In: Integral Methods in Science and Engineering: Techniques and Applications,
C. Constanda and S. Potapenko, eds., Birkhauser-Boston, 2008, pp. 149-160
About traces, extensions and co–normal
derivative operators on Lipschitz domains
S.E. Mikhailov
Brunel University West London, UK, [email protected]
1 Introduction
For a second order partial differential equation
Lu(x) := L(x, ∂x ) u(x) :=
n
X
∂ ³
∂u(x) ´
a(x)
= f (x),
∂xi
∂xi
i=1
x ∈ Ω,
(1)
acting on a function u from the Sobolev space H s (Ω), 12 < s < 32 , the function
derivatives do not generally exist on the boundary in the trace sense but a
co-normal generalized derivative operator can be defined with the help of the
first Green identity. However this definition is related to an extension of the
PDE operator and PDE right hand side from the domain Ω, where they are
prescribed, to the domain boundary, where they are not. Since the extensions
are non-unique, the generalized co-normal derivative appears to be a nonunique operator, which is also non-linear in u unless a linear relation between
u and the PDE right hand side extension is enforced. However, for functions u
from H s (Ω), 21 < s < 32 , that are mapped by the PDE operator into the space
e t (Ω), t ≥ − 1 , one can define a canonical co-normal derivative operator,
H
2
which is unique, linear in u and coincides with the co-normal derivative in the
trace sense if the latter does exist. These notions were developed in [Mik05,
Mik06] for a PDE with an infinitely smooth coefficient on a domain with an
infinitely smooth boundary, and a right hand side from H s−2 (Ω), 1 ≤ s < 32 ,
e t (Ω), t ≥ −1/2.
or extendable to H
In Section 3 of this paper, we generalized the above analysis to the conormal derivative operators on Lipschitz domains for PDE with a Hölder
e s−2 (Ω), 1 < s < 3 . This needed a
coefficient and right hand side from H
2
2
number of auxiliary facts provided in Section 2, some of which might be new
for Lipschitz domains. Particularly, we proved Lemma 1 on unboundedness
of the trace operator, Lemma 2 on boundedness of extension operators from
boundary to the domain, Theorem 1 on characterization of the Sobolev space
2
S.E. Mikhailov
e s (Ω) on the (larger than usual) interval 1 < s < 3 , Lemma
H0s (Ω) = H
2
2
t
4 on characterization of the space H∂Ω
, t > − 32 ; Lemma 5 on existence of
e s : H s (Ω) → H
e s (Ω), − 3 < s < 1 ,
a bounded linear extension operator E
2
2
1
s 6= − 2 .
An analysis of boundary-domain integral and integro-differential equations
extending corresponding results of [Mik05, Mik06] to Lipschitz domain, Hölder
coefficients and wider range of Sobolev spaces, and based on the results of
present paper, is to be published elsewhere.
2 Sobolev spaces, trace operators and extensions
Suppose Ω is a bounded open n–dimensional region of Rn , n ≥ 2, which
boundary ∂Ω is a simply connected, closed, Lipschitz surface.
∞
In what follows D(Ω) = Ccomp
(Ω) denotes the space of Schwartz test
∗
functions, and D (Ω) the space of Schwartz distributions; H s (Rn ) = H2s (Rn ),
H s (∂Ω) = H2s (∂Ω) are the Bessel potential spaces, where s ∈ R is an arbitrary
e s (Ω) the subspace of H s (Rn ), H
e s (Ω) := {g :
real number. We denote by H
s
n
s
g ∈ H (R ), supp g ⊂ Ω}, while H (Ω) denotes the space of restriction on
Ω of distributions from H s (Rn ), H s (Ω) := {g|Ω : g ∈ H s (Rn )}, where g|Ω
denotes the restriction on Ω, and H0s (Ω) is closure of D(Ω) in H s (Ω). We
recall that H s coincide with the Sobolev–Slobodetski spaces W2s for any nons
e s (Ω)), which
negative s. We denote by H∂Ω
the subspace of H s (Rn ) (and H
s
s
n
elements are supported on ∂Ω, i.e., H∂Ω := {g : g ∈ H (R ), supp g ⊂ ∂Ω}.
To introduce generalized co-normal derivatives in the next section, we will
need several facts about traces and extensions in Sobolev spaces on Lipschitz
domain. First of all, it is well known [Cos88, Lemma 3.7], that the trace
1
1
operators τ : H s (Rn ) → H s− 2 (∂Ω) and τ + : H s (Ω) → H s− 2 (∂Ω) are
1
3
continuous for 2 < s < 2 on any Lipschitz domain Ω. We will also use
notation u+ := τ + u.
