A note on the supersolution method for Hardy’s inequality

Bianchi, Francesca; Brasco, Lorenzo; Sk, Firoj; Chiara, Zagati, Anna

doi:10.1007/s13163-023-00460-7

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A Maz'ya-type Poincaré Inequality1

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2023

2024

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Cited by 3 publications

(1 citation statement)

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“…see [7,Lemma 2.6]. If we now use (4.3) to bound the seminorm of ũ and the properties of 𝜂, from (4.5) we get…”

Section: A Maz'ya-type Poincaré Inequalitymentioning

confidence: 99%

An optimal lower bound in fractional spectral geometry for planar sets with topological constraints

Bianchi,

Brasco

2023

Journal of London Math Soc

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“…see [7,Lemma 2.6]. If we now use (4.3) to bound the seminorm of ũ and the properties of 𝜂, from (4.5) we get…”

Section: A Maz'ya-type Poincaré Inequalitymentioning

confidence: 99%

An optimal lower bound in fractional spectral geometry for planar sets with topological constraints

Bianchi,

Brasco

2023

Journal of London Math Soc

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On the sharp Hardy inequality in Sobolev–Slobodeckiĭ spaces

Bianchi,

Brasco,

Zagati

2023

Math. Ann.

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On fractional Hardy-type inequalities in general open sets

Cinti,

Prinari

2024

ESAIM: COCV

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We show that, when sp > N, the sharp Hardy constant hs,p of the punctured space ℝN \ {0} in the Sobolev–Slobodeckiĭ space provides an optimal lower bound for the Hardy constant hs,p(Ω) of an open set Ω ⊂ ℝN. The proof exploits the characterization of Hardy’s inequality in the fractional setting in terms of positive local weak supersolutions of the relevant Euler–Lagrange equation and relies on the construction of suitable supersolutions by means of the distance function from the boundary of Ω. Moreover, we compute the limit of hs,p as s ↗ 1, as well as the limit when p ↗ ∞. Finally, we apply our results to establish a lower bound for the non-local eigenvalue λs,p(Ω) in terms of hs,p when sp > N, which, in turn, gives an improved Cheeger inequality whose constant does not vanish as p ↗ ∞.

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A note on the supersolution method for Hardy’s inequality

Cited by 3 publications

References 40 publications

An optimal lower bound in fractional spectral geometry for planar sets with topological constraints

An optimal lower bound in fractional spectral geometry for planar sets with topological constraints

On the sharp Hardy inequality in Sobolev–Slobodeckiĭ spaces

On fractional Hardy-type inequalities in general open sets

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