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/n², 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/(n²|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 ⁿλ/|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.

Authors with CRIS profile

Enrique Zuazua Iriondo

Involved external institutions

Basque Center of Applied Mathematics (BCAM)

Spain (ES)

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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