1
n
Since the space H s (Rn ) is dense in H 2 (RS
) for s > 12 , we have that the
trace operator is well defined on the set D = 1 <s≤1 H s (Rn ), which is dense
2
1
in H 2 (Rn ).
Lemma 1. For a Lipschitz domain Ω, the norm of the trace operator, τ :
1
H s (Rn ) → H s− 2 (∂Ω) tends to infinity as s → 12 , s > 12 , while the operator
1
τ : H 2 (Rn ) → L2 (∂Ω) is unbounded.
Proof. It is suffice to show that there exists a function v ∈ L2 (∂Ω) and a
sequence wk0 ∈ D such that kwk0 k 12 n ≤ C < ∞ but
H (R )
|hv, τ wk0 i|
→ ∞,
k → ∞.
(2)
Let us first find such a sequence for the half space Ω = Rn+ = {x ∈ Rn : xn >
0}, where x = {x0 , xn }. For a non-zero function v ∈ L2 (∂Ω) and w ∈ D, we
have,
About traces, extensions and co–normal derivative operators
3
hv, τ wi = hv, δn wi = hv ⊗ δn , wi,
where δn = δ(xn ) is the one-dimensional Dirac function. However the Fourier
transform Fx→ξ {v(x0 )δ(xn )} = v̂(ξ 0 ) := Fx0 →ξ0 {v(x0 )}, and we have for s > 21
(cf. the proof of Theorem 3.39 in [McL00]) ,
Z
kv ⊗ δ(xn )k2H −s (Rn ) =
(1 + |ξ|2 )−s |v̂(ξ 0 )|2 dξ = Cs kvk2 1 −s n−1 ,
Rn
H2
(R
)
R∞
1
where Cs = −∞ (1 + η 2 )−s dη, and the substitution ξn = (1 + |ξ 0 |2 ) 2 η was
used. Since kvk2 1 −s n−1 → kvk2L2 (Rn−1 ) 6= 0 and Cs → ∞ as s → 12 , we
H2
have
sup
kwkH s (Ω) =1
(R
)
|hv ⊗ δn , wi| = kv ⊗ δ(xn )kH −s (Rn ) → ∞,
s→
This implies that there exists a sequence wk ∈ D such that kwk k
1
.
2
1
H − 2 (Rn )
≤1
but |hv ⊗ δ(xn ), wk i| → ∞ as k → ∞, which proves the lemma for Ω = Rn+ if
we take wk0 = wk .
Let now Ω be a half-space bounded by a Lipschitz hypograph, Ω = {x ∈
Rn : xn > ζ(x0 )}, where ζ is a Lipschitz function. Then the sequence wk0 (x) =
wk (x0 , xn − ζ(x0 )) will have the necessary properties since H s (Rn ) is invariant
under a Lipschitz change of coordinates if 0 ≤ s ≤ 1.
If Ω is a general Lipschitz domain, then (generally after a rigid rotation
of coordinates) it has a part of the boundary Γ1 ⊂ ∂Ω that is a Lipschitz
hypograph and can be extended to the boundary of a half-space. Choosing
v such that v = 0 on ∂Ω\Γ1 , kvkL2 (Γ1 ) 6= 0, and then wk0 as in the previous
paragraph, we obtain property (2), which completes the proof.
¤
Lemma 2. For a bounded Lipschitz domain Ω there exists a linear bounded
1
extension operator e : H s− 2 (∂Ω) → H s (Rn ), 12 ≤ s ≤ 23 , which is right
1
inverse to the trace operators τ ± , i.e., τ ± eg = g for any g ∈ H s− 2 (∂Ω).
Moreover, kek s− 12
≤ C, where C is independent of s.
s
n
H
(∂Ω)→H (R )
Proof. For Lipschitz domains and 12 < s ≤ 1, the boundedness of the extension
operator is well known, see e.g. [McL00, Theorem 3.37].
To prove it for the whole range 12 ≤ s ≤ 32 , let us consider the classical
3
−1
single layer potential V∆ ϕ with a density ϕ = V∆
g ∈ H s− 2 (∂Ω), solving
the Laplace equation in Ω + with the Dirichlet boundary data g, where V∆
−1
is the direct value of the operator V∆ on the boundary. The operators V∆
:
s− 21
s− 32
s− 32
s
n
H
(∂Ω) → H
(∂Ω) and V∆ : H
(∂Ω) → Hloc (R ) are continuous for
1
3
2 ≤ s ≤ 2 as stated in [JK81b, JK81a, JK82, Ver84, Cos88]. Thus it suffice
−1
to take e = χV∆ V∆
, where χ ∈ D(Rn ) is a cut-off function such that χ = 1
+
≤ C, where C is independent of s,
in Ω̄ . The estimate kek s− 12
(∂Ω)→H s (Rn )
H
then follows by interpolation.
