Torsion functions and the Cheeger problem : a fractional approach.
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Data
2016
Título da Revista
ISSN da Revista
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Resumo
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.
Descrição
Palavras-chave
Fractional cheeger problem, Torsion functions, Fractional, Fractional p-Laplacian
Citação
BUENO, H. P. et al. Torsion functions and the Cheeger problem: a fractional approach. Advanced Nonlinear Studies, v. 16, p. 689-697, 2016. Disponível em: <https://www.degruyter.com/view/j/ans.2016.16.issue-4/ans-2015-5048/ans-2015-5048.xml>. Acesso em: 02 out. 2017.