Optimizing Improved Hardy Inequalities

Filippas, Stathis; Tertikas, Achilles

doi:10.1006/jfan.2001.3900

Cited by 170 publications

(192 citation statements)

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“…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 ( ),…”

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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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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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“…In analogy with the results of [10] and [16], it is then natural to ask whether one can make subcritical or critical Sobolev improvements to the inequality (1.5) via finding lower bounds for I n [u; Ω]. Here we enter into another type of criticality, where the Sobolev critical exponent is formally +∞.…”

Section: Hardy-sobolev Inequalitymentioning

confidence: 98%

“…Actually, it is well known that I[u; Ω] > 0 if u ∈ W 1,2 0 (Ω)\{0}, which suggests the possibility of improving the inequality (1.1) in the form of lower bounds for I[u; Ω]. While for Ω = R n it has been shown that additional correction terms cannot be added (see for example [16], [19] & [14]), if Ω has finite volume such an improvement is possible. The following subcritical Sobolev improvement of the Hardy inequality (1.1) is due to Brezis and Vazquez [10]: If Ω has finite volume, then for any 1 ≤ q < 2 * , there exists C n,q > 0 depending only on n and q, such that I[u; Ω] 1/2 ≥ C n.q vol(Ω) 1/q−1/2 * Ω |u| q dx 1/q for all u ∈ C ∞ c (Ω).…”

Section: Introductionmentioning

confidence: 99%

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A Leray–Trudinger inequality

Psaradakis

Spector

2015

Journal of Functional Analysis

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“…More recently, it was observed by Brezis and Vázquez [8] and others [24] that the inequality can be improved. The story here is the link-discovered by Ghoussoub and Moradifam [26,27]-between various improvements of this inequality confined to bounded domains and Sturm's theory regarding the oscillatory behavior of certain linear ordinary equations.…”

Section: To See Thatmentioning

confidence: 99%

Sobolev inequalities for the Hardy–Schrödinger operator: extremals and critical dimensions

Ghoussoub

Robert

2015

Bull. Math. Sci.

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In this survey paper, we consider variational problems involving the HardySchrödinger operator L γ := − − γ |x| 2 on a smooth domain of R n with 0 ∈ , and illustrate how the location of the singularity 0, be it in the interior of or on its boundary, affects its analytical properties. We compare the two settings by considering the optimal Hardy, Sobolev, and the Caffarelli-Kohn-Nirenberg inequalities. The latter can be stated as:where γ < n 2 4 , s ∈ [0, 2) and 2 (s) := 2(n−s) n−2 . We address questions regarding the explicit values of the optimal constant C := μ γ,s ( ), as well as the existence of non-trivial extremals attached to these inequalities. Scale invariance properties often lead to situations where the best constants μ γ,s ( ) do not depend on the domain, and hence they are not attainable. We consider two different approaches for "breaking the homogeneity" of the problem, and restoring compactness. One approach was initiated by Brezis-Nirenberg, when γ = 0 and s = 0, and by Janelli, when γ > 0 and s = 0. It is suitable for the case where the singularity 0 is in the interior of , and consists of considering lower order perturbations of the critical nonlinearity. The other approach was initiated by Ghoussoub-Kang for γ = 0, s > 0, and by C.S. Lin et al. and Ghoussoub-Robert, when γ = 0, s ≥ 0. It consists of considering domains, where the singularity 0 is on the boundary. Both of these approaches are rich in structure and in challenging problems. If 0 ∈ , then a negative linear perturbation suffices for higher dimensions, while a positive "Hardy-singular interior mass" theorem for the operator L γ is required in lower dimensions. If the singularity 0 belongs to the boundary ∂ , then the local geometry around 0 (convexity and mean curvature) plays a crucial role in high dimensions, while a positive "Hardy-singular boundary mass" theorem is needed for the lower dimensions. Each case leads to a distinct notion of critical dimension for the operator L γ .

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Optimizing Improved Hardy Inequalities

Cited by 170 publications

References 10 publications

On the best possible remaining term in the Hardy inequality

On the best possible remaining term in the Hardy inequality

A Leray–Trudinger inequality

Sobolev inequalities for the Hardy–Schrödinger operator: extremals and critical dimensions

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