¤
4
S.E. Mikhailov
Note that for s = 12 the trace operator τ + is understood in the nontangential sense, and continuity of the operators τ ± was not needed in the
proof.
e s (Ω) for 1 < s < 3 , we will need
To characterize the space H0s (Ω) = H
2
2
the following statement.
Lemma 3. If Ω is a Lipschitz domain and u ∈ H s (Ω), 0 < s < 12 , then
Z
dist(x, ∂Ω)−2s |u(x)|2 dx ≤ Ckuk2H s (Ω) .
(3)
Ω
Proof. Note first that the lemma claim holds true for u ∈ D(Ω), see [McL00,
Lemma 3.32]. To prove it for u ∈ H s (Ω), let first the domain Ω be such that
dist(x, ∂Ω) < C0 < ∞
(4)
for all x ∈ Ω, which holds true particularly for bounded domains. Let
φk ∈ D(Ω) be a sequence converging to u in H s (Ω). If we denote w(x) =
dist(x, ∂Ω)−2s , then w(x) > C0−2s > 0. Since (3) holds for functions from
D(Ω), the sequence φk ∈ D(Ω) is fundamental in the weighted space L2 (Ω, w),
which is complete, implying that φk ∈ D(Ω) converges in this space to a
function u0 ∈ L2 (Ω, w). Since both L2 (Ω, w) and H s (Ω) are continuously
imbedded in the non-weighted space L2 (Ω), the sequence φk converges in
L2 (Ω) implying the limiting functions u and u0 belong to this space and thus
coincide.
If the condition (4) is not satisfied, let χ(x) ∈ D(Rn ) be a cut-off function
such that 0 ≤ χ(x) ≤ 1 for all x, χ(x) = T1 near ∂Ω, while w(x) < 1 for
x ∈ supp (1 − χ). Then (4) is satisfied in Ω supp χ(x) and
Z
Z
Z
(1 − χ(x))w(x)|u(x)|2 dx +
χ(x)w(x)|u(x)|2 dx ≤
Ω
Ω
Z
p
2
2
kukL2 (Ω) +
w(x)| χ(x)u(x)| dx
Ω
p
≤ kuk2H s (Ω) + Ck χ(x)uk2H s (Ω) ≤ C1 kuk2H s (Ω) .
w(x)|u(x)|2 dx =
Ω
due to the previous paragraph.
Lemma 3 allows now extending the following statement known for
s ≤ 1, see [McL00, Theorem 3.40(i)], to a wider range of s.
Theorem 1. If Ω is a Lipschitz domain and
e s (Ω) = {u ∈ H s (Ω) : τ + u = 0}.
H
1
2
< s <
3
2,
1
2
¤
<
then H0s (Ω) =
e s (Ω), for 1 < s < 3 is well known,
Proof. The first equality, H0s (Ω) = H
2
2
see e.g. [McL00, Theorem 3.33]. The theorem claim for 12 < s ≤ 1 is stated
e s (Ω), then evidently γ + u = 0 since
in [McL00, Theorem 3.40(ii)]. If u ∈ H
s
e
D is dense in H (Ω) and the trace operator γ + is bounded in H s (Rn ) for
About traces, extensions and co–normal derivative operators
5
1
2
< s < 32 . To prove the theorem for 1 < s < 32 , it remains to prove that
the extension, ũ, of any u ∈ H s (Ω) such that γ + u = 0, by zero outside Ω,
belongs to H s (Rn ). To do this, we remark first of all that ũ ∈ H 1 (Rn ) due to
[McL00, Theorem 3.40(ii)], and then make estimates similar to those in the
proof of [McL00, Theorem 3.33],
Z
Z
|∇ũ(x) − ∇ũ(y)|2
dx dy
2(s−1)+n
Rn Rn |x − y|
Z Z
|∇u(x) − ∇u(y)|2
= kuk2W 1 (Ω) +
dx dy
2(s−1)+n
2
Ω Ω |x − y|
Z
Z
Z
Z
|∇u(y)|2
|∇u(x)|2
dx
dy
+
dx dy
+
2(s−1)+n
2(s−1)+n
Ω Rn \Ω |x − y|
Rn \Ω Ω |x − y|
Z
|ws−1 (x)∇u(x)|2 dx
= kuk2W2s (Ω) + 2
kũk2H s (Rn )
∼
kũk2W 1 (Rn )
2
+
Ω
where
Z
ws−1 (x) :=
Rn \Ω
dy
,
|x − y|2(s−1)+n
x ∈ Ω,
and W2s (Ω) is the Sobolev-Slobodetski space. Introducing polar coordinates
with x as an origin, we obtain, ws−1 (x) ≤ C dist(x, ∂Ω)−2(s−1) for x ∈ Ω.
