弯曲能量的定量估计及其在非局部变分问题中的应用
Quantitative estimates for bending energies and applications to non-local variational problems
AI总结：
研究带电弹性液滴玩具模型的变分问题，通过弯曲能量定量估计证明小电荷下球或同心环为唯一极小化子，大电荷下不存在极小化子。
摘要：
我们讨论了一个变分模型，由周长、弯曲和 Riesz 相互作用能量的加权和给出，可以被视为带电弹性液滴的玩具模型。不同的贡献对强局部化和最大分散结构具有竞争偏好。我们研究了能量景观对“电荷”大小（即 Riesz 相互作用能量权重）的依赖性。在二维情形下，我们首先证明对于具有小弹性能量的单连通集，弹性亏缺控制等周亏缺。基于此结果，我们证明对于小电荷，完整变分模型的唯一极小化子是球或同心环。我们通过大电荷的不存在性结果补充了这些陈述。在三维情形下，我们证明了具有小 Willmore 能量构型的面积和直径界，并证明对于足够小的电荷，球是我们变分模型的唯一极小化子。
英文摘要：
We discuss a variational model, given by a weighted sum of perimeter, bending and Riesz interaction energies, that could be considered as a toy model for charged elastic drops. The different contributions have competing preferences for strongly localized and maximally dispersed structures. We investigate the energy landscape in dependence of the size of the 'charge', i.e. the weight of the Riesz interaction energy. In the two-dimensional case we first prove that for simply connected sets of small elastic energy, the elastic deficit controls the isoperimetric deficit. Building on this result, we show that for small charge the only minimizers of the full variational model are either balls or centered annuli. We complement these statements by a non-existence result for large charge. In three dimensions, we prove area and diameter bounds for configurations with small Willmore energy and show that balls are the unique minimizers of our variational model for sufficiently small charge.
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