Basic Properties of Sobolev Spaces
1.5 More on Extension of Functions in Sobolev Spaces
1.5.3 Extension with Zero Boundary Data
Let G and Ω be bounded domains in Rn, Ω ∈ EVpl. Let ˚Vpl(G) denote the completion of D(G) with respect to the norm in Vpl(G). If ¯Ω ⊂ G then, multiplying the operator E : Vpl(G)→Vpl(Rn) by a truncating functionη ∈ D(G), η = 1 on Ω, we obviously obtain a linear continuous operator ˚E : Vpl(Ω) → ˚Vpl(G). If Ω ⊂ G and the boundaries ∂G, ∂Ω have a nonempty intersection, then proving the existence of ˚E becomes a nontrivial problem.
Making no attempt at a detailed study, we shall illustrate possibilities arising here by an example borrowed from the paper by Havin and the author [568].
In that paper, the above-formulated problem appeared in connection with certain problems of approximation in the mean by analytic functions.
LetΩandGbe plane domains such thatΩ∈EVp1andΩ⊂G, and let the origin be the only common point of intersection of the disk BR={eiθ : 0≤ < R} withG\Ω¯ (see Fig.9). IfR is sufficiently small, thenBR∩(G\Ω) is¯ the union of two disjoint domainsω1andω2. We assume that the intersection of any circle |z| = , ∈ (0, R), with each domain ωj(j = 1,2) is a single arc. Let this arc be given by the equation z = eiθ with θ ∈ (αj(), βj()), whereαjandβjare functions satisfying a Lipschitz condition on [0, R], and let eiαj()∈∂Ω,eiβj()∈∂G. Further, let∂j() =βj()−αj(),lj() =δj().
Fig. 9.
Theorem. The following properties are equivalent.
(1)The function u∈L1p(Ω)can be extended to a function in ˚Vp1(G).
(2) R
0
|u(eiαj())|p
[lj()]p−1 d <∞. (1.5.6) (Here u(eiαj()) is the boundary value of u at the point eiαj()∈∂Ω. This boundary value exists almost everywhere on∂Ω.)
Proof. SinceΩ∈EVp1, then to prove that (2) implies (1) we may assume thatuhas already been extended to a function inVp1(B), whereB is a disk containing ¯G.
Let η satisfy a Lipschitz condition on the exterior of |z| =R, η = 0 on R2\G,η= 1 onΩandη(eiθ) = 1−(θ−αj())/δj() foreiθ∈ωj,j= 1,2.
Clearly, forθ∈(αj(), βj()), u
eiθ
−u
eiαj()≤ βj()
αj()
∂u
∂θ
eiθdθ
≤ βj()
αj()
(∇u)
eiθdθ
≤
δj()(p−1)/p βj() αj()
(∇u)
eiθpdθ 1/p
.(1.5.7) Thus
ωj
|u(eiθ)|p
[lj()]p ddθ≤c
∇upLp(ωj)+ R
0
|u(eiαj())|p [lj()]p−1 d
.
We can easily deduce that the preceding inequality impliesuη∈˚Vp1(G).
Ifu∈˚Vp1(G) then by (1.5.7) R
0
|u(eiαj())|p
[lj()]p−1 d≤ ∇upLp(ωj).
Sincelj()≤2π, it follows that the condition (1.5.6) withp≥2 cannot be valid for all u ∈ Vp1(Ω) and hence the operator ˚E does not exist. The same holds for 1 ≤ p < 2 provided lj() = O(1+ε), ε > 0. In fact, the functionu∈Vp1(Ω), defined near 0 by the equalityu(eiθ) =1+δ−2/p with 0< δ < ε(p−1)/pdoes not satisfy (1.5.6).
Now let 1 ≤ p < 2 and lj() ≥ c , c > 0. Using an estimate similar to (1.5.7), we arrive at
R 0
u
eiαj()p d p−1 ≤c
∇upLp(ωj)+−1up
Lp(ωj)
,
which together with Hardy’s inequality (1.5.2) shows that (1.5.6) is valid for all u ∈ Vp1(Ω). Consequently, for p∈ [1,2) and lj() ≥ c the operator ˚E
exists.
1.5.4 Comments to Sect. 1.5
P. Jones [404] introduced a class of so-called (ε, δ) domains and showed that these domains belong toEVpl forp∈[1,∞] andl= 1,2, . . .. Forε∈(0,∞), δ∈(0,∞],Ω⊂Rn is an (ε, δ) domain if any pointsx, y∈Ωwith|x−y|< δ can be joined by a rectifiable arc γ⊂Ωsuch that
(γ)≤ |x−y|/ε, dist(z, ∂Ω)≥ε|x−z||y−z|/|x−y|,
where (γ) is the length of γ and z ∈ γ an arbitrary point. It should be noted that Jones’ bounded extension operator Vpl(Ω)→Vpl(Rn) depends on l (while that of Stein does not). Any domain in C0,1 with compact closure is an (ε, δ) domain for some ε, δ and for n= 2 the class of simply connected (ε, δ) domains coincides with the class of quasidisks [404]. It is interesting to observe that multidimensional domains with isolated inward cusps satisfy Jones’ theorem and hence lie in EVpl for all p ≥ 1 and l = 1,2, . . . (cf.
