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SMOOTH APPROXIMATION OF SOBOLEV FUNCTIONS ON PLANAR DOMAINS

By Wayne Smith, Alexander Stanoyevitch and David A. Stegenga Abstract. We examine two related problems concerning a planar domain Ω. The first is whether Sobolev functions on Ω can be approximated by global C ∞ functions, and the second is whether approximation can be done by functions in C ∞ (Ω) which, together with all derivatives, are bounded on Ω. We find necessary and sufficient conditions for certain types of domains, such as starshaped domains, and we construct several examples which show that the general problem is quite difficult, even in the simply connected case.

1. INTRODUCTION Let Ω be a bounded domain (i.e., a bounded open connected set) in Rn . For a number p, 1 ≤ p < ∞, and a nonnegative integer k, we let W k,p (Ω) denote the Sobolev space of those (real-valued) Lp -integrable functions u on Ω whose distributional partial derivatives Dα u, |α| ≤ k, also lie in Lp (Ω). The Sobolev space W k,p (Ω) becomes a Banach space when endowed with the norm X kukW k,p (Ω) = kukLp (Ω) + kDα ukLp (Ω) . |α|≤k

By a Sobolev function, we mean any function which belongs to some Sobolev space. The Sobolev functions are the “functions” of the modern theories of partial differential equations and consequently have been extensively studied for their own intrinsic properties. For good general references on the Sobolev spaces and their functions, we cite: [13], [1], [10], and Chapter 7 of [5]. We employ the familiar notation C ∞ (Ω) for the space of infinitely differentiable functions on Ω and C ∞ (Ω) for the space of restrictions of functions in C ∞ (= C ∞ (Rn )) to Ω. We also let Cb∞ (Ω) denote those C ∞ (Ω)-functions u for which Dα u is bounded on Ω for all α. Of course, C ∞ (Ω) is contained in Cb∞ (Ω) provided that Ω is a bounded domain. A fundamental approximation theorem for Sobolev spaces is the well known result of Meyers and Serrin [11] which states that C ∞ (Ω) is dense in W k,p (Ω) for all k ≥ 1 and 1 ≤ p < ∞. In this paper, we study the questions of which domains satisfy the following closely related properties: Property 1. C ∞ (Ω) is dense in W k,p (Ω). Property 2. Cb∞ (Ω) is dense in W k,p (Ω). 1991 Mathematics Subject Classification. Primary 46E35 Secondary 30C60, 30E10, 41A99. Key words and phrases. Sobolev spaces, uniform domains, C ∞ -approximation. The first author is supported in part by a grant from the National Science Foundation. Typeset by AMS-TEX

2 BY WAYNE SMITH, ALEXANDER STANOYEVITCH AND DAVID A. STEGENGA

Property 2 is a weaker requirement than Property 1. Property 1 has been shown to hold for domains satisfying the segment condition which says that, to every point x ∈ ∂Ω, there corresponds a positive number ηx and nonzero vector yx with the following property: if |z − x| < ηx and z ∈ Ω then Ω contains the open segment {z + tyx : 0 < t < 1}. Indeed, if Ω satisfies the segment condition, then C ∞ (Ω) is dense in W k,p (Ω) for all values of k and p (see, Theorem 3.18 on page 54 of [1]; also see [2]). Another class of domains for which Property 1 has an affirmative answer are the uniform domains. Definition. A domain Ω is a uniform domain if and only if there is a constant MΩ such that: given any two points x, y ∈ Ω there is a rectifiable curve γ in Ω with endpoints {x, y} and such that: (1.1) The length of gamma, |γ|, is less than MΩ |x − y|, and (1.2) the distance to ∂Ω for any point z ∈ γ is larger than MΩ−1 times the shorter of the two arc lengths of γ(x, z) and γ(z, y). Indeed, Jones introduced a class of subdomains of Rn which he called –δ domains and showed [6] that every Sobolev function on such a domain is the restriction of a global Sobolev function (of the same order and exponent). Hence, Meyers and Serrin approximation, applied to Rn , implies that Property 1 holds on –δ domains. When the parameter δ is infinity, the corresponding class of domains coincides with the class of uniform domains. For a proof of this latter fact, see Theorem 2.10 in [12]; see also the proof of Theorem 7 in [4]. Our main results concern bounded starshaped domains in the plane, that is domains for which there is a point x0 ∈ Ω such that for all x ∈ Ω the line segment [x0 , x] ⊂ Ω , and interior segment domains. Definition. A bounded domain Ω in Rn is called an interior segment domain, if to every point x ∈ ∂Ω, there corresponds a positive number ηx and nonzero vector yx with the following property: if |z − x| < ηx and z ∈ Ω then Ω contains the segment {z + tyx : 0 ≤ t ≤ 1}. Definition. Let µ be a positive measure on Rn . If x ∈ Rn and E ⊂ Rn satisfies µ(E ∩ B(x, r)) > 0 for every ball, centered at x with arbitrary radius r > 0, then we say that x is a µ-limit point of E. It may at first appear that there is very little difference between the segment condition and the interior segment condition. However, the segment condition implies that ∂Ω is locally the graph of a continuous function, whereas it is possible that the n-dimensional Lebesgue measure of the boundary, mn (∂Ω), is positive for a domain satisfying the interior segment property. In fact, this possibility is a crucial aspect of our work. The following three theorems are the main results of this paper. Theorem A. Let Ω be a bounded domain in R2 . If Ω is starshaped or if Ω satisfies the interior segment condition, then Cb∞ (Ω) is dense in W k,p (Ω) for all k = 1, 2, . . . and all 1 ≤ p < ∞. Theorem B. Let Ω be a bounded domain in R2 which is either starshaped or satisfies the interior segment condition. If every z ∈ ∂Ω is an m2 -limit point of Ωc , then C ∞ (Ω) is dense in W k,p (Ω) for all k = 1, 2, . . . and all 1 ≤ p < ∞.

