Abstract
In this article, we prove that the following weighted Hardy inequality
holds with optimal Hardy constant \({\big(\frac{|d-p|}{p}\big)^{p}}\) for all \({u \in W^{1,p}_{\mu,0}(\Omega)}\) if the dimension d ≥ 2, 1 < p < d, and for all \({u \in W^{1,p}_{\mu,0}(\Omega{\setminus}\{0\})}\) if p > d ≥ 1. Here we assume that Ω is an open subset of \({\mathbb{R}^{d}}\) with \({0 \in \Omega}\) , A is a real d × d-symmetric positive definite matrix, c > 0, and
If p > d ≥ 1, then we can deduce from (1) a weighted Poincaré inequality on \({W^{1,p}_{\mu,0}(\Omega \setminus\{0\})}\) . Due to the optimality of the Hardy constant in (1), we can establish the nonexistence (locally in time) of positive weak solutions of a p-Kolmogorov parabolic equation perturbed by a singular potential in dimension d = 1, for 1 < p < + ∞, and when Ω = ]0, + ∞[.
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Both authors have been supported by the DAAD-MIUR grant, Project Vigoni-ID 54710868.
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Hauer, D., Rhandi, A. A weighted Hardy inequality and nonexistence of positive solutions. Arch. Math. 100, 273–287 (2013). https://doi.org/10.1007/s00013-013-0484-5
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DOI: https://doi.org/10.1007/s00013-013-0484-5