A modular Poincaré-Wirtinger type inequality on Lipschitz domains for Sobolev spaces with variable exponents
arXiv (Cornell University)(2023)
摘要
In the context of Sobolev spaces with variable exponents,
Poincaré–Wirtinger inequalities are possible as soon as Luxemburg norms are
considered. On the other hand, modular versions of the inequalities in the
expected form
∫_Ω|f(x)-⟨ f⟩_Ω|^p(x) d
x⩽ C ∫_Ω|∇ f(x)|^p(x)d x,
are known to be false. As a result, all available
modular versions of the Poincaré- Wirtinger inequality in the
variable-exponent setting always contain extra terms that do not disappear in
the constant exponent case, preventing such inequalities from reducing to the
classical ones in the constant exponent setting. Our contribution is threefold.
First, we establish that a modular Poincaré–Wirtinger inequality
particularizing to the classical one in the constant exponent case is indeed
conceivable. We show that if Ω⊂ℝ^n is a bounded
Lipschitz domain, and if p∈ L^∞(Ω), p ≥ 1, then for every
f∈ C^∞(Ω̅) the following generalized Poincaré–Wirtinger
inequality holds
∫_Ω|f(x)-⟨ f⟩_Ω|^p(x) d
x≤ C ∫_Ω∫_Ω|∇ f(z)|^p(x)/|z-x|^n-1d zd x,
where ⟨ f⟩_Ω denotes the mean of f over Ω, and
C>0 is a positive constant depending only on Ω and
p_L^∞(Ω). Second, our argument is concise and constructive
and does not rely on compactness results. Third, we additionally provide
geometric information on the best Poincaré–Wirtinger constant on Lipschitz
domains.
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关键词
sobolev spaces,lipschitz domains,inequality,e-wirtinger
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