SMOOTH APPROXIMATION

3

Here Ωc denotes the complement R2 \ Ω. The next theorem shows that the converse to Theorem B holds in much greater generality. In particular, it shows that the density condition involving m2 -limit points of Ωc , is necessary for C ∞ to be dense in W k,p (Ω), for finitely connected domains Ω. Theorem C. Let Ω be a domain in R2 . Suppose that z ∈ ∂Ω, that z is not a m2 limit point of Ωc and that z is a limit point of the set of nondegenerate components of ∂Ω; then C ∞ (Ω) is not dense in W k,p (Ω), for any k ≥ 1 and any 1 ≤ p < ∞. Example 1. Let R denote the open rectangle (0, 2) × (0, 1) and let E be any proper closed subset of [0, 1]. Put Ω = R \ ([0, 1] × E). It follows that Ω is an interior segment domain and hence Cb∞ (Ω) is dense in W k,p (Ω), by Theorem A. Moreover, by Theorem B and Theorem C, a necessary and sufficient condition that C ∞ is dense in W k,p (Ω) is that every point of E is an m1 -limit point for E. We construct an example showing that Theorem A and Theorem B both fail in dimension 3, and as a result we focus our attention on planar domains. Since uniform domains have Property 1 and since bounded simply connected planar uniform domains are the same as quasidisks, Jones asked whether every Jordan domain has Property 1, see [7]. Recently, Lewis [9] answered this question in the affirmative for the first order Sobolev spaces, i.e., for k = 1. On the other hand, there are remarkably simple examples for which C ∞ (Ω) is not dense W k,p (Ω), but R2 \ Ω is not connected; see [3] and [8]. In light of the above, it might seem reasonable that if Ω is a bounded simply connected planar domain, R2 \ Ω is connected and each point of Ωc is an m2 -limit point of Ωc , then C ∞ (Ω) is dense in W k,p (Ω) . This turns out to be very far from the truth. We construct several examples in Section 7 which show that this is in fact false. Each of the examples are simply connected domains but the presence of what we call two-sided boundary points is a key feature. For a domain Ω, we say that a point x ∈ ∂Ω is a two-sided boundary point for Ω provided that there is a δ(x) > 0 such that for each δ, 0 < δ < δ(x), the set B(x, δ) ∩ Ω has at least two components whose closures contain x. A better question is the following: Question 1. If Ω is a simply connected planar domain without two-sided boundary points and for which all points in Ωc are m2 -limit points of Ωc , then does it follow that C ∞ (Ω) is dense in W k,p (Ω)? In Sections 2 and 4 we prove special cases of Theorem A and Theorem B, with some preliminary work for the latter being in Section 3. The proofs Theorem A and Theorem B are found in Section 5. Section 6 contains Theorem C as well as another necessary condition for the approximation of Sobolev functions, and the examples mentioned above appear in Section 7. 2. Bounded Meyers and Serrin Approximation The approximation result of Meyers and Serrin states that C ∞ (Ω) is dense in W k,p (Ω), where Ω is any domain in Rn . However, it is not always possible to approximate a Sobolev function with a C ∞ (Ω) function which has bounded derivatives on Ω. We construct a domain for which Cb∞ (Ω) is not dense in W k,p (Ω) in Section 7. Our main result in this section is a geometric condition which guarantees this type of approximation.


We first describe a basic interior segment domain. Let Ψ be a domain in Rn−1 , I = (a, b) be a finite interval and η > 0. Denote by e1 the first coordinate unit vector. We say that a domain Ω which is contained in the product I × Ψ is a basic interior segment domain provided (b − η, b) × Ψ ⊂ Ω and for all other points x ∈ Ω, x + te1 is also in Ω, for 0 ≤ t ≤ η. For example, the unit square in R2 minus the product [0, 1/2] × E is a basic interior segment domain for any closed set E ⊂ [0, 1]. See also Figure 1.

x

x+ηe1

Ψ(x1)

a

b-η

x1

b

Figure 1. A basic interior segment domain in the plane. Theorem 1. Let Ω ⊂ (a, b)×Ψ ⊂ Rn be a basic interior segment domain for n > 1 and assume that Ψ and each component of each open set Ψ(x1 ) = {x ∈ Rn−1 | (x1 , x) ∈ Ω}

a < x1 < b

is a uniform domain in Rn−1 . Then Cb∞ (Ω) is dense in W k,p (Ω). Lemma 2.1. Let Ω1 and Ω2 be uniform domains with finite diameters d(Ω1 ) and d(Ω2 ). Then the product Ω1 × Ω2 is a uniform domain. Proof. Let (x1 , x2 ) and (y1 , y2 ) be points in Ω1 × Ω2 , and assume without loss of generality that |x1 − y1 | ≤ |x2 − y2 |. Choose a point z1 ∈ Ω1 such that |x1 − z1 | =

|x2 − y2 |d(Ω1 ) . 2d(Ω2 )

By considering points on a curve from x1 to z1 satisfying (1.1) and (1.2), we can find w1 ∈ Ω1 such that |x2 − y2 |d(Ω1 ) |x1 − w1 | = 4d(Ω2 )


5

and the distance from w1 to ∂Ω1 is at least |x1 − w1 |/MΩ1 . Let γ1 be the curve in Ω1 from x1 to y1 formed by joining curves from x1 to w1 and from w1 to y1 that satisfy (1.1) and (1.2), and let γ2 be the curve in Ω2 from x2 to y2 satisfying (1.1) and (1.2). Now let zi (t), 0 ≤ t ≤ Li , be the arclength parameterizations of γi with zi (0) = xi , i = 1, 2. It follows that C0 = L2 /L1 is bounded from above and away from 0 by constants depending only on d(Ωi ) and MΩi , i = 1, 2. Consider the curve σ given by (z1 (t), z2 (C0 t)), 0 ≤ t ≤ L1 , from (x1 , x2 ) to (y1 , y2 ) in Ω1 × Ω2 . The arclength of the part of σ corresponding to a ≤ t ≤ b is comparable to b−a, and the distance from (z1 (t), z2 (C0 t)) to ∂(Ω1 ×Ω2 ) is comparable to the minimum of the distances from zi to ∂Ωi . Thus, the fact that each γi satisfies (1.2) implies that σ also satisfies this condition. Lastly, the length of σ is comparable to the sum of the lengths of the γi ’s, and hence to |x2 − y2 |. It follows that σ satisfies (1.1) and that Ω1 × Ω2 is a uniform domain. Remark. By letting Ω1 be an interval and Ω2 a half plane, we see that it is necessary to assume that the domains are bounded. Observe that if Ω ⊂ (a, b) × Ψ is a basic interior segment domain, then there is an upper semicontinuous (usc) function f : Ψ → R such that for each x0 ∈ Ψ: (x1 , x0 ) ∈ Ω

if and only if f (x0 ) < x1 < b.

We say that f is the defining function for Ω. To make the proof of Theorem 1 easier to follow, we first prove a special case: Lemma 2.2. Suppose Ω ⊂ (a, b) × Ψ satisfies the hypotheses of Theorem 1, and further assume that the defining function for Ω, f : Ψ → R, assumes only finitely many values. Then Cb∞ (Ω) is dense in W k,p (Ω). Proof. The proof is by induction on the number of values f assumes. First consider the case that f is constant. Then, Ω = J × Ψ for some interval J. Since intervals are uniform domains, Lemma 2.1 implies Ω is a uniform domain and we are done because, as noted before, uniform domains have Property 1. Now suppose that the defining function f has m values, and let u ∈ W k,p (Ω). Let am be the largest value, so that Ω0 = (am , b) × Ψ is the part of Ω with x1 > am . Since Ω0 is a uniform domain, there is a global Sobolev function U whose restriction to Ω0 is u. By subtracting U , which clearly can be approximated by functions in Cb∞ (Ω), we see that we may assume u is identically zero on Ω0 . We decompose Ω into the disjoint union Ω = (Ω ∩ Ω0 ) ∪ Ω1 ∪ Ω2 . . . , where the Ωk ’s, for k ≥ 1, are the domains determined by the components of the open set Ψ(am ) in Rn−1 . By hypothesis, each component of each cross-section is a uniform domain and furthermore, the defining function for each Ωk , with k > 0, has fewer than m values. Since ∪Ωk has finite volume, there is an integer K such that we may redefine u to be identically zero on Ωk for all k > K and still have the modified function v within of u in the Sobolev norm.