Then, taking into account that ∇u ∈ H s−1 (Ω) and k∇ukH s−1 (Ω) ≤ kukH s (Ω) ,
we have by Lemma 3,
kuk2He s (Ω) = kũk2H s (Rn ) ≤ kuk2W2s (Ω) + 2Ckuk2H s (Ω) ≤ Cs kuk2H s (Ω) .
¤
Lemma 4. Let Ω be a Lipschitz domain in Rn .
t
(i) If t ≥ − 12 , then H∂Ω
= {0}.
3
1
t
(ii) If − 2 < t < − 2 , then H∂Ω
is the set of distributions g on Rn of the
form
hg, wiRn = hv, τ wi∂Ω ∀ w ∈ H −t (Rn ),
(5)
1
where τ is the trace operator, v ∈ H t+ 2 (∂Ω) and kvk t+ 21
≤ CkgkH t (Rn )
H
(∂Ω)
with C independent of t.
Proof. We will follow an idea in the proof of Lemma 3.39 in [McL00], extending
it from a half-space to a Lipschitz domain Ω.
Let Ω + = Ω and Ω − = Rn \Ω̄. For any φ ∈ D(Rn ), let us define
(
φ(x) if x ∈ Ω ± ,
φ± (x) =
0
otherwise.
e −t (Ω ± ) (see e.g. [McL00, Theorems 3.33, 3.40]
Let t > − 12 . Then φ± ∈ H
1
for − 2 < t ≤ 0, for greater t it then follows by embedding), kφ − φ+ −
6
S.E. Mikhailov
φ− kH −t (Rn ) = 0, and there exist sequences φk± ∈ D(Ω ± ) converging to φ±
e −t (Ω ± ) as k → ∞. Hence hg, φi = limk→∞ hg, φ+ + φ− i = 0 for any
in H
k
k
t
g ∈ H∂Ω
, t > − 21 , proving (i) for such t.
1
t
Let us prove (ii). For g ∈ H∂Ω
, − 32 < t < − 12 , let us define v ∈ H t+ 2 (∂Ω)
as
1
hv, φi∂Ω := hg, eφiRn ∀ φ ∈ H −t− 2 (∂Ω),
1
where e : H −t− 2 (∂Ω) → H −t (Ω) is a bounded extension operator, whose
existence is proved in Lemma 2. Observe that
|hv, φi| ≤ kgkH t (Rn ) kφk
1
H −t− 2 (∂Ω)
kek
1
H −t− 2 (∂Ω)→H −t (Rn )
,
so kvk t+ 12
≤ kek −t− 12
kgkH t (Rn ) ≤ CkgkH t (Rn ) , where C
H
(∂Ω)
H
(∂Ω)→H −t (Rn )
is independent of t due to Lemma 2. We also have that
hg, wiRn − hv, τ wi∂Ω = hg, ρiRn
where
∀ w ∈ H −t (Rn ),
ρ = w − eτ w ∈ H −t (Rn ).
e −t (Ω ± ) and thus
Then we have τ ρ = 0, which due to Theorem 1 means ρ ∈ H
±
±
e
there exist sequences φk ∈ D(Ω ) converging to ρ in H −t (Ω ± ), implying
t
hg, ρiRn = 0 since g ∈ H∂Ω
, and thus ansatz (5).
−1
It remains to deal with the case t = − 21 in (i). Let g ∈ H∂Ω2 . Since
−1
t
H∂Ω2 ⊂ H∂Ω
for t < − 12 , ansatz (5) is valid for g. However, owing to Lemma
1
1 the norm of the trace operator τ : H −t (Rn ) → H −t− 2 (∂Ω) tends to infinity
1
as t → − 2 , which means v should be zero.