Example 1.5.1/3). Fain [267] and Shvartsman [698] extended Jones’ theorem to anisotropic Sobolev spaces, and Chua [188] extended the result of Jones to weighted Sobolev spaces. We also mention here the paper by Rogers [680], where a bounded extension operator Vpl(Ω) → Vpl(Rn), independent of l, p was constructed for an (ε, δ) domain.
Romanov [681] showed that the role of the critical value 2 in Theorem1.5.2 is related to the particular domain dealt with in this theorem. He constructed a planar fractal domain Ω such that Ω ∈ EVp1 for p∈[1, q) andΩ /∈ EVp1
forp≥q. In [803], S. Yang constructed an example of a homeomorphism of Rn such that the image of Rn+ is inEVp1 for allp >1, but does not satisfy P.
Jones’ condition [404]. In particular, it is not a quasidisk ifn= 2.
P. Shvartsman [700] described a class ofEWpl domainsΩ⊂Rn whenever p > n. Suppose that there exist constants C and θ such that the following condition is satisfied: for everyx, y∈Ω such that |x−y| ≤θ, there exists a rectifiable curveγ⊂Ωjoining xtoy such that
γ
dist(z, ∂Ω)1−np−1 ds(z)≤C|x−y|p−np−1. (1.5.8) ThenΩ is anEWqldomain for everyl≥1 and everyq≥p.
Forl= 1 andq > pthis result was proved by Koskela [454].
Buckley and Koskela [148] showed that if a finitely connected bounded do- mainΩ⊂R2 is aEWp1 domain for somep >2, then there exists a constant C > 0 such that for every x, y ∈ Ω there exists a rectifiable curve γ ⊂Ω satisfying inequality (1.5.8) (withn= 2). Combining this result with Shvarts- man’s theorem [700] one obtains the following fact. Let 2< p <∞and letΩ be a finitely connected bounded planar domain. Then Ω is anEWp1 domain if and only if for everyx, y∈Ωthere exists a rectifiable curveγ⊂Ωjoining xtoy such that
γ
dist(z, ∂Ω)1−p1 ds(z)≤C|x−y|pp−1−2, (1.5.9) with some C >0.
Here we mention some results by Poborchi and the author [575] on the extension of Sobolev functions from cusp domains. A typical domain with the vertex of an outward cusp on the boundary is
Ω=
x= (y, z)∈Rn:z∈(0,1), |y|< ϕ(z)
, n≥2,
where ϕ is an increasing Lipschitz continuous function on [0,1] such that ϕ(0) = limz→0ϕ(z) = 0. Cusp domains are generally not in EVpl, but it is possible to extend the elements ofVpl(Ω) to some weighted Sobolev space on Rn.
Let σ be a bounded nonnegative measurable function on Rn, which is separated away from zero on the exterior of any ball centered at the origin.
By Vp,σl (Rn) we mean the weighted Sobolev space with norm uVp,σl (Rn)=
l
k=0
σ∇kuLp(Rn).
Clearly this weighted space is wider than Vpl(Rn). The following assertion gives precise conditions on the weightσ.
LetΩ have an outward cusp as above. In order that there exist a linear continuous extension operator Vpl(Ω)→ Vp,σl (Rn) it is sufficient and ifσ(x) depends only on|x|and is nondecreasing in the vicinity of the origin, then it is also necessary that
σ(x)≤
c(ϕ(|x|)/|x|))min{l,(n−1)/p} iflp=n−1, c(ϕ(||xx||))l|log(ϕ(||xx||))|(1−p)/p iflp=n−1,
in a neighborhood of the origin, where c is a positive constant independent ofx.
Let p ∈ (1,∞) and l = 1,2, . . .. We now give sharp conditions on the exponent q∈[1, p) such that there is a linear continuous extension operator Vpl(Ω)→Vql(Rn) for the same Ω. This extension operator exists if and only
if 1
0
tβ ϕ(t)
n/(β−1)
dt t <∞, where
1/q−1/p=l(β−1)/
β(n−1) + 1
forlq < n−1 and
1/q−1/p= (n−1)(β−1)/np forlq > n−1.
In the case lq = n−1 the factor |log(ϕ(t)/t)|γ should be included into the integrand above with
γ= (1−1/q)/(1/p−1/q), β= (np−q)/
q(n−1) .
Example. Power cusp.Letϕ(z) =c zλ,λ >1. A linear continuous exten- sion operator Vpl(Ω)→Vql(Rn) forq < pexists in the following cases.
1◦ lq < n−1 and
q−1> p−1+l(λ−1)/
1 +λ(n−1) . 2◦ lq=n−1 and either the same inequality as in 1◦ holds or
q−1=p−1+l(λ−1)/
1 +λ(n−1)
with 2q−1<1 +p−1. 3◦ lq > n−1 andq−1> p−1(1 + (λ−1)(n−1)/n).
Various results on extensions of functions inWpl(Ω), with deterioration of the class can be found in Burenkov’s survey [156]. Finally we note that prop- erties of extension domains for functions with bounded variation are discussed in Chap. 9 of the present book.