Ψ(am )

Ω

1

Ω

a

Ω

2

Ω

3

am

0

b

Figure 2. Decomposing Ω in order to apply induction. Since v is 0 on Ω0 , for 0 < λ < b − am we can define vλ (x) = v(x + λe1 ) on Ω, by extending v to be 0 for x1 ≥ b. By the Lp -continuity of translation, we can choose λ sufficiently small so that vλ is within of v in Sobolev norm. Note that there is a neighborhood of am such that vλ is 0 whenever x1 is in that neighborhood. By induction, we can approximate the function vλ restricted to Ωk by a function wk ∈ Cb∞ (Ωk ) which differs from vλ in the Sobolev norm by an amount that is less than /K. Moreover, we may assume that each wk vanishes for x1 in a neighborhood of am , since vλ is 0 there. It follows that the function w which is zero on Ω \ (Ω1 ∪ · · · ∪ ΩK ) and w = wk on Ωk , 1 ≤ k ≤ K, is a well defined function in Cb∞ (Ω) which differs from u in the Sobolev norm by an amount that is less than C. This completes the proof. We now ready to complete the proof of Theorem 1. Proof of Theorem 1. Let u ∈ W k,p (Ω) and > 0. We may assume that u ∈ C ∞ (Ω) by the Meyers-Serrin Theorem. Put λ0 = η/2 and let φ ∈ C ∞ (R) satisfy φ = 0 for x1 ≤ b−η and φ = 1 for x1 ≥ b−λ0 . Write u = u1 +u2 , where u1 (x) = φ(x1 )u(x) for x1 > b −η and u1 (x) = 0 for x1 ≤ b −η. Now, we see that u1 ∈ W k,p ((a, b) ×Ψ) and hence by Lemma 2.1, Jones’ Extension Theorem and Meyers-Serrin approximation it follows that u1 can be approximated by functions in Cb∞ (Ω). Thus, it suffices to approximate u2 . But u2 = (1 − φ)u = 0 for all x1 ≥ b − λ0 . We may therefore assume, without loss of generality, that our original function u satisfies u = 0 for all b > x1 > b − λ0 . For 0 < λ < λ0 , we can now define uλ (x) = u(x + λe1 ), by extending u to be zero for x1 > b − λ0 , and get a W k,p function on the shifted domain (Ω − λe1 ) ∪ Ω


7

with norm the same as the norm of u on Ω. By the Lp -continuity of translation we again get that there exists λ1 > 0 with ku − uλ1 kW k,p (Ω) < .

(2.1)

Let Ω1 = (Ω−λ1 e1 )∪Ω and suppose that there is a domain Ω0 with Ω ⊂ Ω0 ⊂ Ω1 with the Cb∞ approximation property. Then we could find v ∈ Cb∞ (Ω0 ) such that kuλ1 − vkW k,p (Ω0 ) < . It would follow that ku − vkW k,p (Ω) ≤ ku − uλ1 kW k,p (Ω) + kuλ1 − vkW k,p (Ω0 ) < 2 , and hence u could be approximated by a function in Cb∞ (Ω). Hence the theorem will be proved once such a domain is constructed. Let f : Ψ → R be the defining function for Ω. We define Ω0 by f (x0 ) λ1 < x1 < b}. Ω = {(x1 , x ) | x ∈ Ψ and λ1 0

0

0

Here [t] denotes the greatest integer less than or equal to t. Clearly Ω ⊂ Ω0 ⊂ Ω1 , and since f is usc it follows that Ω0 is an open set. If we let f 0 : Ψ → R be the corresponding usc function which defines Ω0 then we see that f 0 takes on only finitely many values. Therefore Lemma 2.2 allows us to conclude that Ω0 has the required Cb∞ approximation property, and this completes the proof. 3. A Smooth Cantor-Type Sneak Function The Cantor function is a function φ : [0, 1] → R which is continuous, satisfies φ = 0 a.e. and yet φ manages to sneak from the value zero to the value one. We construct a similar function which is smooth and does most of its increasing on a Cantor set of positive measure. We note that the desired function in the theorem can not be constructed by the usual mollification of the characteristic function of K. 0

Theorem 2. Let K be a closed subset of the open interval I = (a, b). Assume that K has positive measure. If > 0, k ∈ {1, 2, . . . }, and 1 ≤ p < ∞, then there is a function φ ∈ C ∞ (R) satisfying (3.1)

φ(x) = 0 for x ≤ a, φ(x) = 1 for x ≥ b

and Z (3.2) I\K

|φ(j) |p dx <

j = 1, . . . , k .


Proof. First consider the case k = 1. Denote the measure of a set E by |E|. There is an open set U with K ⊂ U ⊂ I and |U \ K| < |K|p . Let ψ ∈ C ∞ (R) satisfy 0 ≤ ψ ≤ 1, ψ = 1 on K and ψ = 0 on U c . Put  0 if x ≤ a   R x ψ(t) (3.3) φ(x) = R dt if x > a .   a ψ dt I

so that φ clearly satisfies (3.1). Moreover, p Z Z ψ(t) 0 p R |φ | dx = dt ψ dt I\K

I\K

I

1 ≤ |K|p

Z ψ p dt

U\K

≤

|U \ K| < , |K|p

which proves (3.2). Now assume, by induction, that the theorem has been proved for some k ≥ 1. There exists an open set U ⊂ I containing K with 2p |U \ K| < |K|p . Since K is compact, U has n interval components Ji , i = 1, . . . , n, which cover K and without loss of generality such that Ki = K ∩ Ji has positive measure. We now apply the induction hypothesis to appropriate subintervals of Ji to produce a function that first goes from 0 to 1 on an interval Ji,1 ⊂ Ji satisfying |Ji,1 ∩K| = |Ki |/4, remains constant on an interval Ji,2 satisfying |Ji,2 ∩K| = |Ki |/2, and then decreases from 1 to 0 on an interval Ji,3 satisfying |Ji,3 ∩ K| = |Ki |/4. Thus, we can find ψ ∈ C ∞ (R) such that (3.4) (3.5)

0 ≤ ψ ≤ 1 and ψ = 0 on U c , Z Z 1 ψ dx ≥ ψ dx ≥ |Ki | , i = 1, . . . , n 2 Ji

and

Ji,2

Z

(3.6)

|ψ (j) |p dx
0. Theorem 3. Let Ω ⊂ R2 be a basic interior segment domain. A sufficient condition for C ∞ (Ω) to be dense in W k,p (Ω) is that every boundary point of Ω be a m2 -limit point of Ωc . Lemma 4.1. Let R = (0, a) × (0, b) be a rectangle and suppose that f ∈ C k+1 (R) ˜ the rectangle with |Dα f (z)| ≤ M for all z ∈ R and |α| ≤ k + 1. Denote by R ˜ with F = f on R and (0, a) × (0, 2b). Then, there is a function F ∈ C k (R) |Dα F (z)| ≤ CM,

˜ z ∈ R,

|α| ≤ k

where C is some absolute constant. Proof of the lemma. See the proof of Lemma 6.37 in [5], page 136. Proof of Theorem 3. Let Ω be a basic interior segment domain with boundary satisfying the above density condition. Using the same techniques as in the proof of Theorem 1, we see that it suffices to consider a domain Ω whose defining function assumes only m values. Let am be the maximum value of f which is now defined on some interval Ψ. Let u ∈ W k,p (Ω) and let > 0. Again, as in the proof of Theorem 1, we may assume that u = 0 on the strip (am , b) × Ψ and in fact, we may assume that u = 0 if x > am − λ for some λ > 0. Decompose Ω into the disjoint union Ω = (Ω ∩ Ω0 ) ∪ Ω1 ∪ . . . as in the proof of Theorem 1. The Ωj ’sPare determined by the interval components of the open set Ψ(am ) in R1 . Since m2 (Ωj ) converges (where m2 is two dimensional Lebesgue measure), there is an integer J such that we may redefine u to be identically zero on ΩJ+1 ∪ . . . and still have the modified function u1 within of u in the W k,p (Ω) norm. By Theorem 1, applied to each Ωj , there is a function v ∈ Cb∞ (Ω) which is within of u1 in the Sobolev norm and vanishes on all of Ω except possibly on the Ωj ’s. Also, v = 0 for all x > am − λ/2. Put M=

sup z∈Ω,|α|≤k

which is finite since v ∈ Cb∞ (Ω).

