足够宽的带状区域唯一极小化平面周期分数阶周长
Sufficiently wide strips uniquely minimize the planar periodic fractional perimeter
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中文总结 AI 辅助
该论文研究平面周期分数阶周长的极小化问题，证明足够宽的带状区域是唯一极小化者，并通过定量下界处理凹性困难，适用于任意非负可积剖面。
中文摘要 AI 辅助
我们研究了具有给定水平周期和每周期面积的平面集合的每周期分数阶周长。我们证明了，在所有可测竞争者中，足够宽的带状区域是唯一极小化者，且唯一性成立至垂直平移和零测集。对于周期为1的情形，半宽度至少为$14$即可对所有分数阶指数$s\in(0,1)$成立。证明首先将每个垂直截面替换为具有相同长度的居中区间。然后，我们建立了周长超出量的定量下界，该下界用所得半宽度剖面与其均值的偏差来表示。主要困难在于带状区域的周长是其宽度的凹函数。我们证明，在平均宽度较大时，不等截面的相互作用成本支配了相应的凹性亏空，并对每个非常数剖面留下正余量。对小水平间距和大水平间距的估计使得充分宽度界不依赖于$s$。该论证适用于任意非负可积剖面，包括无界剖面和在正测度集上取零值的剖面。
英文摘要
We study the fractional perimeter per period of planar sets with prescribed horizontal period and area per period. We prove that sufficiently wide strips are the unique minimizers among all measurable competitors, up to vertical translation and null sets. For period one, a half-width of at least $14$ suffices for every fractional exponent $s\in(0,1)$. The proof first replaces each vertical section by a centered interval of the same length. We then establish a quantitative lower bound for the perimeter excess in terms of the deviation of the resulting half-width profile from its mean. The main difficulty is that the perimeter of a strip is a concave function of its width. We show that, at large mean width, the interaction cost of unequal sections dominates the corresponding concavity deficit and leaves a positive remainder for every nonconstant profile. Estimates at small and large horizontal separations make the sufficient width bound independent of $s$. The argument applies to arbitrary nonnegative integrable profiles, including unbounded profiles and profiles that vanish on sets of positive measure.
发表机构
- Dong-A University（东亚大学）
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