¤
Lemma 5. Let Ω be a Lipschitz domain and let − 32 < s < 12 , s 6= − 12 . There
e s : H s (Ω) → H
e s (Ω).
exists a bounded linear extension operator E
e s (Ω) = H s (Ω), see e.g. [McL00, Theorems 3.33
Proof. If 0 ≤ s < 12 , then H
s
e can be taken as the identity operator.
and 3.40], which implies E
1
e −s (Ω) = H −s (Ω) as in the above paragraph, we
Let − 2 < s < 0. Since H
s
−s
∗
e
e s (Ω). The asterisk denotes the
have H (Ω) = [H (Ω)] = [H −s (Ω)]∗ = H
s
e can be taken as the identity operator.
dual space. This implies E
Let now − 23 < s < − 12 . For s in this range, the trace operator τ + :
1
H −s (Ω) → H −s− 2 (∂Ω) is bounded due to [Cos88, Lemma 3.6] (see also
[McL00, Theorem 3.38]), and there exists a bounded extension operator e :
1
H −s− 2 (∂Ω) → H −s (Ω), see Lemma 2. Then due to Theorem 1, (I − eτ + ) is a
e −s (Ω). Thus any functional
bounded projector from H −s (Ω) to H0−s (Ω) = H
e s (Ω) such
v ∈ H s (Ω) can be continuously mapped into the functional ṽ ∈ H
+
−s
that ṽu = v(I − eτ )u for any u ∈ H (Ω). Since ṽu = vu for any u ∈
e −s (Ω), we have, E
e s = (I − eτ + )∗ : H s (Ω) → H
e s (Ω) is a bounded extension
H
operator.
¤
About traces, extensions and co–normal derivative operators
7
es :
Note that Lemma 4 implies uniqueness of the extension operator E
1
1
s
e
H (Ω) → H (Ω) for − 2 < s < 2 , which coincides in this case with the
identity operator and will be called canonical extension operator. For − 32 <
1
s < − 12 , on the other hand, the extension operator e : H −s− 2 (∂Ω) → H −s (Ω)
es :
in the proof of Lemma 5 is not unique, implying non-uniqueness of E
s
s
e
H (Ω) → H (Ω).
s
3 Partial differential operator extensions and co-normal
derivatives
For u ∈ H s (Ω), s > 23 , and a ∈ C 0,α (Ω), α > 12 , we can denote by T + the
corresponding co–normal derivative operator on ∂Ω in the sense of traces,
T + (x, n+ (x), ∂x ) u(x) :=
n
X
µ
a+ (x) n+
i (x)
i=1
∂u(x)
∂xi
¶+
,
where n+ (x) is the outward (to Ω) unit normal vectors at the point x ∈ ∂Ω,
∂x = (∂1 , ∂2 , ..., ∂n ) and ∂j := ∂/∂xj (j = 1, 2, ..., n).
For simplicity, we will further consider only bounded Lipschitz domains.
We will need the following particular case of a statement from [Gri85, Theorem
1.4.1.1].
Theorem 2. Let Ω be a bounded Lipschitz domain and v ∈ C k,α (Ω) with
k+α ≥ |s| when s is an integer while k+α > |s| when s is not an integer. Then
vu ∈ H s (Ω) for every u ∈ H s (Ω), and there exists a constant K = K(v, s)
such that
kvukH s (Ω) ≤ KkukH s (Ω) .
(6)
For u ∈ H s (Ω) and v ∈ H 2−s (Ω), 12 < s < 23 , a ∈ C 0,α (Ω), α > |s − 1| if
s 6= 1, and α = 0 if s = 1, let us define a bilinear form,
E(u, v) :=
3
X
ha∂i u, ∂i viΩ ,
i=1
where h · , · iΩ denote the duality brackets between the spaces H s−1 (Ω) and
H 1−s (Ω).
Let u ∈ H s (Ω), 12 < s < 32 . Then the Lu is understood as the following
distribution,
hLu, viΩ := −E(u, v) ∀v ∈ D(Ω).
(7)
e 2−s (Ω), the above formula defines a
Since the set D(Ω) is dense in H
s
s−2
e 2−s (Ω)]∗ , 1 < s < 3 ,
bounded operator L : H (Ω) → H
(Ω) = [H
2
2
e 2−s (Ω).
hLu, viΩ := −E(u, v) ∀v ∈ H
(8)
8
1
2
S.E. Mikhailov
e s−2 (Ω) = [H 2−s (Ω)]∗ ,
Let us consider also the operator L̂ : H s (Ω) → H
3
< s < 2 , defined as,
hL̂u, viΩ := −E(u, v) ∀v ∈ H 2−s (Ω),
(9)
which is evidently bounded. For any u ∈ H s (Ω), the functional L̂u belongs to
e s−2 (Ω) and is an extension of the functional Lu ∈ H s−2 (Ω) from domain
H
e 2−s (Ω) to domain of definition H 2−s (Ω).
of definition H
The extension is not unique, and any functional of the form
L̂u + g,
s−2
g ∈ H∂Ω
(10)
provides another extension. On the other hand, any extension of the domain
e 2−s (Ω) to H 2−s (Ω) has evidently
of definition of the functional Lu from H
form (10). The existence of such extensions is provided by Lemma 5.
