论文标题
$ \ mathbb {r}^1 $ with log-concave密度的等级$ n $ bubble问题的解决方案
Solution to the isoperimetric $n$-bubble problem on $\mathbb{R}^1$ with log-concave density
论文作者
论文摘要
我们在$ \ mathbb {r}^1 $上研究了类似的密度函数$ f $，从而影响了面积和周长的测量方式。我们检查了对称，径向增加并满足两个其他条件的密度函数：它们具有零密度（在原点），并且满足“ log-concavity”需求$ \ left [\ log f \ right]'\ leq 0 $。在这些条件下，我们发现等等$ n $ bubbles满足了常规结构，并且可以通过任意$ n $确定。这概括了对密度函数$ | x |^p $的最新工作，并且与没有这种常规结构的对数 - 串联密度函数形成鲜明对比。
We study the isoperimetric problem on $\mathbb{R}^1$ with a prescribed density function $f$ that affects how area and perimeter are measured. We examine density functions that are symmetric, radially increasing, and satisfy two additional conditions: they have a point of zero density (at the origin), and they satisfy a "log-concavity" requirement $\left[ \log f \right]'' \leq 0$. Under these conditions, we find that isoperimetric $n$-bubbles satisfy a regular structure and can be identified for arbitrary $n$. This generalizes recent work done on the density function $|x|^p$, and stands in contrast to log-convex density functions which have no such regular structure.
