Removable Sets for Sobolev Functions

Basic Properties of Sobolev Spaces

1.1.18 Removable Sets for Sobolev Functions

If such an operator exists for a domain then, by definition, this domain belongs tothe class EV_p^l.

Thus, domains with smooth boundaries are contained inEV_p^l. The bounded domains of the class C^0,1 turn out to have the same property. The last as- sertion is proved in Stein [724], §3, Ch.VI (cf. also Comments to the present section).

In Sect. 1.5 we shall return to the problem of description of domains in EV_p^l.

1.1 The SpacesLp(Ω),Vp(Ω) andWp(Ω) 29 denote the projection of a pointx∈Rⁿ on the coordinate hyperplane orthog- onal to thexi axis. By assumptions, each set p_i(F) has (n−1)-dimensional Lebesgue measure zero. Thus, almost every straight line that is parallel to the x_i axis is disjointed fromF. According to Fubini’s theorem, we have

Ω

η ∂u

∂x_i dx=

Ω

dx

(x)

η∂u

∂x_idx_i, 1≤i≤n,

whereΩ =pi(Ω), x =pi(x) and (x) is the intersection ofΩ with the line x = const. Note that(x) is in Ω\F for almost allx∈Ω. An application of Theorem 1.1.3/2 leads to

Ω

dx

(x)

η ∂u

∂xi

dxi=−

Ω

dx

(x)

u∂η

∂xi

dxi−

Ω

u∂η

∂xi

dx.

Hence (1.1.28) follows for |α|=l= 1. The general case can be concluded by induction onl. Indeed, letl≥2 and let (1.1.28) hold foru∈V_p^l−1(Ω\F) with

|α| ≤l−1. Ifu∈V_p^l(Ω\F), |α|=l, andD^α =D_iD^β for some 1≤i≤n, the left-hand side of (1.1.28) equals

−

Ω

∂u

∂xi

D^βηdx, (1.1.29)

sinceu∈V_p¹(Ω\F). By the induction hypothesis, expression (1.1.29) coincides with the right-hand side of (1.1.28) (because ∂u/∂xi ∈ V_p^l−1(Ω\F)). This

completes the proof.

Let us show that the conditionHn−1(F) = 0 in the above theorem cannot be replaced by the finiteness ofHn−1(F).

Example.LetΩ={x∈R²:|x|<2} and letF be the segment{x∈Ω: x₂ = 0,|x₁| ≤1}. We introduce the set S ={x∈Ω :|x₁| ≤ 1, x₂ >0} and define the functionuonΩby

u(x) =

0 onΩ\S,

exp(−(1−x²₁)⁻¹) onS.

We haveH₁(F) = 2, u∈V_p^l(Ω\S) for anyl, butu /∈V_p^l(Ω).

1.1.19 Comments to Sect.1.1

The space W_p^l(Ω) was introduced and studied in detail by Sobolev [711–

713]. (Note that as early as in 1935 he also developed a theory of distribu- tions in (C₀^l)^∗ [710].) The definitions of the spaces L^l_p(Ω) and ˙L^l_p(Ω) are borrowed from the paper by Deny and Lions [234]. The proofs of Theo- rems 1.1.2, 1.1.5/1, 1.1.12, 1.1.13/1, and 1.1.13/2 follow the arguments of

this paper where similar results are obtained forl = 1. Theorem1.1.5/1 was also proved by Meyers and Serrin [599]. Concerning the contents of Sect.1.1.3 we note that the mollification operator Mε was used by Leray in 1934 [487]

and independently by Sobolev in 1935 [709]. A detailed exposition of proper- ties of mollifiers including those with a variable radius can be found in Chap. 2 of Burenkov’s book [155]. For a history of the mollifiers see Naumann [624].

The property of absolute continuity on almost all straight lines parallel to coordinate axes served as a foundation for the definition of spaces similar to L¹_p in papers by Levi [489], Nikod´ym [637], Morrey [612], and others. The example of a domain for whichW₂²(Ω)=V₂²(Ω) (see Sect.1.1.4) is due to the author. The example considered at the beginning of Sect. 1.1.6is borrowed from the paper by Gagliardo [299]. Remark1.1.6and the subsequent example are taken from the paper by Kolsrud [440]. Theorem 1.1.6/1 is given in the textbook by Smirnov [705] and Theorem 1.1.6/2 was proved in the above- mentioned paper by Gagliardo. The topic of Sect.1.1.6was also discussed by Fraenkel [285] and Amick [46]. The following deep approximation result was obtained by Hedberg (see [370]).

Theorem. Let Ω be an arbitrary open set in Rⁿ and let f ∈W˚_p^l(Ω) for some positive integer l and somep,1< p <∞. Then there exists a sequence of functions{ων}ν≥1,0≤ων ≤1, such that suppων is a compact subset ofΩ, ωνf ∈(L_∞∩W˚_p^l)(Ω), and

νlim→∞f−ωνfW_p^l(Ω)= 0.

An ingenious approximation construction for functions in a two-dimension- al bounded Jordan domain was proposed by Lewis in 1987 [492] (see also [491]). He proved that C^∞( ¯Ω) is dense in W_p¹(Ω) for 1< p < ∞. Unfortu- nately, this construction does not work for higher dimensions, for p= 1, and for the space W_p^l(Ω) with l > 1. If a planar domain is not Jordan, it may happen that forl >1 bounded functions in the spaceL^l_p(Ω) are not dense in L^l_p(Ω) (Maz’ya and Netrusov [572], see Sect.1.7in the sequel).