Now we can extend the definition from [McL00, Lemma 4.3] of the generalized co–normal derivative to a range of Sobolev spaces.
Lemma 6. Let Ω be a bounded Lipschitz domain, 12 < s < 32 , a ∈ C 0,α (Ω),
α > |s − 1| if s 6= 1, and α = 0 if s = 1, u ∈ H s (Ω), and Lu = f˜|Ω in Ω
e s−2 (Ω). Let us define the generalized co–normal derivative
for some f˜ ∈ H
3
+
s−
Te (f˜, u) ∈ H 2 (∂Ω) as
D
E
Te+ (f˜, u) , w
∂Ω
:= hf˜, e+ wiΩ + E(u, e+ w)
= hf˜ − L̂u, e+ wiΩ
3
∀ w ∈ H 2 −s (∂Ω), (11)
3
where e+ : H 2 −s (∂Ω) → H 2−s (Ω) is a bounded extension operator.
Then Te+ (f˜, u) is independent of e+ ,
kTe+ (f˜, u)k
3
H s− 2 (∂Ω)
≤ C1 kukH s (Ω) + C2 kf˜kHe s−2 (Ω) ,
(12)
and the first Green identity holds in the following form,
D
E
Te+ (f˜, u) , v +
= hf˜, viΩ +E(u, v) = hf˜− L̂u, viΩ ∀ v ∈ H 2−s (Ω). (13)
∂Ω
The lemma proof is available in [McL00, Lemma 4.3] for s = 1. Taking into
account that Lemma 2 provides existence of a bounded extension operator
3
e+ : H 2 −s (∂Ω) → H 2−s (Ω) in the whole range 12 < s < 23 , the proof from
[McL00, Lemma 4.3] works verbatim (with the appropriate change of the
Sobolev space indices and invoking Theorem 1 and Lemma 4) for all s in this
range.
Note that because of the involvement of f˜, the generalized co-normal
derivative Te+ (f˜, u) is generally non-linear in u. It becomes linear if a linear
relation is imposed between u and f˜ (including behavior of the latter on the
About traces, extensions and co–normal derivative operators
9
e s−2 (Ω). For example, f˜|
boundary ∂Ω), thus fixing an extension of f˜|Ω into H
Ω
can be extended as fˆ = L̂u 6= f˜. Then obviously, Te+ (fˆ, u) = Te+ (L̂u, u) = 0.
In fact, for a given function u ∈ H s (Ω), 12 < s < 23 , any distribution
3
t+ ∈ H s− 2 (∂Ω) may be nominated as a co-normal derivative of u, by an
e s−2 (Ω).
appropriate extension f˜ of the distribution Lu ∈ H s−2 (Ω) into H
This extension is again given by the second Green formula (13) re-written as
follows,
­
®
hf˜, viΩ := t+ , τ + v ∂Ω − E(u, v) = hτ +∗ t+ + L̂u, viΩ ∀ v ∈ H 2−s (Ω). (14)
3
e s−2 (Ω) is dual to the trace operaHere the operator τ +∗ : H s− 2 (∂Ω) → H
3
+∗ +
+ +
tor, hτ t , viΩ := ht , τ vi∂Ω for all t+ ∈ H s− 2 (∂Ω) and v ∈ H 2−s (Ω).
e s−2 (Ω) and is an
Evidently, the distribution f˜ defined by (14) belongs to H
e s−2 (Ω) since τ + v = 0 for v ∈ H
e 2−s (Ω).
extension of the distribution Lu into H
+
e
To analyze apart from T (L̂u, u) another case, when the co-normal derivative operator becomes linear, let us consider a subspace H s,t (Ω; L∗ ) of H s (Ω).
Definition 1. Let s ∈ R and L∗ : H s (Ω) → D∗ (Ω) be a linear operator. For
t ≥ − 12 , we introduce a space H s,t (Ω; L∗ ) := {g : g ∈ H s (Ω), L∗ g|Ω =
e t (Ω)} endowed with the norm kgkH s,t (Ω;L ) := kgkH s (Ω) +
f˜g |Ω , f˜g ∈ H
∗
kf˜g kHe t (Ω) .
e t (Ω), t ≥ − 1 , in the above definition is an extenThe distribution f˜g ∈ H
2
sion of the distribution L∗ g|Ω ∈ H t (Ω), and the extension is unique (if it does
exist) due to the Lemma 4. The uniqueness implies the norm kgkH s,t (Ω;L∗ ) is
well defined. Note that another subspace of such kind, where L∗ g|Ω belongs
to Lp (Ω) instead of H t (Ω) was presented in [Gri85, p. 59].
