Isoperimetric bubbles in spaces with density $r^p + a$
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912286082334720 |
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| author | Gwynne, Martyn Cox, Simon |
| author_facet | Gwynne, Martyn Cox, Simon |
| contents | Least perimeter solutions for a region with fixed mass are sought in ${\mathbb{R}^d}$ on which a density function $ρ(r) = r^p+a$, with $p>0, a>0$, weights both perimeter and mass. On the real line ($d=1$) this is a single interval that includes the origin. For $p \le 1$ the isoperimetric interval has one end at the origin; for larger $p$ there is a critical value of $a$ above which the interval is symmetric about the origin. In the case $p=2$, for $d=2$ and $3$, the isoperimetric region is a circle or sphere, respectively, that includes the origin; the centre moves towards the origin as $a$ increases, with constant radius, and then remains centred on the origin for $a$ above the critical value as the radius decreases. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_17177 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Isoperimetric bubbles in spaces with density $r^p + a$ Gwynne, Martyn Cox, Simon Optimization and Control 28A75, 52B60 Least perimeter solutions for a region with fixed mass are sought in ${\mathbb{R}^d}$ on which a density function $ρ(r) = r^p+a$, with $p>0, a>0$, weights both perimeter and mass. On the real line ($d=1$) this is a single interval that includes the origin. For $p \le 1$ the isoperimetric interval has one end at the origin; for larger $p$ there is a critical value of $a$ above which the interval is symmetric about the origin. In the case $p=2$, for $d=2$ and $3$, the isoperimetric region is a circle or sphere, respectively, that includes the origin; the centre moves towards the origin as $a$ increases, with constant radius, and then remains centred on the origin for $a$ above the critical value as the radius decreases. |
| title | Isoperimetric bubbles in spaces with density $r^p + a$ |
| topic | Optimization and Control 28A75, 52B60 |
| url | https://arxiv.org/abs/2503.17177 |