The Hardy Inequality and the Asymptotic Behaviour of the Heat Equation with an Inverse-Square Potential

Vázquez, Juan Luís; Zuazua, Enrique

doi:10.1006/jfan.1999.3556

Cited by 369 publications

(357 citation statements)

References 17 publications

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“…For example, one can derive various Sobolev embedding theorems in the L p setting, which can then be used to prove the existence of solutions of the Cauchy problem for elliptic and parabolic PDEs (see e.g. [9,22,36,15,44,55,39]), to study the asymptotic behaviour of solutions [2,57], as well as their stability [10,11]. They are present in probability theory (see e.g.…”

Section: Introductionmentioning

confidence: 99%

On a variant of the Hardy inequality between weighted Orlicz spaces

Kałamajska

Pietruska-Pałuba

2009

Studia Math.

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Abstract. Let M be an N -function satisfying the ∆2-condition, and let ω, ϕ be two other functions, with ω ≥ 0. We study Hardy-type inequalitieswhere u belongs to some set R of locally absolutely continuous functions containing C ∞ 0 (R+). We give sufficient conditions on the triple (ω, ϕ, M ) for such inequalities to be valid for all u from a given set R. The set R may be smaller than the set of Hardy transforms. Bounds for constants are also given, yielding classical Hardy inequalities with best constants.

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Section: Introductionmentioning

confidence: 99%

On a variant of the Hardy inequality between weighted Orlicz spaces

Kałamajska

Pietruska-Pałuba

2009

Studia Math.

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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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“…Our approach is quite different from that in [5,14], in some sense simpler and easier to be adapted for the weighted versions. Following the idea used in [8], we convert the problem from R N to one defined on a cylinder C ¼ R Â S NÀ1 : From there an inequality similar to the classical one-dimensional Hardy inequality on ð0; NÞ is used to tackle the technical part of the proof.…”

Section: Introductionmentioning

confidence: 96%