An isoperimetric inequality in the plane with a log-convex density

I. McGillivray

doi:10.1007/s11587-018-0382-z

An isoperimetric inequality in the plane with a log-convex density

I. McGillivray^*

^*Corresponding author for this work

Research output: Contribution to journal › Article (Academic Journal) › peer-review

6 Citations (Scopus)

366 Downloads (Pure)

Abstract

Given a positive lower semi-continuous density f on (Formula presented.) the weighted volume (Formula presented.) is defined on the (Formula presented.)-measurable sets in (Formula presented.). The f-weighted perimeter of a set of finite perimeter E in (Formula presented.) is written (Formula presented.). We study minimisers for the weighted isoperimetric problem (Formula presented.)for (Formula presented.). Suppose f takes the form (Formula presented.) where (Formula presented.) is a non-decreasing convex function. Let (Formula presented.) and B a centred ball in (Formula presented.) with (Formula presented.). We show that B is a minimiser for the above variational problem and obtain a uniqueness result.

Original language	English
Pages (from-to)	1-58
Number of pages	58
Journal	Ricerche di Matematica
Early online date	21 Mar 2018
DOIs	https://doi.org/10.1007/s11587-018-0382-z
Publication status	E-pub ahead of print - 21 Mar 2018

Keywords

Generalised mean curvature
Isoperimetric problem
Log-convex density

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AB - Given a positive lower semi-continuous density f on (Formula presented.) the weighted volume (Formula presented.) is defined on the (Formula presented.)-measurable sets in (Formula presented.). The f-weighted perimeter of a set of finite perimeter E in (Formula presented.) is written (Formula presented.). We study minimisers for the weighted isoperimetric problem (Formula presented.)for (Formula presented.). Suppose f takes the form (Formula presented.) where (Formula presented.) is a non-decreasing convex function. Let (Formula presented.) and B a centred ball in (Formula presented.) with (Formula presented.). We show that B is a minimiser for the above variational problem and obtain a uniqueness result.

KW - Generalised mean curvature

KW - Isoperimetric problem

KW - Log-convex density

UR - https://www.scopus.com/pages/publications/85044241344

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DO - 10.1007/s11587-018-0382-z

M3 - Article (Academic Journal)

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SN - 0035-5038

SP - 1

EP - 58

JO - Ricerche di Matematica

JF - Ricerche di Matematica

ER -

An isoperimetric inequality in the plane with a log-convex density

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