Improved multipolar hardy inequalities - Cazacu, Cristian; Zuazua, Enrique (Studies in Phase Space Analysis with Applications to PDEs, 2013)
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 RN, if and only if λ≤(N-2)2/n2, independently of the number n and location of the singularities xiϵRN, 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.
CRIS-Team Scopus Importer:2021-06-15
Keywords: Hardy inequalities; Multipolar potentials; Schrödinger operators