Multipolar potentials and weighted Hardy inequalities. (English) Zbl 1542.35016
Summary: In this paper we state the following weighted Hardy type inequality for any functions \(\varphi\) in a weighted Sobolev space and for weight functions \(\mu\) of a quite general type
\[
c_{N, \mu} \int_{\mathbb{R}^N}V\, \varphi^2\mu(x)dx\le \int_{\mathbb{R}^N}|\nabla \varphi|^2\mu(x)dx +C_\mu \int_{\mathbb{R}^N}W \varphi^2\mu(x)dx,
\]
where \(V\) is a multipolar potential and \(W\) is a bounded function from above depending on \(\mu \). Our method is based on introducing a suitable vector-valued function and an integral identity that we state in the paper. We prove that the constant \(c_{N, \mu}\) in the estimate is optimal by building a suitable sequence of functions.
MSC:
| 35A23 | Inequalities applied to PDEs involving derivatives, differential and integral operators, or integrals |
| 35K15 | Initial value problems for second-order parabolic equations |
| 35K65 | Degenerate parabolic equations |
| 35B25 | Singular perturbations in context of PDEs |
| 34G10 | Linear differential equations in abstract spaces |
| 47D03 | Groups and semigroups of linear operators |
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