The problem of the approximation of Sobolev functions on planar domains byC^∞ functions, which together with all derivatives are bounded onΩ, was treated by Smith, Stanoyevitch, and Stegenga in [707], where some interesting counterexamples are given as well.

In [346] Hajlasz and Mal´y studied the approximation of mappingsu:Ω→ R^mfrom the Sobolev space [W_p¹(Ω)]^mby a sequence{uν}ν≥1of more regular mappings in the sense of convergence

Ω

f(x, u_ν,∇u_ν) dx→

Ω

f(x, u,∇u) dx for a large class of nonlinear integrands.

During the last two decades a theory of variable exponent Sobolev spaces W^1,p(·)(Ω) was developed, where the Lebesgue L^p-space is replaced by the

1.1 The SpacesLp(Ω),Vp(Ω) andWp(Ω) 31 Lebesgue space with variable exponent p(x). This topic in not touched in the present book, but we refer to Kov´acˇik and R´akoˇsnik [461] where such spaces first appeared and to the surveying papers by Diening, H¨ast¨o and Nekvinda [236], Kokilashvili and Samko [439] and S. Samko [690]. In partic- ular, the density of C₀^∞(Rⁿ) in W^1,p(·)(Rⁿ) was proved by S. Samko [688]

under the so-called log-condition onp(x), standard for the variable exponent analysis:

p(x)−p(y)≤ const

|log|x−y||

for small |x−y|. This question is nontrivial because of impossibility to use mollifiers directly. The Hardy type inequality in variable Lebesgue spaces are studied by Rafeiro and Samko [669].

In connection with Sect. 1.1.7 see the books by Morrey [613], Reshet- nyak [677], and the article by the author and Shaposhnikova [578], see also Sect. 9.4 in the book [588].

The condition of being starshaped with respect to a ball and having the cone property were introduced into the theory ofW_p^l spaces by Sobolev [711–

713]. Lemma 1.1.9/2 was proved by Glushko [312]. The example given in Sect.1.1.9is due to the author; another example of a Lipschitz domain that does not belong toC^0,1can be found in the book by Morrey [613]. Properties of various classes of domains appearing in the theory of Sobolev spaces were investigated by Fraenkel in [285]. Fraenkel’s paper [286] contains a thorough study of the conditions on domains Ω guaranteeing the embedding of the spaceC¹( ¯Ω) inC^(0,α)( ¯Ω) whenα >0.

Integral representations (1.1.8) and (1.1.10) were obtained by Sobolev [712, 713] and used in his proof of embedding theorems. Various generalizations of such representations are due to Il’in [396, 397], Smith [706] (see also the book by Besov, Il’in, and Nikolsky [94]), and to Reshetnyak [675]. We follow Burenkov [154] in the proof of Theorem 1.1.10/1.

The Poincar´e inequality for bounded domains that are the unions of do- mains of the class C was proved by Courant and Hilbert [216]. Properties of functions in L^l_p(Ω) for a wider class of domains were studied by J.L. Li- ons [499].

Stanoyevitch showed [720] (see [721] for the proof) that the best constant in the one-dimensional Poincar´e inequality

u−u¯L_p(−1,1)≤C∇uL_p(−1,1)

is equal to 1 ifp= 1 and

(p)^1/pp^1/p

Γ(1/p)Γ(1/p) = psin(π/p) π(p−1)^1/p,

if 1 < p < ∞. Moreover, a unique extremal function exists if and only if 1< p≤ ∞.

Theorem1.1.16on equivalent norms inW_p^l(Ω) is due to Sobolev [713, 714].

The extension procedure described at the beginning of Sect.1.1.17was pro- posed by Hestenes [378] for the spaceC^k( ¯Ω) (see also Lichtenstein [494]). The same method was used by Nikolsky [638] and Babich [59] forW_p^l(Ω). The fact that a space-preserving extension for functions in W_p^l(Ω) (1< p <∞) onto Rⁿ is possible for domains of the classC^0,1was discovered by Calder´on [163].

His proof is based on the integral representation (1.1.8) along with the theo- rem on the continuity of the singular integral operator in Lp. A method that is appropriate forp= 1 andp=∞was given by Stein [724]. The main part of his proof was based on the extension of functions defined in a neighborhood of a boundary point. Then, using a partition of unity, he constructed a global extension. For the simple domain

Ω=

x= (x, xn) :x∈Rⁿ⁻¹, xn> f(x) ,

where f is a function onRⁿ⁻¹satisfying a Lipschitz condition, the extension ofuis defined by

u^∗(x, xn) = _∞

1

u

x, xn+λδ(x, xn)

ψ(λ) dλ, xn < f(x).

Here δ is an infinitely differentiable function, equivalent to the distance to

∂Ω. The functionψis defined and continuous on [1,∞), decreases asλ→ ∞ more rapidly than any power of λ⁻¹, and satisfies the conditions

_∞

1

ψ(λ) dλ= 1,

_∞

1

λ^kψ(λ) dλ= 0, k= 1,2, . . . .

More information on extension operators acting on Sobolev spaces can be found in Sect.1.5.

Theorem1.1.18on removable sets for Sobolev functions is due to V¨ais¨al¨a [771] (see also Reshetnyak [677, Chap. 1, 1.3]). Koskela showed thatW_p¹ re- movability of sets lying in a hyperplane depends on their thickness measured in terms of a so-called pporosity [455].