|Dα v|,


Let am−1 be the second largest value for the defining function f . For each j = 1, . . . , J, the intersection Ωj ∩ {(x, y) ∈ R2 : am−1 < x < am } is a rectangle (am−1 , am )×I j , where the intervals I j are components of Ψ(am ). Now of course the intervals {I j } are disjoint by the construction, but moreover, the density condition implies that their closures are disjoint. Let I j = (cj , dj ) and assume that the intervals are ordered so that dj−1 < cj . For dj−1 < y < cj , observe that either the point (am , y) ∈ ∂Ω or else the segment (am−1 , b) × {y} is in Ω. Consequently, ∂Ω ∩ [{am } × (dj , dj + η)] and ∂Ω ∩ [{am } × (cj − η, cj )] must both have positive linear measure by the density condition, for any η > 0. We now extend the definition of v to the whole strip S = (am−1 , b) × Ψ using Theorem 2. Fix η, with 0 < η < 12 min{cj − dj−1 } and η < /(JM p (am −am−1 )). Now apply Theorem 2 in each of the intervals (cj −η, cj ), (dj , dj + η) to construct a function ψ ∈ C ∞ (R1 ) with 0 ≤ ψ ≤ 1 and such that ψ = 1 on ∪(cj , dj ), ψ = 0 off ∪(cj − η, dj + η) and Z |ψ (l) |p dx < , l = 1, . . . , k, JM p Kc

where K = ∂Ω ∩ {x = am }. By Lemma 4.1 there is a function v1 which is C ∞ on the set J

G = ∪ (am−1 , am ) × (cj − η, dj + η) j=1

and whose restriction to ∪Jj=1 (am−1 , am ) × (cj , dj ) is v. Moreover, |Dα v1 | ≤ CM for all |α| ≤ k. Now put v2 (x, y) = v1 (x, y)ψ(y) for (x, y) in Ω ∪ G and v2 = 0 on S \ G. Since v = 0 for all x > am − λ/2 we see that v2 is a C ∞ -function on S ∪ Ω. Finally, we observe that kv − v2 kW k,p (Ω) = kv2 kW k,p (Ω∩G\(Ω1 ∪···∪ΩJ )) = kv1 ψkW k,p (Ω∩G\(Ω1 ∪···∪ΩJ )) =

J X

kv1 ψkW k,p ([(am−1 ,am )×(cj −η,cj )]∩Ω)

j=1

+

J X

kv1 ψkW k,p ([(am−1 ,am )×(dj ,dj +η)]∩Ω) ,

j=1

and we will show this quantity is small due to the choice of η. Consider a typical term in the derivative of the product v1 ψ. We estimate as follows: Z am Z |Dα v1 ψ (l) (y)|p dydx am−1

(cj −η,cj )\K

Z ≤ CM (am − am−1 ) p

Kc

≤ CM p

C , = p JM J

|ψ (l) (y)|p dydx


11

for all l > 0. Similarly, for l = 0, we get Z

am am−1

Z |Dα v1 ψ(y)|p dydx ≤ CM p (am − am−1 )η < (cj −η,cj )\K

C , J

by the choice of η. The same estimate applies when the integration is done over (am−1 , am ) × (dj , dj + η). Adding these estimates, we see that kv − v2 kW k,p (Ω) is smaller than a constant (depending on k) times . This proves that a Sobolev function on Ω can be approximated by a function w ∈ C ∞ (Ω ∪ S). Consider such a function w. As argued before, we may as well assume that w = 0 on S, because S is a uniform domain. We may also assume that w = 0 for x in a neighborhood of am−1 . By induction, wj = w |Ωj can be approximated, within /J in the Sobolev norm, by a global C ∞ -function which also vanishes for x ≥ am−1 and such that wj |Ωk ≡ 0 for j 6= k. Adding these functions gives a function in C ∞ (Ω) which is within of u in the W k,p (Ω) norm. This completes the proof. 5. Approximation on Starshaped and Interior Segment Domains In this section we prove Theorems A and B of the introduction. Our method is based on the following principle. First, there are general homeomorphisms which preserve the density conditions of Properties 1 and 2 (of the introduction) as well as Lebesgue density points. For example, it is clear that bilipschitz maps enjoy these properties, more generally, so do the so-called quasi-isometric maps which are studied in Chapter 1 of [10]. By invoking such homeomorphisms in conjunction with standard partition of unity arguments, we can obtain results similar to those of Theorems 1 and 3 for different sorts of domains. Theorem 4. Let Ω be a bounded domain in R2 . If Ω satisfies the interior segment condition, then Cb∞ (Ω) is dense in W k,p (Ω) for all k = 1, 2, . . . and all p, 1 ≤ p < ∞. Proof. For each z ∈ ∂Ω let ηz > 0 and yz 6= 0 be as in the definition of the interior segment condition. Choose points z1 , z2 , . . . , zN ∈ ∂Ω so that the balls N {B(zi , ηzi /2)}i=1 cover the compact set ∂Ω. We now fix an index i, 1 ≤ i ≤ N , and introduce new (orthogonal) coordinates (x1 , x2 ) so that yzi lies on the x1 -axis. Next, replacing the given cover by a cover of smaller balls if necessary, we may assume that for some rectangle Ri = { (x1 , x2 ) | ai < x1 < bi and ci < x2 < di } we have Ω ∩ B(zi , ηzi /2) ⊂ Ri and Ωi = Ω ∩ Ri is a basic interior segment domain. Finally, choose a domain R0 containing the closure of Ω \ ∪Ri and which is itself compactly contained in

12 BY WAYNE SMITH, ALEXANDER STANOYEVITCH AND DAVID A. STEGENGA N

Ω. We now select a partition of unity {λi }i=0 of Ω subordinate to the covering {R0 , R1 , . . . , RN }. This means that there are functions λi ∈ C ∞ for which X λi ≥ 0, λi ≡ 0 on Ric , and λi ≡ 1 on Ω. With this we are ready to approximate. So assume that u ∈ W k,p (Ω) and > 0 are given. As before, the Meyers-Serrin theorem permits us to assume that u ∈ C ∞ (Ω). For each i > 0, Theorem 1 provides us with a function fi ∈ Cb∞ (Ωi ) with k fi − ui kW k,p (Ωi ) < . N (We use the notation ui to denote the restriction of the function u to the Ωi .) Letting i for i > 0, we consider the function P f0 = u and extending∞fi to be 0 off ΩP f= fi λi . Certainly f ∈ Cb (Ω) and since λi ≡ 1 on Ω, we have that k f − u kW k,p (Ω) ≤

