Improved Multipolar Hardy Inequalities
In this paper we prove optimal Hardy-type inequalities for Schrödinger operators with positive multi-singular inverse square potentials of the form $$A_{\lambda } := -\Delta- \lambda \displaystyle\sum _{1\leq i
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Improved Multipolar Hardy Inequalities Cristian Cazacu and Enrique Zuazua
Abstract In this paper we prove optimal Hardy-type inequalities for Schrödinger operators with positive multi-singular inverse square potentials of the form Aλ := −Δ − λ
|xi − x j |2 , 2 2 1≤i< j≤n |x − xi | |x − x j |
∑
λ > 0.
More precisely, we show that Aλ is nonnegative in the sense of L2 quadratic forms in RN , if and only if λ ≤ (N − 2)2 /n2 , independently of the number n and location of the singularities xi ∈ RN , 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 − ∑ni=1 λ /|x − xi |2 , up to lower order L2 -reminder terms. By means of the optimal results obtained for Aλ , we also build some examples of bounded domains Ω in which these lower order terms can be removed in H01 (Ω ). In this way C. Cazacu () BCAM–Basque Center for Applied Mathematics, Mazarredo, 14, E-48009 Bilbao, Basque Country, Spain Departamento de Matemáticas, Universidad Autónoma de Madrid, E-28049 Madrid, Spain e-mail: [email protected] E. Zuazua BCAM–Basque Center for Applied Mathematics, Mazarredo, 14, E-48009 Bilbao, Basque Country, Spain Ikerbasque, Basque Foundation for Science, Alameda Urquijo 36-5, Plaza Bizkaia, E-48011, Bilbao, Basque Country, Spain e-mail: [email protected] M. Cicognani et al. (eds.), Studies in Phase Space Analysis with Applications to PDEs, Progress in Nonlinear Differential Equations and Their Applications 84, DOI 10.1007/978-1-4614-6348-1__3, © Springer Science+Business Media New York 2013
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C. Cazacu and E. Zuazua
we obtain new lower bounds for the optimal constant in the standard multi-singular Hardy inequality for the operator Bλ in bounded domains. The best lower bounds are obtained when the singularities xi are located on the boundary of the domain. Key words: Hardy inequalities, Multipolar potentials, Schrödinger operators 2010 Mathematics Subject Classification: 35J10, 26D10, 46E35, 35Q40, 35J75.
3.1 Introduction This paper is concerned with a class of Schrödinger operators of the form −Δ +V (x) with multipolar Hardy-type singular potentials like V ∼ ∑i αi /|x − xi |2 , αi ∈ R, xi ∈ RN , N ≥ 3. The study of such singular potentials is motivated by applications to various fields as molecular physics [26], quantum cosmological models such as the Wheeler–DeWitt equation (see, e.g., [6]), and combustion models [21]. The singularity of inverse square potentials cannot be considered as a lower perturbation of the Laplacian since it has homogeneity -2, being critical from both a mathematical and a physical viewpoint. Pote
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