Some sharp Hardy inequalities on spherically symmetric domains

Details Some sharp Hardy inequalities on spherically symmetric domains Some sharp Hardy inequalities on spherically symmetric domains

2008-07-29

arxiv.org/abs/0807.4692v1

Mathematics

Analysis of PDEs

15 pages

Scientific paper

We prove some sharp Hardy inequalities for domains with a spherical symmetry. In particular, we prove an inequality for domains of the unit $n$-dimensional sphere with a point singularity, and an inequality for functions defined on the half-space $\R_+^{n+1}$} vanishing on the hyperplane $\{x_{n+1}=0\}$, with singularity along the $x_{n+1}$-axis. The proofs rely on a one-dimensional Hardy inequality involving a weight function related to the volume element on the sphere, as well as on symmetrization arguments. The one-dimensional inequality is derived in a general form.

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