X

kλi (f − u) kW k,p (Ωi ) X ≤C k f − u kW k,p (Ωi ) i>0

< C. Here C is a constant depending only on k, p, and the λi ’s; the second inequality results from the Leibniz rule. The proof is thus complete. Remark. We observe that in the application of Theorem 1 in the above proof, it was automatically true that the basic interior segment domains Ωi satisfied the uniform domain requirements because, the components of Ψ(x1 ) are just intervals in R1 . The situation for n > 2 is more complicated and in fact the theorem is no longer true; see Section 7. The situation is similar in the next three theorems. Theorem 5. Let Ω be a bounded domain in R2 which satisfies the interior segment condition. If every z ∈ ∂Ω is an m2 -limit point of Ωc , then C ∞ (Ω) is dense in W k,p (Ω) for all k = 1, 2, . . . and all p, 1 ≤ p < ∞. Proof. The proof is the same as that of Theorem 4 except for the use of Theorem 3 in place of Theorem 1. Theorem 6. Let Ω be a bounded starshaped domain in R2 . Then Cb∞ (Ω) is dense in W k,p (Ω) for all k = 1, 2, . . . and all p, 1 ≤ p < ∞. Proof. For convenience, we assume that Ω is starshaped with respect to the origin. Choose numbers r and R, 0 < r < R < ∞, such that ∂Ω ⊂ {z | r < |z| < R }. We next cover Ω with the following three open sets: V1 = { z : r/2 < |z| < R, arg {z/|z|} ∈ (π/4, 7π/4) } , V2 = { z : r/2 < |z| < R, arg {z/|z|} ∈ (−3π/4, 3π/4) } V3 = { z : |z| < r}.

and


13

Suitable modifications of the exponential function provide us with conformal maps Φi : Ri → Vi , i = 1, 2, where Ri is a rectangle. Note that these mappings Φi have the property that (for a fixed k) all of the partial derivatives of Φi and of its of order at most k are uniformly bounded. Letting inverse Φ−1 i Ω i = Ω ∩ Vi ,

(i = 1, 2)

we conclude there exists a constant C such that 1 ku◦Φi kW k,p (Φi (Ωi )) ≤ kukW k,p (Ωi ) ≤ Cku◦Φi kW k,p (Ωi ) . C Suppose now we are given a function u ∈ W k,p (Ω) ∩C ∞ (Ω) along with a number > 0. For i = 1, 2, it is clear that Φi (Ωi ) is a basic interior segment domain, hence by invoking Theorem 1 we obtain functions fi ∈ Cb∞ (Φi (Ωi )) such that k fi − ui ◦Φi kW k,p (Φi (Ωi )) < . Hence,

k fi ◦Φ−1 i − ui kW k,p (Ωi ) < C.

If we select a partition of unity {λ1 , λ2 , λ3 } subordinate to the open cover { V1 , V2 , V3 } of Ω then, just as in the proof of Theorem 4, one can show that the function −1 f = f1 ◦Φ−1 1 λ1 + f2 ◦Φ2 λ2 + uλ3 is in Cb∞ (Ω) and satisfies k f − u kW k,p (Ω) < C. This completes the proof of Theorem 6 and as before we also have the following theorem: Theorem 7. Let Ω be a bounded starshaped domain in R2 . If every z ∈ ∂Ω is a m2 -limit point of Ωc , then C ∞ (Ω) is dense in W k,p (Ω) for all k = 1, 2, . . . and all p, 1 ≤ p < ∞. We end this section with another geometric sufficient condition for globally smooth approximation. This theorem will, under certain circumstances, allow us to ignore finitely many “bad points” of a domain in which we want to approximate Sobolev functions. We will make use of it in some of our constructions of examples in Section 7. To facilitate the statement and proof of this result, we begin with a definition and a lemma. Definition. Suppose that we are given a bounded domain Ω ∈ R2 , with distinct points ζ1 , ζ2 , . . . , ζN ∈ ∂Ω, and a positive number δ < min{ |ζi − ζj |/2 | i 6= j }. We then define the δ-modification of Ω with respect to { ζ1 , ζ2 , . . . , ζN } as Ω(δ; ζ1 , ζ2 , . . . , ζN ) = Ω \

N [ j=1

B(ζj , δ).


Lemma 5.1. Suppose that 1 ≤ p ≤ 2 and that and δ are positive numbers. Then there exist numbers δ1 and δ2 with 0 < δ1 < δ2 < δ and a function φ ∈ C ∞ (R2 ), 0 ≤ φ ≤ 1, satisfying (i) φ ≡ 0 on B(0, δ1 ), (ii) RR φ ≡ 1 off B(0, δ2 ), and (iii) |∇φ|p dxdy < . R2

Proof. Case 1: 1 ≤ p < 2. Choose the positive number δ1 to satisfy n δ1 < min

p

2−p

o /3π, δ/2 .

Next, put δ2 = 2δ1 and consider the Lipschitz radial function φ0 : R2 → R1 defined by  0 if |x| < δ1   φ0 (x) = 1 if |x| > δ2 and   otherwise. (|x| − δ1 )/δ1 Integration in polar coordinates yields ZZ

Zδ2 |∇φ0 | dxdy = 2π p

R2

δ1

r dr = 3πδ12−p < . δ1p

Taking φ to be a suitable mollification of φ0 , we get a smooth function φ satisfying (iii) which equals 1 off B(0, δ) and vanishes in some neighborhood of the origin. By appropriately decreasing δ1 and increasing δ2 we see that this φ has all of the desired properties. Case 2: p = 2. Let δ2 be any positive number less than δ and choose δ1 to be another positive number satisfying δ1 exp

2π < δ2 .

Next we consider the Lipschitz radial function φ0 : R2 → R1 defined by     φ0 (x) =

0

1    log(δ1 /|x|) log(δ1 /δ2 )

if |x| < δ1 if |x| > δ2 and otherwise.

The proof can be then as before, since an integration using polar coorRR completed 2 dinates again yields |∇φ0 | dxdy < . R2


15

Theorem 8. Let Ω ⊂ R2 be a bounded domain, let 1 ≤ p ≤ 2, and let ζ1 , ζ2 , . . . , ζN be distinct points of ∂Ω. If there exists a null sequence of positive numbers, {δi }∞ i=1 , such that for each i, the boundary of Ω(δi ; ζ1 , ζ2 , . . . , ζN ) consists of finitely many disjoint Jordan curves, then C ∞ (R2 ) is dense in W 1,p (Ω).

Figure 3. A domain to which Theorem 8 is applicable. Proof. For general domains (in Rn ) it is known that W 1,p (Ω) ∩ C ∞ (Ω) ∩ L∞ (Ω) is dense in W 1,p (Ω) for all p, 1 ≤ p < ∞; see Corollary 3.1.2 on page 162 in [10]. Thus, we need only show that we can approximate Sobolev functions which are bounded and smooth. We now fix an exponent p, 1 ≤ p ≤ 2, along with a smooth bounded function u in the corresponding Sobolev space. For each ζj , 1 ≤ j ≤ N , > 0 and δ > 0, we let φζj ,,δ denote the function φ(x − ζj ), where φ is the function satisfying the properties (i)–(iii) of Lemma 5.1. Observe that (by using the product rule for differentiation) the function v=u

