Trace and extension theorems for BV-functions

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 1
References 4
1. Introduction
1.1. Definition. We say a sequence u k ∈ BV (Ω) converges in the BV
sense to u ∈ BV (Ω), if
∫
(1.1)
|u − u k | → 0
and
(1.2)
∫
Ω
Ω
∫
|Du k | →
Ω
|Du|.
1.2. Lemma. Let Ω = B 1 + (0) be the upper half ball and let u ∈ BV (Ω)
have compact support with respect to the spherical boundary. Define for 0 < h
(1.3) u h (ˆx, x n ) = u(ˆx, x n h),
then u h is defined in {x n > −h} and the mollification u h,ɛ with an even
mollifier is well defined in Ω if 0 < ɛ < ɛ 0 (h). There are sequences (h k , ɛ k )
converging to zero such that u hk ,ɛ k
converge in the BV sense to u.
Proof. Let η i ∈ Cc
∞ (Ω), ‖(η i )‖ ≤ 1, then
∫
∫
∫
(1.4)
u h,ɛ D i η i = uD i η−h,ɛ i ≤
if ɛ ≤ ɛ 0 (h), since
Ω
(1.5) η i −h,ɛ ∈ C ∞ c (Ω)
Ω
Ω
|Du|,
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 ɛ, where we may consider only η i 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
u hk ,ɛ k
, or any other sequence u k ∈ C 1 ( ¯Ω) converging in the BV sense to u,
define a unique trace of u, tr u, on the hyperplane such that the divergence
theorem holds
∫
∫ ∫
(1.6)
D i uη = − uD i η + uην i
Ω
Ω
∂Ω
for all η ∈ Cc 1 (Ω ɛ ∪ ∂Ω), cf. Remark 1.6.
Extending u as an even function to the lower half space
{
(1.7) ũ(ˆx, x n u(ˆx, x n ) x n > 0,
) =
u(ˆx, −x n ) x n < 0,
there holds
(1.8)
and
(1.9)
∫
B 1(0)
∫
∫
|Dũ| ≤ 2 |Du|
Ω
∂Ω
|Dũ| = 0.
Proof. Let η i ∈ C 1 (B 1 (0)), then
∫
∫
∫
D i ũη i = D i uη i +
(1.10)
hence
(1.11)
B 1(0)
B + 1
∫
= −
B 1(0)
∫
Γ
B − 1
∫
D i ũη i +
Γ
Γ
D i ũη i
∫
∫
ũD i η i + (u + − u + )η i ν i +
D i ũη i = 0.
Γ
D i ũη i
Moreover, if ‖(η i )‖ ≤ 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 R n with compact support such that, if the extended function is
called ũ,
∫ ∫
(1.12)
|Dũ| ≤ c (|Du| + |u|)
R n Ω
and
∫
(1.13)
|Dũ| = 0.
∂Ω

TRACES AND EXTENSIONS OF BV -FUNCTIONS 3
Proof. Let U k be a finite open covering of ¯Ω with the usual conditions for
the boundary neighbourhoods and let η k be a subordinate finite partition of
unity, then each function uη k can be extended to R n with compact support
such that, if ũ k is the extended function,
∫
∫
(1.14)
|Dũ k | ≤ c (|Du| + |u|).
R n Ω
The function
(1.15) ũ = ∑ k
ũ 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 u k ∈ C 1 ( ¯Ω) converging in the BV sense to u, at least in a boundary
neighbourhood, then tr u k converges in L 1 (∂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 ( ¯Ω), then
(1.16) tr u = u |∂Ω
Proof. It suffices to prove the uniqueness of the trace, since u can be extended
to R n , and hence also be approximated by mollifications.
Let u k and ũ k be C 1 ( ¯Ω) approximations of u and let v k = u k − ũ k , then
∫
(1.17) lim |v k | = 0,
Ω ɛ
where Ω ɛ is a boundary strip of width ɛ such that
∫
(1.18)
|Du| = 0
{d=ɛ}
and
(1.19) lim sup
∫
|Dv k | ≤ 2
Ω ɛ
∫
|Du|.
Ω ɛ
Now, there holds for any ɛ > 0
∫
∫
(1.20)
|v k | ≤ c |Dv k | + c ɛ |v k |,
∂Ω
∫Ω ɛ Ω ɛ
hence
∫
∫
(1.21) lim sup |v k | ≤ 2c lim sup
∂Ω
|Du| = 0,
Ω ɛ
where the last equality is due to the fact that |Du| is a bounded measure in
Ω ɛ0 and the Ω ɛ are monotone ordered such that
⋂
(1.22)
Ω ɛ = ∅.
0

4 CLAUS GERHARDT
□
1.6. Remark. By approximation we also obtain
∫
∫ ∫
(1.23)
D i uη = − uD i η + uην i
Ω
for all η ∈ C 1 c (Ω ɛ ∪ ∂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 = R n , namely,
∫
∫
(1.24)
|Du| = sup{ uD i η i : g ij 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/
∂Ω