International Journal of Analysis and Applications

Volume 19, Number 4 (2021), 503-511

URL: https://doi.org/10.28924/2291-8639

DOI: 10.28924/2291-8639-19-2021-503

TRACE RESULT FOR SOBOLEV EXTENSION DOMAINS

DJAMEL AIT-AKLI∗, ABDELKADER MERAKEB

L2CSP, Mouloud Mammeri university of Tizi-Ouzou, 15000, Tizi-Ouzou, Algeria

∗Corresponding author: djamel.aitakli@ummto.dz

Abstract. In this paper, we establish the existence and continuity of a trace operator for functions of the

Sobolev space W 1,p(Ω) with 1 < p < ∞ on the boundary of a domain Ω that has the Sobolev W 1,p−extension

property. First, we prove the existence and the continuity of such an operator when it is applied to the

elements of the subspace of the up to boundary smooth functions by using a uniform estimate. The essential

ingredients used in the proof of this estimate are Green’s representation of a function on a disk as well as

Banach’s isomorphism theorem. Finally, we conclude the trace result using the density of smooth functions

in W 1,p(Ω). The presented proof fully exploits the extensibility hypothesis of the domain Ω. The relevance

of the result lies in the existence of extension domains which are not Lipschitz and under this point of view

it constitutes a generalization of the usual trace theorem.

1. Introduction

The trace operator, when applied to the functions of the Sobolev space W 1,p(Ω) defined on a Lipschitz

domain Ω, is a standard notion in the theory of Sobolev spaces. More precisely, it is well established that

the trace operator is well defined on the boundary of a Lipschitz domain and, moreover, it is continuous,

cf. [1]. The result established in this paper generalizes this fact to the case of a non-Lipschitz domain, namely

to W 1,p(Ω)-extension domains. The regularity of this class of domains is intermediate between Lipschitz

regularity and Jordan regularity.

Received March 7th, 2021; accepted April 12th, 2021; published 0, 2021.

2010 Mathematics Subject Classification. 46E35, 35J57, 26A16, 46B25.

Key words and phrases. Sobolev spaces; W 1,p−extension domains; Green representation; trace inequality; density of smooth

functions.

©2021 Authors retain the copyrights

of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.

503

https://doi.org/10.28924/2291-8639
https://doi.org/10.28924/2291-8639-19-2021-503


Int. J. Anal. Appl. 19 (4) (2021) 504

The functions of the Sobolev space W 1,p(Ω) defined on such a domain constitute a relevant class of

functions in the following sense: there exists domains whose boundary is parameterized by a continuous

function that is not Lipschitz but admit the property of (1,p)− extension. Indeed, Maz’ya has constructed,

cf. [2], an example of a Jordan domain, Ω, such that the boundary ∂Ω is not Lipschitz in the neighborhood

of at least one of its points and such that the property of (1,p)− extension is only valid for p < 2. We

emphasize that a Lipschitz domain is always an extension domain.

We give, herein, a proof of the existence and the boundedness of a trace operator defined on the Sobolev

space W 1,p(Ω) where Ω is a W 1,p−extension domain, 1 < p < ∞. The proof that we present in the framework

of extension domains fully exploits the domain’s extensibility hypothesis.

We start by introducing some preliminary notions as well as the tools and results that are essential to

carry out the proof of the main result.

2. Classical facts and tools

Let Ω ⊂ R2 be a domain. We denote by Int(Ω) the topological interior of Ω. Throughout this section,

and only otherwise explicitly stated, p is a real number with 1 < p < ∞. The usual Sobolev space W 1,p(Ω)

is defined by

W 1,p(Ω) := {u ∈ Lp(Ω), ∇u ∈ Lp(Ω)}.

We further assume that Ω has the property of W 1,p−extension. Recall that the property of W 1,p−extension

of a domain Ω means that a Sobolev extension operator can be defined on W 1,p(Ω), we recall its definition:

Definition 2.1. We say that a domain Ω ⊂ R2 is a W 1,p−extension domain, if there is a linear operator

P defined by

P : W 1,p(Ω) → W 1,p(R2)

u → Pu.

such that P is continuous i.e. there exists cext > 0 such that ∀u ∈ W 1,p(Ω) we have

(2.1) ||Pu||W1,p(R2) ≤ cext||u||W1,p(Ω)

where cext depends only on Ω and (Pu)(x) = u(x) almost everywhere in Ω.

