On the isoperimetric problem with perimeter density  $r^p$

Gyula Csató

doi:10.3934/cpaa.2018129

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2018, Volume 17, Issue 6: 2729-2749. Doi: 10.3934/cpaa.2018129

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$L_p$

-Lipschitz theory for parabolic equations with time measurable pseudo-differential operators

On the isoperimetric problem with perimeter density $r^p$

Gyula Csató^,

Universitat Politècnica de Catalunya, member of BGSMath, Barcelona, Spain

* Corresponding author
* Corresponding author

Received: November 30, 2016

Revised: June 30, 2017

Published: June 2018

The author is supported by FONDECYT grant 11150017.

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Abstract

In this paper the author studies the isoperimetric problem in ${\mathbb{R}}^n$ with perimeter density $|x|^p$ and volume density 1. We settle completely the case $n = 2$ , completing a previous work by the author: we characterize the case of equality if $0≤p≤1$ and deal with the case $-∞<p<-1$ (with the additional assumption $0∈Ω$ ). In the case $n≥3$ we deal mainly with the case $-∞<p<0$ , showing among others that the results in 2 dimensions do not generalize for the range $-n+1<p<0.$

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Mathematics Subject Classification: Primary: 49Q10, 49Q20; Secondary: 26D10.

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Figure 1. Construction of $\Omega_i$ and $r_{i, j}$

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Figure 2. the domain $\Omega_{\epsilon}$

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