N Y

φζj ,,δ

j=1

can be made as close as we wish to u in W 1,p (Ω)−norm by choosing and δ appropriately small. Also, for a sufficiently large index i, we observe that v vanishes on ∪N j=1 B(ζj , δi ) ∩ Ω. Now by assumption, the domain Ω(δi ; ζ1 , ζ2 , . . . , ζN ) is a finitely connected domain whose boundary consists of finitely many disjoint Jordan curves {Γl }. One of these boundary components, call it Γ0 , will contain Ω(δi ; ζ1 , ζ2 , . . . , ζN ) in its interior, while the others will contain this domain in their exteriors. We appeal to a result of Lewis [9], which says that for any (bounded) Jordan domain D in the plane, C ∞ (R2 ) is dense in W 1,p (D) for any p, 1 ≤ p < ∞. The proof can now be completed by a partition of unity argument similar to the ones used above. For the truncation of v to an Ω−neighborhood of Γ0 , we use the Lewis result directly to approximate on the interior of Γ0 . For the trunctation of


v to an Ω−neighborhood of Γl , l 6= 0, we again use the Lewis result, only after a preliminary (complex analytic) inversion with respect to any point in the interior of Γl . 6. Necessary Conditions In this section we establish some necessary conditions for approximating Sobolev functions. We begin by showing that it is necessary for every limit point of the set of nondegenerate components of ∂Ω to be a m2 -limit point of Ωc in order to have C ∞ (Ω) dense in W k,p (Ω). The following result concerning continua in the plane will be needed. Proposition 6.1. Let K be a compact connected subset of R2 containing at least two distinct points and let r0 be a positive number. Then at least one of the following holds: (i) There exists a ∈ K and r ∈ (0, r0 ) such that B(a, r) \ K has at least two components; (ii) K contains a nontrivial circular arc. Remark. The example K = B(0, 1) shows that (i) may not occur. Also, as the proof will show, if (ii) fails then the set of radii r for which the condition (i) holds will be a dense subset of the interval [0, (1/2)diam K]. Proof. Since the statements are invariant under translations and dilations, we may assume that 0 ∈ K and that there is a point z0 ∈ K \ B(0, 2). Consider K0 = K ∩ ∂B(0, 1). If K0 contains a nontrivial arc, then (ii) is satisfied and nothing further is required. Thus we assume that K0 has empty interior relative to the circle ∂B(0, 1). Each z ∈ K0 is contained in a disk B(z, (z)) such that (z) < min{1/2, r0 } and such that ∂B(z, (z)) is disjoint from K0 . Since K0 is compact, it is contained in finitely many such disks, {∆1 , . . . , ∆n } with the property that no ∆j is contained in ∪i6=j ∆i . If ∆j \ K has at least two components, for some 1 ≤ j ≤ n, then we are done since (i) is satisfied. Suppose to the contrary, that each of the open sets ∆j \ K is connected, for 1 ≤ j ≤ n. We show that this leads to a contradiction. Assume first that the ∆j ’s are disjoint. For each j, 1 ≤ j ≤ n, there is a simple curve γj ⊂ ∆j \ K that connects the two points of the set ∂∆j ∩ ∂B(0, 1). Let {Aj } be the n subarcs which are the components of the set ∂B(0, 1) \ ∪∆j . Each Aj is a circular arc disjoint from K. Combining the γj ’s with the Aj ’s we get a closed Jordan curve Γ disjoint from K. See Figure 4 below. Each portion γj of Γ can be continuously deformed within ∆j to the circular arc ∂B(0, 1)∩∆j and consequently Γ can be deformed to the unit circle ∂B(0, 1) within the annulus 1/2 ≤ |z| ≤ 3/2. Hence the winding number of Γ about 0 is one, while the winding number of Γ about z0 is zero. Whence 0 and z0 lie in different components of K, but this violates the connectedness of K. This contradiction proves the proposition is the disjoint case. However, the general case is just slightly more difficult and is left to the reader. This completes the proof.


17

z0 A

1

∆3

∆1 γ

γ

3

|z|=1/2

1

z=0

|z|=1

A

3

A

2

γ

2

∆2

|z|=3/2

Figure 4. Contructing the curve Γ in the proof of Proposition 6.1. Theorem C. Let Ω be a domain in R2 . Suppose that z ∈ ∂Ω, that z is not a m2 limit point of Ωc and that z is a limit point of the set of nondegenerate components of ∂Ω; then C ∞ (Ω) is not dense in W k,p (Ω), for any k ≥ 1 and any 1 ≤ p < ∞. Remark. The non degeneracy assumption on the boundary components is needed, at least when p > 2. See Example 7.5. Proof. Since W k,p −norms increase with k, it suffices to establish the theorem for k = 1. Fix a p with 1 ≤ p < ∞. Let z be a limit point of the nondegenerate components of ∂Ω and t > 0, and assume that m2 (B(z, t) \ Ω) = 0. Then there exists w ∈ B(z, t/2) belonging to a nondegenerate component of ∂Ω such that m2 (B(w, t/2) \ Ω) = 0. Applying Proposition 6.1 with K the component of B(w, t/4) \ Ω that contains z, we see that there is a disk B(a, r) with the property that B(a, r/2)∩Ω has at least two components and m2 (B(a, r)\Ω) = 0. Indeed, this is immediate if case (i) of Proposition 6.1 holds, and if case (ii) holds, then B(a, r) will have the required properties provided a is an interior point of the circular arc and r is sufficiently small. Let B(a, r/2) ∩ Ω = U1 ∪ U2 , where U1 is a component of the set. By changing coordinates if necessary, we assume that S1 = [0, s] × [0, s] ⊂ U1 ∩ B(a, r/2) and S2 = [0, s] × [b s, s + sb] ⊂ U2 ∩ B(a, r/2), where s, b s > 0 and s + sb ≤ 1. Also, put R = [0, s]×[0, s+b s] and observe that m2 (R\Ω) = 0. Let ϕ ∈ C ∞ (R2 ) be identically 1 on B(a, r/2) and zero on R2 \B(a, r), and set u = ϕ·χU1 . Then u ∈ W 1,p (Ω), and we will show that u cannot be approximated to arbitrary accuracy in this space by functions in C ∞ (R2 ).


To this end, suppose that vn ∈ C ∞ (R2 ) and lim ku − vn kW 1,p (Ω) = 0. Since u is identically 1 on S1 and 0 on S2 it follows that ZZ ZZ 1 1 |vn | dxdy − 2 |vn | dxdy = 1 . lim 2 n→∞ s s S1

S2

On the otherhand, since |∇u| = 0 on R ∩ Ω and m2 (R ∩ Ω) = m2 (R) we have that ZZ ZZ ZZ 1 1 1 |vn | dxdy − 2 |vn | dxdy ≤ 2 |vn (x, y) − vn (x, y + sb)| dxdy s2 s s S1

S2

S1

1 ≤ 2 s

ZZ Z

Z

S1

0

s b

|∇vn (x, y + τ )| dτ dxdy

Z 1 s s+bs |∇vn | dydx ≤ s 0 0 ZZ 1 = |∇(vn − u)| dxdy. s R∩Ω

By applying H¨older’s inequality, we get that the last term above is dominated by s−1/p ku − vn kW 1,p (Ω) , which tends to zero as n tends to infinity. This contradiction proves that C ∞ (Ω) is not dense in W 1,p (Ω). This completes the proof of Theorem C. The next proposition provides another useful necessary condition for approximation of Sobolev functions. The geometric condition on the domain is simpler than that in the Theorem C, however it is necessary to also assume the existence of a certain Sobolev function on the domain. The proposition is closely related to an example of Amick [3]. It is an easy consequence of the Sobolev embedding theorem (see Theorem 5.4, Part II in [1]) and the argument used to prove Lemma 2 in [3]. We omit the proof. This proposition will be used in the next section to construct examples. Proposition 6.2. Let Ω be a domain in R2 containing congruent triangles T1 , T2 ⊂ Ω with disjoint interiors and a common vertex. If p > 2 and there exists u ∈ W 1,p (Ω) such that u |T1 ≡ 1 and u |T2 ≡ −1, then C(R2 ) ∩ W 1,p (Ω) is not dense in W 1,p (Ω). 7. Examples We begin this section by constructing, for each p > 2, a domain Ω ⊂ R2 for which the functions in C ∞ (Ω) with bounded gradients are not dense in W 1,p (Ω). Of course, by Theorem A, Ω will not satisfy the interior segment condition. We ˜ ⊂ R3 which does satisfy the interior segment then use Ω to produce a domain Ω condition and for which all points in Ωc are m3 limit points of Ωc , and yet functions ˜ with bounded gradients are not dense in W 1,p (Ω). ˜ Thus any extension in C ∞ (Ω) n of Theorem A or Theorem B to R will require further geometric hypotheses on the domain.


