# Improved multipolar hardy inequalities

Cazacu C, Zuazua E (2013)

**Publication Type:** Book chapter / Article in edited volumes

**Publication year:** 2013

**Publisher:** Springer US

**Edited Volumes:** Studies in Phase Space Analysis with Applications to PDEs

**Series:** Progress in Nonlinear Differential Equations and Their Application

**Book Volume:** 84

**Pages Range:** 35-52

**ISBN:** 978-1-4614-6348-1

**DOI:** 10.1007/978-1-4614-6348-1_3

### Abstract

In this paper we prove optimal Hardy-type inequalities for Schrödinger operators with positive multi-singular inverse square potentials of the form (Formula presented.) More precisely, we show that A λ is nonnegative in the sense of L ^{2} quadratic forms in R^{N}, if and only if λ≤(N-2)^{2}/n^{2}, independently of the number n and location of the singularities xiϵR^{N}, where N ≥ 3 denotes the space dimension. This aims to complement some of the results in Bosi et al. (Comm. Pure Appl. Anal. 7:533–562, 2008) obtained by the “expansion of the square” method. Due to the interaction of poles, our optimal result provides a singular quadratic potential behaving like (n−1)(N −2)^{2}/(n^{2}|x−xi|^{2}) at each pole xi. Besides, the authors in Bosi et al. (Comm. Pure Appl. Anal. 7:533–562, 2008) showed optimal Hardy inequalities for Schrödinger operators with a finite number of singular poles of the type Bλ:=-Δ-Σi=1 ^{n}λ/|x-xi|^{2}, up to lower order L ^{2}-reminder terms. By means of the optimal results obtained for A λ, we also build some examples of bounded domains Ω in which these lower order terms can be removed in H 0 ^{1}(Ω). In this way we obtain new lower bounds for the optimal constant in the standard multi-singular Hardy inequality for the operator B λ in bounded domains. The best lower bounds are obtained when the singularities xi are located on the boundary of the domain.

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### How to cite

**APA:**

Cazacu, C., & Zuazua, E. (2013). Improved multipolar hardy inequalities. In Massimo Cicognani, Ferruccio Colombini, Daniele Del Santo (Eds.), *Studies in Phase Space Analysis with Applications to PDEs.* (pp. 35-52). Springer US.

**MLA:**

Cazacu, Cristian, and Enrique Zuazua. "Improved multipolar hardy inequalities." *Studies in Phase Space Analysis with Applications to PDEs.* Ed. Massimo Cicognani, Ferruccio Colombini, Daniele Del Santo, Springer US, 2013. 35-52.

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