(2016) Bueno, Hamilton Prado; Ercole, Grey; Macedo, Shirley da Silva; Pereira, Gilberto A.
Let Ω be a Lipschitz bounded domain of ℝN, N ≥ 2. The fractional Cheeger constant hs(Ω),
0 < s < 1, is defined by
hs(Ω) = inf
E⊂Ω
Ps(E)
|E|
, where Ps(E) = ∫
ℝN
∫
ℝN
|χE(x) − χE(y)|
|x − y|
N+s
dx dy,
with χE denoting the characteristic function of the smooth subdomain E. The main purpose of this paper is
to show that
lim
p→1
+
|ϕ
s
p
|
1−p
L∞(Ω)
= hs(Ω) = lim
p→1
+
|ϕ
s
p
|
1−p
L
1(Ω)
,
where ϕ
s
p
is the fractional (s, p)-torsion function of Ω, that is, the solution of the Dirichlet problem for the
fractional p-Laplacian: −(∆)
s
p u = 1 in Ω, u = 0 in ℝN \ Ω. For this, we derive suitable bounds for the first
eigenvalue λ
s
1,p
(Ω) of the fractional p-Laplacian operator in terms of ϕ
s
p
. We also show that ϕ
s
p minimizes the
(s, p)-Gagliardo seminorm in ℝN, among the functions normalized by the L
1
-norm.
