1,784 research outputs found
A remark on the radial minimizer of the Ginzburg-Landau functional
Denote by the Ginzburg-Landau functional in the plane and let
be the radial solution to the Euler equation associated
to the problem . Let be a smooth, bounded domain with the same area as . Denoted by
we prove \min_{v \in \mathcal{K}} E_\varepsilon (v,\Omega)\le
E_\varepsilon (\tilde u_\varepsilon,B_1). $
Some sharp Hardy inequalities on spherically symmetric domains
We prove some sharp Hardy inequalities for domains with a spherical symmetry.
In particular, we prove an inequality for domains of the unit -dimensional
sphere with a point singularity, and an inequality for functions defined on the
half-space } vanishing on the hyperplane , with
singularity along the -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.Comment: 15 page
Weighted isoperimetric inequalities in cones and applications
This paper deals with weighted isoperimetric inequalities relative to cones
of . We study the structure of measures that admit as
isoperimetric sets the intersection of a cone with balls centered at the vertex
of the cone. For instance, in case that the cone is the half-space
and the measure is
factorized, we prove that this phenomenon occurs if and only if the measure has
the form , for some , . Our
results are then used to obtain isoperimetric estimates for Neumann eigenvalues
of a weighted Laplace-Beltrami operator on the sphere, sharp Hardy-type
inequalities for functions defined in a quarter space and, finally, via
symmetrization arguments, a comparison result for a class of degenerate PDE's
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