The reader may refer to [2] for a detailed discussion of this class of domains. We recall a density result

established in ( [3], p.261). This result is valid in the case where Ω is an extension domain or more generally

a Jordan domain:

Theorem 2.1. Let Ω ⊂ R2 be a domain whose boundary is a Jordan curve, then C∞(Ω) is dense in W 1,p(Ω)

with 1 < p < ∞.



Int. J. Anal. Appl. 19 (4) (2021) 505

We recall the property of continuity of the trace operator, on the Lipschitz (or weakly Lipschitz) boundary

of a domain D ⊂ R2, defined for the Sobolev space W 1,p(D), cf. [1]:

Proposition 2.1. Let D ⊂ R2 be a Lipschitz domain. There exists CD > 0, depending merely on D and p,

such that

(2.2) ||u||
W

1− 1
p
,p

(∂Ω)
≤ CD||u||W1,p(D),

for all u ∈ W 1,p(D).

It should be noted that proposition 2.1 does not apply in principle neither to an extension domain nor to

the more general Jordan domain given the primordial assumption of the Lipschitz character of the ambient

domain D.

Let us also recall a trace result for the functions of the Sobolev space W 1,1(Dr) where Dr ⊂ R2 is a disk

of radius r > 0, cf. ( [4], estimate 7.1):

Proposition 2.2. For all u ∈ W 1,1(Dr) we have

(2.3)

∫
∂Dr

|u|dσ ≤
2

r

∫
Dr

|u|d x +
∫
Dr

|∇u|d x.

In addition, we recall the integral representation of a function u ∈ C2(Dr) ∩ C0(∂Dr) on the disk Dr,

cf. [5]. For such a function, this representation writes: for all x ∈ Int(Dr) we have

u(x) =

∫
Dr

∆u(y)G(x,y)dy +

∫
∂Dr

u(y)K(x,y)dσ(y),(2.4)

where G is the Green function of the disk, Dr, associated with the Laplace operator and defined by:

G(x,y) := Γ(x−y) + hx(y),

with Γ(x) = 1
2π

ln |x| being the fundamental solution of the Laplace operator in dimension two and hx

is the harmonic function in y that equals −Γ(x − y) for x ∈ Int(Dr) and y ∈ ∂Dr. On the other hand

K(x,y) =
∂G(x,y)
∂νy

is the Poisson kernel which in the case of the disk is written:

K(x,y) =
r2 −|x|2

2ω2r|x−y|2
,

where ω2 is the measure of the two-dimensional unit ball. This integral representation is interesting insofar

as it makes it possible to point-wisely express a function using its Laplacian and the values it takes on the

boundary of the disk Dr.



Int. J. Anal. Appl. 19 (4) (2021) 506

3. Principal Results

The main result of this paper is stated in the following theorem, its proof will be given after establishing

an auxiliary lemma.

Theorem 3.1. Let Ω be a domain having the W 1,p−extension property such that 1 < p < ∞, then we can

define a trace operator

T : W 1,p(Ω) → Lp(∂Ω)

u → Tu

that coincides with the restriction operator on the boundary ∂Ω for continuous functions. In addition, T is

continuous i.e. there exists ct > 0 independent on u such that for all u ∈ W 1,p(Ω) we have

(3.1) ||Tu||p,∂Ω ≤ ct||u||W1,p(Ω) ,

where we denote || ||p,∂Ω the usual norm of the Lebesgue space of p−integrable functions.

3.1. Auxiliary lemma. We now state with a proof an auxiliary lemma essential for establishing the main

theorem:

Lemma 3.1. Let 1 < p < 2 and fix a point x0 ∈ R2. We denote Dr0 := D(x0,r0) ⊂ R2 the disk with center

x0 and radius r0 > 0. There exists a constant c(r0,p) > 0 which is independent of both u and x such that

for all u ∈ C∞(Dr0 )

(3.2) ∀x ∈ Dr0
2
, |u(x)| ≤ c(r0,p)||u||W1,p(Dr0 ).

We denote by |x−y|2 the Euclidean distance between the points x and y.

