ∈F \ D n : Q ∩ spt(ψ 0 ) = ∅}.
We estimate using Lemma 2.5 exactly as above (see (4.2)) to obtain for |α| = k (4.5) Again, we estimate separately, Next we note that ∇ k u ǫ ≡ 0 inH i \ ∪ j =i spt(ψ j ) and we compute for |α| = k The terms in the first and second summands have been estimated earlier.
Q ′′ , we estimate now the third one; where we used the facts that for β < α, ∇ α−β j (ψ j ) = 0 and ψ 0 ≡ 0 in H ′ i in the first inequality, Lemma 2.4 in the second inequality since H ′ i ⊂ CQ i for some absolute constant C coming from Lemma 3.2 and in the third inequality we used Lemma 2.5.
Remark 4.1. Note that for each Q ∈ I i we have P Q = P Q i where Q i is the associated square ofH i . We note that any Q ′ ∈ ∂D n occurs in at most three distinct collections P Q i . Moreover any Q ∈ D n+1 appears in only a bounded number of the collections B Q ′′ , where Q ′′ ∈ J n . In particular, any Q ′ ∈ ∂D n appears in only a bounded number of the collections B Q ′′ , where Q ′′ ∈ J n . The bounds are provided by absolute constants coming from volume comparison. Now it follows from equations (4.3), (4.4), (4.5), (4.6) and (4.7) that when |α| = k. By Remark 4.1 we may choose n such that Then, the claim follows from (4.1) and (4.8).
Remark 4.2. We note that when k = 1 we may take the function to be smooth as well as bounded for showing the density of W 1,∞ (Ω) in W 1,p (Ω). This is because truncations approximate the functions in W 1,p (Ω). This allows us to boundealso approximate the L p norm of u. Indeed let u ∈ W 1,p (Ω) ∩ C ∞ (Ω) ∩ L ∞ (Ω) such that u L ∞ ≤ M. Decompose the domain as in the above construction; then choose n large enough such that u W 1,p (Ω\D n−1 ) ≤ ǫ and M|Ω \ D n−1 | < ǫ. Then it follows from estimates in the proof that the function u ǫ defined as above approximates u in W 1,p (Ω) with error given by ǫ. This conclusion is the content of [14].
Finally, let us show how the smooth approximation in Jordan domains is done.
Proof of Corollary 1.3. The argument we need follows the one used to prove [14, Corollary 1.2]. As in [14], given a bounded Jordan domain we approximate it from outside by a nested sequence of Lipschitz and simply connected domains G s which are obtained for example by taking the complement of the unbounded connected component of the union Whitney squares larger than 2 −s from the Whitney decomposition of the complementary Jordan domain of Ω.
Then, we note that for given n, taking s n large enough, we have that the squares in ∂D n are Whitney type sets in G sn , meaning they have diameters comparable to the distance from the boundary of G sn .
Note that G sn ⊂ B(Ω, 2 −sn+5 ) are simply connected. Now the set G sn \C n (recall that C n is a suitable connected component of the interior of the union of the Whitney squares of scale less than 2 −n ) can be decomposed in the same way as Ω \C n was decomposed into the setsH i in Section 3.
We may then follow the argument used in the proof of Theorem 1.1 to obtain an approximating sequence of functions u n in G sn which are in the space W k,∞ (G sn ) ∩ L k,p (G sn ) ∩ C ∞ (G sn ). By multiplying with a smooth cut-off function that is 1 on Ω and compactly supported in G sn , we obtain a sequence of global smooth functions having the desired properties.