Trace and extension theorems for BV-functions (pdf) - Uni Heidelberg

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

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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 (ˆ x, xn ) = u(ˆ x, 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

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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 ¯ converging in the BV sense to u, uhk ,k , or any other sequence uk ∈ C 1 (Ω) 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(ˆ x, xn ) xn > 0, (1.7) u ˜(ˆ x, xn ) = u(ˆ x, −xn ) xn < 0, there holds Z

Z |D˜ u| ≤ 2

(1.8) B1 (0)

|Du| Ω

and Z |D˜ u| = 0.

(1.9) ∂Ω

Proof. Let η i ∈ C 1 (B1 (0)), then Z Z Z Di u ˜η i = Di uη i + (1.10)

B1 (0)

B1−

B1+

Z

i

=−

Di u ˜η i +

Di u ˜η i

Γ

Z

+

+

i

Z

(u − u )η νi +

u ˜Di η + B1 (0)

Z

Γ

Di u ˜η i

Γ

hence Z

Di u ˜η 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 u ˜, Z Z (1.12) |D˜ u| ≤ c (|Du| + |u|) Rn

Ω

and Z |D˜ u| = 0.

(1.13) ∂Ω

TRACES AND EXTENSIONS OF BV -FUNCTIONS

3

¯ with the usual conditions for Proof. Let Uk be a finite open covering of Ω 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 u ˜k is the extended function, Z Z (1.14) |D˜ uk | ≤ c (|Du| + |u|). Rn

Ω

The function (1.15)

X

u ˜=

u ˜k

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 ¯ converging in the BV sense to u, at least in a boundary let uk ∈ C 1 (Ω) 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 ¯ then tr u. If u ∈ BV (Ω) ∩ C 0 (Ω), (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. ¯ approximations of u and let vk = uk − u Let uk and u ˜k be C 1 (Ω) ˜k , 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