Proof. let u ∈ C∞(Dr0 ). There exists f ∈ C∞(Dr0 ) and ud ∈ C∞(∂Dr0 ) such that the function u solves the

problem

(3.3)



−∆u(y) = f(y) in Dr0,

u(y) = ud(y) on ∂Dr0.

Let x ∈ Dr0
2

. By using the integral representation (2.4) applied to the function u at x, we have:

u(x) =

∫
Dr0

f(y)G(x,y) d y +

∫
∂Dr0

ud(y)K(x,y) d σ(y).(3.4)

Given that ∂Dr0 is C
∞−regular, then the Green’s function satisfies G(x,.) ∈ W 1,p(Dr0 ) for 1 < p < 2,

cf. ( [6], estimate 1.5). By using the continuity of the linear form associated with f ∈ W−1,p(Ω), we obtain



Int. J. Anal. Appl. 19 (4) (2021) 507

from estimate (3.4) the following

|u(x)| ≤ ||f||W−1,p(Dr0 )||G(x,.)||W1,p(Dr0 ) + maxy∈∂Dr0
|K(x,y)|

∫
∂Dr0

|ud(y)|d σ(y).

The last term of this inequality does make sense since |x − y|2 ≥ r02 for x ∈ Dr02 and y ∈ ∂Dr0 . By

applying the trace inequality on the boundary of the disc Dr0 , cf. Estimate (2.3), we get

|u(x)| ≤||f||W−1,p(Dr0 )||G(x,.)||W1,p(Dr0 )

+ max
y∈∂Dr0

|K(x,y)|
(

2

r0
||u||L1(Dr0 ) + ||∇u||L1(Dr0 )

)
,

the Holder inequality then yields

|u(x)| ≤||f||W−1,p(Dr0 )||G(x,.)||W1,p(Dr0 )

+ max
y∈∂Dr0

|K(x,y)|cr0
(
||u||p,Dr0 + ||∇u||p,Dr0

)
.

We apply Banach’s isomorphism theorem to the continuous bijective operator L1 defined by

L1 :
(
W−1,p(Dr0 ), || ||W−1,p(Dr0 )

)
→
(
W

1,p
0 (Dr0 ), || ||W1,p(Dr0 )

)
f → L(f) = u1,

with ∆u1 = f. We then deduce the existence of a constant c
′
1 independent of f such that

|u(x)| ≤c′1||u1||W1,p0 (Dr0 )||G(x,.)||W1,p(Dr0 )

+ max
y∈∂Dr0

|K(x,y)|cr0||u||W1,p(Dr0 )(3.5)

with u1 = u−u2 and u2 solves the problem

(3.6)



−∆u2 = 0 in Dr0,

u2 = u
d on ∂Dr0.

By using the Poincaré inequality, the estimate (3.5) then becomes

|u(x)| ≤c1
(
||∇u||p,Dr0 + ||∇u2||p,Dr0

)
||G(x,.)||W1,p(Dr0 )(3.7)

+ max
y∈∂Dr0

|K(x,y)|cr0||u||W1,p(Dr0 ).

Using Proposition 2.1, we can apply the isomorphism theorem to the operator associated with the problem

(3.6) to establish the existence of a constant c2 such that

||∇u2||p,Dr0 ≤ c2||u2||W1−
1
p
,p

(∂Dr0 )
.



Int. J. Anal. Appl. 19 (4) (2021) 508

where c2 is independent of u2 and ∆u2 = 0. The estimate (3.7) becomes

|u(x)| ≤c1
(
||∇u||p,Dr0 + c2||u||W1−

1
p
,p

(∂Dr0 )

)
||G(x,.)||W1,p(Dr0 )

+ max
y∈∂Dr0

|K(x,y)|cr0
(
||u||p,Dr0 + ||∇u||p,Dr0

)
.

Applying the trace theorem in W 1,p(Dr0 ), cf. Proposition 2.1, allows to infer the existence of a constant

c3 > 0 such that

|u(x)| ≤c1
(
||∇u||p,Dr0 + c3(||u||p,Dr0 + ||∇u||p,Dr0 )

)
||G(x,.)||W1,p(Dr0 )

+ max
y∈∂Dr0

|K(x,y)|cr0||u||W1,p(Dr0 ),

then we infer that there exists a constant c(r0,p) which is independent of both u and x such that

∀x ∈ Dr0
2
, |u(x)| ≤c(r0,p)||u||W1,p(Dr0 ).

