TRACES AND EXTENSIONS OF BV -FUNCTIONS
CLAUS GERHARDT
Abstract. We give a simple proof of Gagliardo’s trace and extension
theorem for BV -functions.
Contents
1. Introduction
References
1
4
1. Introduction
1.1. Definition. We say a sequence uk ∈ BV (Ω) converges in the BV
sense to u ∈ BV (Ω), if
Z
(1.1)
|u − uk | → 0
Ω
and
Z
Z
|Duk | →
(1.2)
Ω
|Du|.
Ω
1.2. Lemma. Let Ω = B1+ (0) be the upper half ball and let u ∈ BV (Ω)
have compact support with respect to the spherical boundary. Define for 0 < h
uh (x̂, xn ) = u(x̂, xnh ),
(1.3)
then uh is defined in {xn > −h} and the mollification uh, with an even
mollifier is well defined in Ω if 0 < < 0 (h). There are sequences (hk , k )
converging to zero such that uhk ,k converge in the BV sense to u.
Proof. Let η i ∈ Cc∞ (Ω), k(η i )k ≤ 1, then
Z
Z
Z
i
uh, Di η i =
uDi η−h,
≤
|Du|,
(1.4)
Ω
Ω
Ω
if ≤ 0 (h), since
(1.5)
i
η−h,
∈ Cc∞ (Ω)
Date: May 20, 2011.
2000 Mathematics Subject Classification. 35J60, 53C21, 53C44, 53C50, 58J05.
Key words and phrases. Traces, extensions of BV -functions, Gagliardo’s theorem.
1
2
CLAUS GERHARDT
for those , where we may consider only ηi the support of which is uniformly
compact with respect to the spherical boundary in view of the assumption
on u. This proves the lemma.
1.3. Lemma. Let u, Ω be as above, then the approximating functions
uhk ,k , or any other sequence uk ∈ C 1 (Ω̄) converging in the BV sense to u,
define a unique trace of u, tr u, on the hyperplane such that the divergence
theorem holds
Z
Z
Z
(1.6)
Di uη = −
uDi η +
uηνi
Ω
Ω
∂Ω
for all η ∈ Cc1 (Ω ∪ ∂Ω), cf. Remark 1.6.
Extending u as an even function to the lower half space
(
u(x̂, xn )
xn > 0,
(1.7)
ũ(x̂, xn ) =
u(x̂, −xn ) xn < 0,
there holds
Z
Z
|Dũ| ≤ 2
(1.8)
B1 (0)
|Du|
Ω
and
Z
|Dũ| = 0.
(1.9)
∂Ω
Proof. Let η i ∈ C 1 (B1 (0)), then
Z
Z
Z
Di ũη i =
Di uη i +
(1.10)
B1 (0)
B1−
B1+
Z
i
=−
Di ũη i +
Di ũη i
Γ
Z
+
+
i
Z
(u − u )η νi +
ũDi η +
B1 (0)
Z
Γ
Di ũη i
Γ
hence
Z
Di ũη i = 0.
(1.11)
Γ
Moreover, if k(η i )k ≤ 1, we conclude from [2, Theorem 10.10.3] and (1.10)
the estimate (1.8).
1.4. Theorem. Let Ω be as in the theorem below, then u ∈ BV (Ω) can
be extended to Rn with compact support such that, if the extended function is
called ũ,
Z
Z
(1.12)
|Dũ| ≤ c (|Du| + |u|)
Rn
Ω
and
Z
|Dũ| = 0.
(1.13)
∂Ω
TRACES AND EXTENSIONS OF BV -FUNCTIONS
3
Proof. Let Uk be a finite open covering of Ω̄ with the usual conditions for
the boundary neighbourhoods and let ηk be a subordinate finite partition of
unity, then each function uηk can be extended to Rn with compact support
such that, if ũk is the extended function,
Z
Z
(1.14)
|Dũk | ≤ c (|Du| + |u|).
Rn
Ω
The function
(1.15)
X
ũ =
ũk
k
is then an extension of u with the required properties.
The above theorem is due to [1].
1.5. Theorem. Let u ∈ BV (Ω), where ∂Ω ∈ C 0,1 and compact, and
let uk ∈ C 1 (Ω̄) converging in the BV sense to u, at least in a boundary
neighbourhood, then tr uk converges in L1 (∂Ω) to a uniquely defined function
depending only u and not on the approximating functions; we call this limit
tr u. If u ∈ BV (Ω) ∩ C 0 (Ω̄), then
(1.16)
tr u = u|∂Ω
Proof. It suffices to prove the uniqueness of the trace, since u can be extended
to Rn , and hence also be approximated by mollifications.
Let uk and ũk be C 1 (Ω̄) approximations of u and let vk = uk − ũk , then
Z
(1.17)
lim
|vk | = 0,
Ω
where Ω is a boundary strip of width such that
Z
(1.18)
|Du| = 0
{d=}
and
Z
(1.19)
Z
|Dvk | ≤ 2
lim sup
Ω
Now, there holds for any > 0
Z
Z
(1.20)
|vk | ≤ c
∂Ω
|Du|.
Ω
Z
|Dvk | + c
|vk |,
Ω
Ω
hence
Z
(1.21)
Z
|vk | ≤ 2c lim sup
lim sup
∂Ω
|Du| = 0,
Ω
where the last equality is due to the fact that |Du| is a bounded measure in
Ω0 and the Ω are monotone ordered such that
\
(1.22)
Ω = ∅.
0<≤0
4
CLAUS GERHARDT
1.6. Remark. By approximation we also obtain
Z
Z
Z
(1.23)
Di uη = −
uDi η +
uηνi
Ω
Ω
∂Ω
for all η ∈ Cc1 (Ω ∪ ∂Ω).
1.7. Remark. The definition of BV (Ω), when Ω ⊂ N , where N is a
Riemannian manifold is analogous to the case N = Rn , namely,
Z
Z
(1.24)
|Du| = sup{
uDi η i : gij η i η j ≤ 1, η i ∈ Cc∞ (Ω) }.
Ω
Ω
Even in the Euclidean case it is sometimes desirable to use the invariant
definition of the total variation.
References
[1] Emilio Gagliardo, Caratterizzazioni delle tracce sulla frontiera relative ad alcune classi
di funzioni in n variabili, Rend. Sem. Mat. Univ. Padova 27 (1957), 284–305.
[2] Claus Gerhardt, Analysis II, International Series in Analysis, International Press,
Somerville, MA, 2006.
Ruprecht-Karls-Universität, Institut für Angewandte Mathematik, Im Neuenheimer Feld 294, 69120 Heidelberg, Germany
E-mail address: gerhardt@math.uni-heidelberg.de
URL: http://web.me.com/gerhardt/