19

Example 7.1. Fix p > 2 and fix a sequence {yj } with y0 = 1, 0 < yj+1 < yj for j ≥ 0, and lim yj = 0. Let {bj } be a sequence of positive numbers satisfying bj < yj−1 − yj ,

(7.1)

j≥1

and ∞ X

(7.2)

2(1−p)

yj

bj < ∞ .

j=1

Define Ω ⊂ R2 to be Ω = {(x, y) ∈ (−2, 2) × (0, 2) | y
2. For j ≥ 1, set Rj = [−yj2 , yj2 ] × [yj , yj + bj ] , and note that, by (7.1), the Define a function u on Ω    u(x, y) =   Clearly, u ∈ W 1,p (Ω) since

rectangles {Rj } are disjoint. by 1

if x > 0 and (x, y) ∈ Ω \ ∪Rj

−1 x/yj2

if x < 0 and (x, y) ∈ Ω \ ∪Rj if (x, y) ∈ Rj , j ≥ 1 .

P

yj2 bj yj−2p < ∞ by (7.2).


It is a consequence of Proposition 6.2 that C(R2 ) ∩ W 1,p (Ω) is not dense in W 1,p (Ω) for p > 2. However, we wish to show moreover that, for this domain, functions in C ∞ (Ω) with bounded gradients can not be used to approximate u. Suppose that v ∈ C ∞ (Ω) and |∇v| is bounded. This implies that, for all sufficiently large j, either v(yj2 , yj + bj /2) < 1/4 or v(−yj2 , yj + bj /2) > −1/4. So fix a j for which, without loss of generality, v(yj2 , yj + bj /2) < 1/4. There is no essential difference in the remaining argument in the case that v(−yj2 , yj + bj /2) > −1/4. We introduce polar coordinates with the origin at (yj2 , yj + bj /2) and such that the sector S = {[r, θ] | 0 ≤ r ≤ 1, 0 ≤ θ ≤ π/4} ⊂ Ω. Now, since u is identically 1 on S, by taking sufficiently small, we can assure that if ku − vkLp (Ω) < , then sup{v[r, θ] | 0 ≤ r ≤ 1} > 3/4 for θ ∈ E ⊂ [0, π/4], where |E| > π/8. But then, for θ ∈ E, Z 1 |∇v[r, θ]| dr 1/2 < 0

Z ≤

0

1

|∇v[r, θ]| r dr p

1/p

Z 0

1

1 r 1/(p−1)

dr

(p−1)/p

.

R Thus, since we have assumed that p > 2, if ku − vkLp (Ω) < , then S |∇v|p is bounded below by a positive constant depending only on p. Since ∇u is identically zero on S, this shows that (7.3)

ku − vkW 1,p (Ω) ≥ C0 > 0 ,

if v ∈ C ∞ (Ω), |∇v| is bounded, and p > 2. Remark. We remark that the argument used to establish (7.3) could also be used to give a prove of Proposition 6.2 which does not use the Sobolev Embedding Theorem. Finally, we point out that it is an immediate consequence of Theorem 8, using δ-modifications at the origin, that C ∞ (R2 ) is dense in W 1,p (Ω) for 1 ≤ p ≤ 2. Using Example 7.1, it is easy to produce domains in Rn , n ≥ 3, for which Theorem A and Theorem B both fail. Example 7.2. Now, again for p > 2 fixed, let Ω be the domain constructed above, ˜ c is an m3 -limit point of Ω ˜c ˜ = (0, 1) × Ω ⊂ R3 . Then each point of Ω and define Ω ˜ is a basic interior segment domain (as in Section 2). Let u and Ω ˜(x, y, z) = u(y, z), ˜ and suppose ˜ (x, y, z) ∈ Ω, where u is the function defined above. Let v˜ ∈ C ∞ (Ω) that |∇˜ v| is bounded. Then Z 1 p ku − v˜(·, t)kpW 1,p(Ω) dt ≥ CC0p > 0 , k˜ u − v˜kW 1,p (Ω) ˜ ≥ C 0

˜ of functions in C ∞ (Ω) ˜ with bounded by (7.3). Thus u ˜ is not the limit in W 1,p (Ω) ∞ gradients and hence the global C functions are not dense. Remark. The construction of the domain in Example 7.2 may be modified to produce a domain that is starshaped instead of satisfying the interior segment condition. Thus there is a starshaped domain Ω ∈ R3 such that each point of Ωc is an m3 -limit point of Ωc , and yet Cb∞ (Ω) is not dense in W 1,p (Ω) for p > 2.


21

We next give an example of a planar domain that is bounded, simply connected, the complement of the closure is connected, each point of the complement is an m2 -limit point of the complement, and for which C(R2 ) ∩ W 1,p (Ω) is not dense in W 1,p (Ω) for p > 2. Example 7.3. Let E ⊂ [−1, 1] be a Cantor set of positive length containing the points −1 and 1; that is, E is a perfect set which is nowhere dense. We further require that every point of E be an m1 -limit point of E. Define Ω to be the domain [

Ω = [−2, 2] × [0, 2] \

s[(0, 0), (e, 1)] ,

e∈E

where s[a,b] denotes the closed line segment in R2 from a to b. It is clear that Ω satisfies all the requirements stated for this example. Let u ∈ Cb∞ (Ω) satisfy, for (x, y) ∈ Ω,    u(x, y) =

 

1 −1 0

if x > y and x2 + y 2 < 1/2 if x < −y and x2 + y 2 < 1/2 if − y < x < y or x2 + y 2 > 1 .

Clearly u ∈ W 1,p (Ω), for all p < ∞. Proposition 6.2 now asserts that C(R2 ) ∩ W 1,p (Ω) is not dense in W 1,p (Ω) for p > 2. We point out, however, that if 1 ≤ p ≤ 2, then C ∞ (Ω) is dense in W 1,p (Ω). This can be seen by performing a δ-modification on Ω at the origin, and using the function φ from Lemma 5.1 as was done in the proof of Theorem 8. A polar coordinate-type mapping takes the resulting domain onto a domain satisfying the assumptions of Theorem 5, and so, as in the proof of Theorem 8, C ∞ (Ω) is dense in W 1,p (Ω) for 1 ≤ p ≤ 2. Example 7.4. In this example, Ω satisfies the geometric and density conditions of the previous example but has infinitely many two sided boundary points. Moreover, C ∞ (Ω) is not dense in W k,p (Ω) for any p. It is possible to construct Cantor sets Kn in (−2, 0) for each n = 1, 2, . . . so that the set E = ∪Kn satisfies: 0 < |I ∩ E| < 1 Let f =

P

for all intervals I ⊂ (−2, 0) .