This last estimate is obtained by using the fact that ||G(x,.)||W1,p(Dr0 ) is uniformly bounded for x ∈

Dr0
2

. �

We now present a proof of Theorem 3.1:

3.2. Proof of the main result.

Proof. Let u ∈ C∞(Ω). The function u is obviously Lipschitz on Ω. If we denote L the Lipschitz constant

of the function x → u(x) relatively to the domain Ω, then for all x ∈ ∂Ω and all y ∈ Int(Ω) we have

|u(x) −u(y)| ≤ L|x−y|2,

which immediately implies

(3.8) |u(x)| ≤ L|x−y|2 + |u(y)|,

for all x ∈ ∂Ω and all y ∈ Int(Ω). Let x ∈ ∂Ω be fixed and let (yδ)δ>0 be a sequence of point such that

yδ ∈ Int(Ω) and |yδ −x|2 → 0 when δ → 0. Let’s fix δ > 0. The estimate (3.8) yields

|u(x)| ≤ L|x−yδ|2 + |u(yδ)|,(3.9)

for all δ > 0. Pose ũ = P u with P being an extension operator defined on W
1,p(Ω) whose range is

W 1,p(R2) ⊂ W 1,p(Ω), cf. Definition 2.1. Such an extension operator is well defined.

It should be noted that the function ũ is not necessarily continuous over R2 − Ω i.e. it does not have a

representative continuous function. Indeed, this is due to the fact that 1 − 2
p
< 0 for 1 < p < 2; therefore

the classical Sobolev embedding into Holder spaces doesn’t apply in this case, namely when 1 < p < ∞.

Since Ω is bounded, there exists r0 > 0 and x0 ∈ Ω such that if we denote Dr0 the disk with center x0

and radius r0 then we have Ω ⊂ Dr0 . The set, C∞(D2r0 ), of smooth functions up to the boundary defined



Int. J. Anal. Appl. 19 (4) (2021) 509

on the Lipschitz domain D2r0 ⊂ R2 being dense in W 1,p(D2r0 ), cf. Theorem 2.1, therefore there exists a

sequence (vn)n, vn ∈ C∞(D2r0 ), such that

(3.10) ||vn − ũ||W1,p(D2r0 ) → 0 n →∞.

The estimate (3.10) implies that there exists a subsequence (φ(n))n such that vφ(n)(x) → ũ(x) a.e. in

D2r0 . As vn and ũ are continuous in Ω, this almost everywhere convergence is valid everywhere in Ω i.e.

(3.11) ∀y ∈ Ω, vφ(n)(y) → ũ(y) , when n →∞.

Using (3.9) and (3.11) we get: ∀δ > 0 and ∀� > 0, ∃n(�,yδ) > 0 such that:

|u(x)| ≤ L|x−yδ|2 + |vφ(n(�,yδ))(yδ)| + �,(3.12)

where � > 0 is intended to tend towards zero and (n(�,yδ))�>0 is a sequence of integers which tends to +∞

when � → 0 for all δ > 0. On the other hand, the formula (3.12) and the regularity of vφ(n(�,yδ)) allow to

write

|u(x)| ≤ L|x−yδ|2 + |vφ(n(�,yδ))|∞,Dr0 + �,

for all δ > 0. By applying the lemma 3.1 in the disk D2r0 with center x0, we have

|u(x)| ≤ L|x−yδ|2 + c(r0,p)||vφ(n(�,yδ)||W1,p(D2r0 ) + �(3.13)

where c(r0,p) is the constant appearing in the estimate (3.2). The estimate (3.13) is valid for all � > 0

independently of δ. So, passing to the limit � → 0 and using Estimate (3.10), we obtain

|u(x)| ≤ L|x−yδ|2 + c(r0,p)||ũ||W1,p(D2r0 ),(3.14)

for all δ > 0. Given that all the elements involved in the estimate (3.14) are independent of δ then by letting

δ → 0 we find

|u(x)| ≤ c(r0,p)||ũ||W1,p(D2r0 ).(3.15)

So the estimate (3.15) yields us

|u(x)| ≤ c(r0,p)||Pu||W1,p(R2).(3.16)

Using (2.1), the estimate (3.16) in turn gives

|u(x)| ≤ c(r0,p)cext||u||W1,p(Ω).(3.17)

Note that the estimate (3.17) is valid for all x ∈ Γ := ∂Ω independently of the quantities present at the

right side of this inequality.