If s, p ∈ R, L∗ : H s (Ω) → H p (Ω) is a linear bounded operator, t ≤ p, and
1
− 2 < t < 12 , then, evidently, H s,t (Ω; L∗ ) = H s (Ω) since H p (Ω) ⊂ H t (Ω)
e t (Ω) = H t (Ω).
and H
Lemma 7. Let s, p ∈ R. If a linear operator L∗ : H s (Ω) → H p (Ω) is continuous, then the space H s,t (Ω; L∗ ) is complete for any t ≥ − 12 .
Proof. Let gk be a Cauchy sequence in H s,t (Ω; L∗ ). Then there exists a
e t (Ω) such that f˜g |Ω = L∗ gk |Ω . Since H s (Ω) and
Cauchy sequence f˜gk in H
k
t
e
e t (Ω)
H (Ω) are complete, then there exist elements g0 ∈ H s (Ω) and f˜0 ∈ H
˜
˜
such that kgk − g0 kH s (Ω) → 0, kfgk − f0 kHe t (Ω) → 0 as k → ∞. On the other
hand, kL∗ gk − L∗ g0 kH p (Ω) → 0 since L∗ is continuous. Taking into account
that L∗ gk |Ω = f˜gk |Ω , we obtain
kf˜0 − L∗ g0 kH p (Ω) ≤ kf˜0 − f˜gk kH t (Ω) + kf˜gk − L∗ g0 kH p (Ω) → 0 k → ∞
if p ≤ t, and
10
S.E. Mikhailov
kf˜0 − L∗ g0 kH t (Ω) ≤ kf˜0 − f˜gk kH t (Ω) + kf˜gk − L∗ g0 kH p (Ω) → 0 k → ∞
if t ≤ p. That is, L∗ g0 |Ω = f˜0 |Ω ∈ H t (Ω) in the both cases, which implies
g0 ∈ H s,t (Ω; L∗ ).
¤
Let us return to the operator L from (7) and define the canonical (or
internal) co-normal derivative operator (cf. [Cos88, Lemma 3.2] where a conormal derivative operator acting on functions from H 1,0 (Ω; L) was defined).
Definition 2. Let u ∈ H s,t (Ω; L), s ∈ R, t ≥ − 21 . Then the distribution
e t (Ω), which we
Lu ∈ H t (Ω) can be uniquely extended to a distribution in H
0
will call the canonical extension and denote by L u or still use the notation
Lu if this will not lead to a confusion.
1
Definition 3. For u ∈ H s,− 2 (Ω; L), 12 < s < 32 ; a ∈ C 0,α (Ω), α > |s − 1|
if s 6= 1, and α = 0 if s = 1, we define the canonical (or internal) co-normal
3
derivative T + u ∈ H s− 2 (∂Ω) as
­ +
®
T u , w ∂Ω := hL0 u, e+ wiΩ + E(u, e+ w)
= hL0 u − L̂u, e+ wiΩ
3
∀ w ∈ H 2 −s (∂Ω), (15)
1
where e+ : H s− 2 (∂Ω) → H s (Ω) is a bounded extension operator. The
canonical co-normal derivative T + u is independent of e+ , the operator T + :
1
3
H s,− 2 (Ω; L) → H s− 2 (∂Ω) is continuous, and the first Green identity holds
in the following form ,
D
E
­ +
®
T u , v+
= Te+ (L0 u, u) , v +
= hL0 u, viΩ + E(u, v)
∂Ω
∂Ω
= hL0 u − L̂u, viΩ
∀ v ∈ H 2−s (Ω).
(16)
The independence of e+ and the continuity of the operator T + as well as identity (16) are implied by the definition of the generalized co-normal derivative
in Lemma 6 and Definition 1. Unlike the generalized co-normal derivative,
the canonical co-normal derivative is uniquely defined by the function u and
operator L only, uniquely fixing an extension of the latter on the boundary.
Definitions (11) and (15) imply that the generalized co-normal derivative
1
e s−2 (Ω) of the
of u ∈ H s,− 2 (Ω; L), 12 < s < 32 , for any other extension f˜ ∈ H
t
distribution Lu|Ω ∈ H (Ω) can be expressed as
D
E
­
®
3
Te+ (f˜, u) , w
= T +u , w
+hf˜−L0 u, e+ wiΩ ∀ w ∈ H 2 −s (∂Ω). (17)
∂Ω
∂Ω
2−s
Note that the distributions f˜ − L̂u, L0 u − L̂u and f˜ − L0 belong to H∂Ω
0
2−s
0
s−2
˜
˜
e
since L u, L̂u, f ∈ H
(Ω), while L u|Ω = L̂u|Ω = f |Ω = Lu|Ω ∈ H
(Ω).
