We give necessary and sufficient conditions on a pair of positive radial functions V and W on a ball B of radius R in R n , n ≥ 1, so that the following inequalities hold for all u ∈ C ∞ 0 (B):This characterization makes a very useful connection between Hardy-type inequalities and the oscillatory behaviour of certain ordinary differential equations, and helps in the identification of a large number of such couples (V, W )-that we call Bessel pairs-as well as the best constants in the corresponding inequalities. This allows us to improve, extend, and unify many results-old and new-about Hardy and HardyRellich type inequalities, such as those obtained by Caffarelli et al. (Compos Math 53:259-275, 1984), Brezis and Vázquez (Revista Mat. Univ. Complutense Madrid N.

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Motivated by problems arising in conductivity imaging, we prove existence, uniqueness, and comparison theorems -under certain sharp conditions -for minimizers of the general least gradient problemand ϕ(x, ξ) is a function that, among other properties, is convex and homogeneous of degree 1 with respect to the ξ variable. In particular we prove that if a ∈ C 1,1 (Ω) is bounded away from zero, then minimizers of the weighted least gradient problem inf u∈BV f Ω a|Du| are unique in BV f (Ω). We construct counterexamples to show that the regularity assumption a ∈ C 1,1 is sharp, in the sense that it can not be replaced by a ∈ C 1,α (Ω) with any α < 1.

We present a new proof of the sphere covering inequality in the spirit of comparison geometry, and as a byproduct we find another sphere covering inequality which can be viewed as the dual of the original one. We also prove sphere covering inequalities on surfaces satisfying general isoperimetric inequalities, and discuss their applications to elliptic equations with exponential nonlinearities in dimension two. The approach in this paper extends, improves, and unifies several inequalities about solutions of elliptic equations with exponential nonlinearities.

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.