Int. J. Anal. Appl. 19 (4) (2021) 510

Now, let us denote t ∈ [a,b] → x(t) = (t,γ(t)) the parametric function representation of the curve Γ. So,

according to (3.17), we have:

|u(t,γ(t))|p ≤ (c(r0,p)cext)p||u||
p
W1,p(Ω)

,(3.18)

for all t ∈ [a,b]. By definition of the line integral, we have∫
Γ

|u|p ds :=
∫ b
a

|u◦γ|pdsγ.(3.19)

Thus, by integrating the two sides of (3.18) with respect to the curvilinear abscissa dsγ and using (3.19),

we find ∫
Γ

|u|p ds ≤ (c(r0,p)cext)p||u||
p
W1,p(Ω)

∫ b
a

1 dsγ,

it yields ∫
∂Ω

|u|p ds ≤ (c(r0,p)cext)p||u||
p
W1,p(Ω)

|∂Ω|.(3.20)

Set ct := [(c(r0,p)cext)
p|∂Ω|]

1
p . This constant does not depend on u but merely on Ω and on the exponent

p. From (3.20), we have for all u ∈ C∞(Ω)

||u||p,∂Ω ≤ ct||u||W1,p(Ω).(3.21)

Currently, we conclude the estimate (3.1). According to Theorem 2.1, for any u ∈ W 1,p(Ω) there exists

(ul)n∈N, ul ∈ C∞(Ω), such that

||ul −u||W1,p(Ω) → 0.

Thus, by applying (3.21) to the elements of the sequence (ul)l, we have for all l ∈ N the following estimate:

||ul||p,∂Ω ≤ ct||ul||W1,p(Ω).(3.22)

Since (ul)l is a Cauchy sequence in the normed space W
1,p(Ω) then estimate (3.22) implies that it is also

a Cauchy sequence in the normed space Lp(∂Ω). But Lp(∂Ω) is complete, then there exists u∗ ∈ Lp(∂Ω)

such that

||u∗||p,∂Ω ≤ ct||u||W1,p(Ω).

Finally, by setting Tu := u∗, we have

||Tu||p,∂Ω ≤ ct||u||W1,p(Ω)



Int. J. Anal. Appl. 19 (4) (2021) 511

for all u ∈ W 1,p(Ω). This defines a continuous trace operator on the space Lp(∂Ω) on the boundary ∂Ω

T : W 1,p(Ω) → Lp(∂Ω)

u → Tu

�

Remark 3.1. The relevance of the established result lies in the fact that it is valid for functions of the space

W 1,p(Ω) which are not necessarily continuous for 1 < p < 2. The existence of the trace for this category of

functions is not obvious. The example of the domain constructed by Mazya in [2] illustrates perfectly not

only the relevance but also the generality which results from the fact that we have considered the domain Ω

having the W 1,p−extension property.

Acknowledgment. The authors would like to thank very much the anonymous referees for reviewing the

article as well as the editorial staff for their support during the submission process.

Conflicts of Interest: The author(s) declare that there are no conflicts of interest regarding the publication

of this paper.

References

[1] E. Gagliardo, Caratterizzazioni delle tracce sulla frontiera relative ad alcune classi di funzioni in n variabili, Ren. Sem.

Mat. Univ. Padova. 27 (1957), 284–305.

[2] V.G. Maz’ya, Extension of functions from Sobolev spaces, J. Math. Sci. 22 (1983), 1851–1855.

[3] J.L. Lewis, Approximation of Sobolev functions in Jordan domains, Ark. Mat. 25 (1987), 255-264.

[4] G. Auchmuty, Sharp boundary trace inequalities, Proc. R. Soc. Edinburgh Sect. A: Math. 144 (2014), 1-12.

[5] L.C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, Americam Mathematical Society, Providence,

1998.

[6] D. Mitrea, I. Mitrea, On the Regularity of Green Functions in Lipschitz Domains, Commun. Part. Differ. Equ. 36 (2010),

304–327.


	1. Introduction
	2. Classical facts and tools
	3. Principal Results
	3.1. Auxiliary lemma
	3.2. Proof of the main result
	Acknowledgment

	References