An improved Hardy-Sobolev inequality in W1,p and its application to Schrödinger operators

Adimurthi,; Esteban, Maria J.

doi:10.1007/s00030-005-0009-4

Cited by 79 publications

(143 citation statements)

References 8 publications

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“…For every u ∈ H 1 0 ( ) This inequality and its various improvements are used in many contexts, such as in the study of the stability of solutions of semilinear elliptic and parabolic equations (1,2), the analysis of the asymptotic behavior of the heat equation with singular potentials (3), as well as in the study of the stability of eigenvalues in elliptic problems such as Schrödinger operators (4). Now, it is well known that ( n−2 2 ) 2 is the best constant for the inequality shown as Eq.…”

mentioning

confidence: 99%

On the best possible remaining term in the Hardy inequality

Ghoussoub

Moradifam

2008

Proc. Natl. Acad. Sci. U.S.A.

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We give a necessary and sufficient condition on a radially symmetric potential V on a bounded domain of R n that makes it an admissible candidate for an improved Hardy inequality of the following type. For every u ∈ H 1 0 ( ) improved Hardy inequality | oscillatory behavior of ordinary differential equations L et be a bounded domain in R n , n ≥ 3, with 0 ∈ . The classical Hardy inequality asserts that for all u ∈ H 1 0 ( )This inequality and its various improvements are used in many contexts, such as in the study of the stability of solutions of semilinear elliptic and parabolic equations (1, 2), the analysis of the asymptotic behavior of the heat equation with singular potentials (3), as well as in the study of the stability of eigenvalues in elliptic problems such as Schrödinger operators (4). Now, it is well known that ( n−2 2 ) 2 is the best constant for the inequality shown as Eq. 1, and that this constant is, however, not attained in H 1 0 ( ). So, one could anticipate improving this inequality by adding a nonnegative correction term to the right-hand side of the inequality shown as Eq. 1, and indeed, several sharpened Hardy inequalities have been established in recent years (3, 5), mostly triggered by the following improvement of Brezis and Vázquez (1). For all u in H 1 0 ( ),The constant λ in Eq. 2 is given bywhere ω n and | | denote the volume of the unit ball and , respectively, and z 0 = 2.4048 . . . is the first zero of the Bessel function J 0 (z). Moreover, λ is optimal when is a ball but is-again-not achieved in H 1 0 ( ). This led to one of the open problems mentioned in ref. 1 (Problem 2), which is whether the two terms on the right-hand side of the inequality shown as Eq. 2 (i.e., the coefficients of |u| 2 ) are just the first two terms of an infinite series of correcting terms.This question was addressed by several authors. In particular, Adimurthi et al. (6) proved that for every integer k, there exists a constant c depending on n, k, and such that for all u ∈ H 1 0 ( ),dx, . e(k−times)). Here, we have used the notation log (1) (.) = log(.) and log (k) (.) = log(log (k−1) (.)) for k ≥ 2.Also motivated by the question of Brezis and Vázquez, Filippas and Tertikas proved in ref. 5 that the inequality can be repeatedly improved by adding to the right-hand side specific potentials that lead to an infinite series expansion of Hardy's inequality. More precisely, by defining iteratively the following functions,they prove that for any D ≥ sup x∈ |x|, the following inequality holds for any u ∈ H 1 0 ( ):Moreover, they proved that 1 4 is the best constant, which again is not attained in H 1 0 ( ). In this article, we show that all the above results-and morefollow from a specific characterization of those potentials V that yield an improved Hardy inequality. Here are our main results.

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confidence: 99%

On the best possible remaining term in the Hardy inequality

Ghoussoub

Moradifam

2008

Proc. Natl. Acad. Sci. U.S.A.

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“…The constant is well known to be the optimal one. For α = n in Lemma 2.1 we obtain (see also [3], [5] and [29,Lemma 17.4])…”

Section: Remark 22 the Classic Multidimensional Hardy Inequalitymentioning

confidence: 89%

“…More generally, in analogy with versions of Hardy's inequality for p = n, it has been extended to p = n ≥ 2 by ( [3], [5] & [7]), and can be stated as follows: If Ω is a bounded domain in R n ; n ≥ 2, then 5) with the best possible constant in case 0 ∈ Ω. If we define the Leray difference…”

Section: Hardy-sobolev Inequalitymentioning

confidence: 99%

A Leray–Trudinger inequality

Psaradakis

Spector

2015

Journal of Functional Analysis

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“…∈ Ω * (1) where · · denotes the scalar product in R . If Ω is convex or star-shaped domain with respect to some interior ball centered at zero then Ω satisfies (1), see [8, Section 1.1.8], so one can take Ω * = Ω.…”

Section: Introductionmentioning

confidence: 99%