Estimation optimale du gradient du semi-groupe de la chaleur sur le ...

498 J. Van Schaftingen / Journal of Functional Analysis 236 (2006) 490–516
For every s>0, the set {x ∈ Rn : ϕ(x) > s} is open and bounded. Moreover, by Sard’s lemma,
for almost every s>0, for every y ∈ ϕ−1 ({s}), ∇ϕ(y) = 0. Hence ∂{x ∈ Rn : ϕ>s} is smooth
and
∇u(x) dx
=
u(y)ν(y)dH n−1
(y)
{x∈R n : ϕ>s}
∂{x∈R n : ϕ(x)>s}
AH n−1 ∂ x ∈ R n : ϕ(x) > s .
A similar reasoning for s s + H n−1 ∂ x ∈ R n : ϕ(x) < −s ds
= A |∇ϕ| dx. ✷
2.5. Geometric characterization of Vn−1(R n )
Proposition 2.11. If n 2, for every u ∈ C(R n ),
uDn−1(R n ) = sup
γ ∈C 1 (S 1 ;R n )
1
˙γ u
L1 (S1 )
γ(t)
˙γ(t)dt
.
The proof repeats the argument of Bourgain and Brezis for the equivalence between the inequality
(1.7) and
u γ(t) ˙γ(t)dtCs,p˙γ L1 (S1 ) uWs,p (Rn ),
for every u ∈ W s,p (R n ) with sp = n [3].
Proof. First note that
S 1
uDn−1(R n ) = sup
f ∈D(R n ;R n )
div f =0
f L 1 (R n ;R n ) 1
S 1
uf d x
.
Let ρ ∈ D(B(0, 1)) be such that ρ 0 and
R n ρdx= 1, and let ρε(x) = ρ(x/ε)/ε n . Define
R n
1
fε(x) =
ρ
˙γ L1 (S1 )
γ(t)−x ˙γ(t)dt.
S 1