The following lemma and corollary give conditions when the canonical
co-normal derivative coincides with the classical co-normal derivative T + u =
∂u +
a+ ( ∂n
) if the latter does exist in the trace sense.
About traces, extensions and co–normal derivative operators
11
s,− 12
Lemma 8. Let a ∈ C 0,α (Ω), α > 12 , u ∈ H
(Ω; L), 12 < s < 32 ,
2
and uk ∈ H (Ω) be a sequence such that kuk − uk s,− 21
→ 0. Then
H
(Ω;L)
Pn
+
+
k j=1 (a∂j uk ) nj − T uk s− 23
→ 0 as k → ∞.
H
(∂Ω)
Proof. Using the canonical first Green identity (16) for u and the classical
first Green identity (follows e.g. from [McL00, Lemma 4.1]) for uk , we have
3
for any w ∈ H 2 −s (∂Ω),
­ +
®
T u, w
∂Ω
= E(u, e+ w) + hL0 u, e+ wiΩ
= E(u − uk , e+ w) + E(uk , e+ w) + hL0 u, e+ wiΩ
Z X
n
= E(u − uk , e+ w) +
(a∂j uk )+ nj wdΓ − hL0 uk , e+ wiΩ + hL0 u, e+ wiΩ
∂Ω j=1
*
+
= E(u − uk , e w) +
n
X
(a∂j uk )+ nj , w
+
j=1
+ hL0 (u − uk ), e+ wiΩ
∂Ω
*
→
n
X
(a∂j uk )+ nj , w
j=1
+
,
∂Ω
as k → ∞. It was taken into account that a∂j uk ∈ H (Ω), < p < α, p ≤ 32 ,
due to Lemma 2. Since T + u is uniquely determined by u, this implies existence
of the limit in the right hand side and its independence of the sequence uk . ¤
The sequence uk mentioned in Lemma 8 does always exist due to the
following statement.
p
1
2
Lemma 9. D(Ω) is dense in H s,t (Ω; L), s ∈ R, − 21 < t < 12 .
Proof. We modify appropriately the proof from [Gri85, L. 1.5.3.9] given for
another space of such kind.
For every continuous linear functional l on H s,t (Ω; L) there exist f˜ ∈
−s
e
H (Ω) and g ∈ H −t (Ω) such that
l(u) = hf˜, uiΩ + hg, LuiΩ .
To prove the lemma claim, it suffice to show that any l which vanishes on
D(Ω), will vanish on any u ∈ H s,t (Ω; L). Indeed, if l(φ) = 0 for any φ ∈ D(Ω),
then
hf˜, φiΩ + hg, LφiΩ = 0.
(18)
e −t (Ω) (cf. the proof of Lemma 5),
Extending g outside Ω by zero to g̃ ∈ H
equation (18) can be rewritten as
hf˜, φiRn + hg̃, LφiRn = 0.
This means Lg̃ = −f˜ in Rn in the sense of distributions. Then the ellipe 2−s (Ω).
ticity of L implies g̃ ∈ H 2−s (Rn ) and consequently g̃ ∈ H
12
S.E. Mikhailov
e q (Ω), q =
Let now gk ∈ D(Ω) be a sequence converging to g̃ in H
−t
s−2
e (Ω) and H
e
max{−t, s − 2}, and thus in H
(Ω), as k → ∞. Then for
any u ∈ H s,t (Ω; L), we have,
l(u) = lim {h−Lgk , uiΩ + hgk , LuiΩ } = 0.
(19)
k→∞
Thus l is identically zero.
¤
Corollary 1. If a ∈ C 0,α (Ω), α >
(a∂j u)+ nj .
1
2,
and u ∈ H q (Ω), q >
3
2,
then T + u =
1
Proof. If u ∈ H q (Ω), q > 32 , then u ∈ H q,t (Ω) ⊂ H s,t (Ω; L) ⊂ H s,− 2 (Ω; L)
for any t ∈ (− 12 , min{α, q − 1} − 1) and any s ∈ ( 12 , 32 ). Hence a sequence
uk ∈ D(Ω) such that kuk − ukH q (Ω) → 0 as k → ∞, satisfies the hypothesis
of Lemma 8 for any s ∈ ( 12 , 32 ). On the other hand, due to Theorem 2,
k
n
X
[a∂j (uk − u)]+ nj k
j=1
3
H s− 2
(∂Ω)
≤k
n
X
[a∂j (uk − u)]+ nj kL2 (∂Ω) ≤
j=1
max |a| k(uk − u)kH p (Ω) ≤ max |a| k(uk − u)kH q (Ω) → 0,
k → ∞.
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