2−n χKn , so that f is an upper semicontinuous function and Ω = (−2, 2) × (−1, 1) \

[

s[(x, −f (x)), (x, f (x))]

x∈(−2,0)

is an open set satisfying the above conditions. If u ∈ W k,p (Ω) and u(x, y) is equal to 1 for (x, y) ∈ Ω with y > 0 and x < x0 < 0 and u(x, y) = −1 for (x, y) ∈ Ω with y < 0 and x < x0 < 0, then u can not be approximated by a C ∞ (Ω) function. (See [8] for a similar example.)


u=1

y=0

u =-1

x=-2

x0

x=0

x=2

Figure 6. C ∞ (Ω) is not dense in W k,p (Ω) for any p. The domain Ω in the next example also satisfies the geometric condition from Proposition 6.2, but now C ∞ (R2 ) will fail to be dense in W 1,p (Ω) for 1 ≤ p < 2, in contrast to Example 7.3. Example 7.5. Using a function similar to the argument function it is possible to 2 construct a function √ f on the open equilateral triangle T ⊂ R with vertices (0, 0), (1, 0) and (1/2, 3/2) such that: (i) R0 ≤ f ≤ 1; (ii) T |∇f |p dxdy = C(p) < ∞, 1 ≤ p < 2; (iii) f extends to be smooth on T \ {(0, 0), (1, 0)}, with f (x, 0) = 0, 0 < x < 1, and f (x, y) = 1 for (x, y) ∈ ∂T and y > 0. Fix a p with 1 ≤ p < 2. Let I0 be the interval (−2, −1) and I1 the interval (1, 2), and let E ⊂ [−1, 1] be a Cantor set with |E| > 0 and such that the complementary intervals, [−1, 1] \ E = ∪∞ j=2 Ij , satisfy (7.4)

∞ X

|Ij |2−p < ∞.

j=2

Denote by Tj the open equilateral triangle in the upper-half plane with one edge Ij . Since Tj can be obtained from T by a translation and a dilation, these operations can be used with the f described above to define a function fj on Tj such R function p that 0 ≤ fj ≤ 1, Tj |∇fj | dx = C(p)|Ij |2−p and the natural analog of (iii) holds


23

for fj on Tj as well.

u=1 T0 y=0

T1

T2 u=f2 I2

u=f0 I0

u=f1 I1

u =-1

x=-2

x=2

x=0

Figure 7. C ∞ (Ω) is not dense in W k,p (Ω) for 1 ≤ p < 2. Let Ω ⊂ R2 be the domain (−2, 2) × (−1, 1) \ [E × {0}]. Observe that Theorem C does not apply to Ω since all but one of the boundary components of Ω are degenerate. Nevertheless, C ∞ (Ω) fails to be dense in W 1,p (Ω) for 1 ≤ p < 2. To see this, let u ∈ C(Ω) be such that  if (x, y) ∈ Tj , j ≥ 0   fj (x, y) 1 if (x, y) ∈ Ω \ ∪∞ u(x, y) = j=0 Tj and y > 0   −u(x, −y) if (x, y) ∈ Ω and y < 0 . Then

Z |∇u| dxdy = 2 p

Ω

=2

∞ Z X j=0 ∞ X

|∇fj |p dxdy

Tj

C(p)|Ij |2−p

j=0

< ∞, by (7.4). Thus, using the absolute continuity of f on horizontal and vertical lines from (iii), by Theorem 1.1.3.2 in [10] and the inequality |u| ≤ 1, we have that u ∈ W 1,p (Ω). R Let v ∈ C ∞ (Ω). An easy argument shows that if Ω |u − v|p dxdy is sufficiently small, then Z 1 |∇v(x, y)| dy ≥ 1, x ∈ F, −1


where F ⊂ E satisfies |F | ≥ |E|/2. Thus by H¨older’s inequality, Z

Z |∇(u − v)| dxdy =

|∇v|p dxdy

p

(E×[−1,1])∩Ω

E×[−1,1]

≥

|E| 1 , 2 2p−1

R if Ω |u − v|p dxdy is sufficiently small. Hence C ∞ (Ω) is not dense in W 1,p (Ω) for 1 ≤ p < 2. In case p > 2, we can appeal to the Sobolev embedding theorem to show that C ∞ (Ω) is dense in W 1,p (Ω). Indeed, by the Sobolev embedding theorem (see Theorem 5.4, Part II on page 98 in [1]), the restriction of any function u ∈ W 1,p (Ω)∩C ∞ (Ω) to either of the rectangles Ω∩{y > 0} or Ω∩{(x, y) | y < 0} must extend to be a continuous function on the corresponding closure. Since Ω ∩ {y = 0} is dense in [−2, 2] × {0}, it follows that u extends to be uniformly continuous on the rectangle Ω. To see that this extension of u is in W 1,p ((−2, 2) × (−1, 1)), we again appeal to Section 1.1.3 in [10]. The original function u must have been absolutely continuous on almost every vertical line inside Ω ∩ {y > 0} or Ω ∩ {y < 0}, and so the continuity of the extended function shows that the same is true for this function on almost every vertical line inside (−2, 2) × (−1, 1). Hence the extension of u is in W 1,p ((−2, 2) × (−1, 1)), and the desired density result now follows from the corresponding property for rectangles which, for example, trivially satisfy the segment condition.


25

References [1] [2] [3] [4] [5] [6] [7]

[8] [9] [10] [11] [12] [13]

R.A. Adams, Sobolev Spaces, Academic Press, New York, NY, (1975). S. Agmon, Lectures on Elliptic Boundary Value Problems, Van Nostrand-Reinhold, Princeton, NJ, (1965). Amick, C. J., Approximation by smooth functions in Sobolev spaces, Bull. London Math. Soc. 11 (1979), 37–40. Gehring, F. W., Uniform domains and the ubiquitous quasidisk, Jber. d. Dt. Math.Verein. 89 (1987), 88–103. Gilbarg, D. and Trudinger, N.S., Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin Heidelberg, (1983). Jones, P. W., Quasiconformal Mappings and Extendability of Functions in Sobolev Spaces, Acta Math. 147 (1981), 71 – 88. Jones, P. W., Approximation by smooth functions in Sobolev spaces, in Linear and Complex analysis Problem Book, Berlin, Heidelberg, 438, Springer-Verlag Lecture Notes in Mathematics #1043 (1984). Kolsrud, T., Approximation by smooth functions in Sobolev spaces, a counterexample, Bull. London Math. Soc. 13 (1981), 167–169. Lewis, J. L., Approximation of Sobolev functions in Jordan domains, Arkiv f¨ or Mat. 25 (1987), 255–264. V.G. Maz’ja, Sobolev Spaces, Springer-Verlag, Berlin Heidelberg, (1985). Meyers, N. and Serrin, J., H = W , Proc. Nat. Acad. Sci. 51, 1055–1056. V¨ ais¨ al¨ a, J., Uniform domains, Tˆ ohoku Math. J. 40, 101–118. W. P. Ziemer, Weakly Differentiable Functions (1989), Springer-Verlag GTM#120, New York.

Dated: 5/31/1994 Department of Mathematics, University of Hawaii, Honolulu, Hawaii 96822 E-mail address: [email protected], [email protected], [email protected]