液滴模型中类Delaunay的紧致平衡态
Delaunay-like compact equilibria in the liquid drop model
AI总结：
针对液滴模型，本文提出了一类大体积下几何形状类似“珍珠项链”且接近常平均曲率 Delaunay unduloid 曲面的紧致嵌入非最小化平衡解。
摘要：
液滴模型由 Gamow 于 1928 年及 Bohr-Wheeler 于 1938 年引入，用于对原子核进行建模。该模型描述了将原子核维系在一起的表面张力与对应于质子间排斥的 Coulomb 力之间的竞争。更确切地说，该问题在于寻找 $\mathbb{R}^3$ 中的曲面 $Σ=\partial Ω$，使其在体积约束 $|Ω| = m$ 下成为能量\n$$\nE(Ω) = {\rm Per\,} (Ω) + \frac 12 \int_Ω\int_Ω\frac {dxdy}{|x-y|}\n$$\n的临界点。项 ${\rm Per\,} (Ω) $ 对应于 $Σ$ 的表面积。相关的 Euler-Lagrange 方程为\n$$\nH_Σ(x) + \int_{Ω} \frac {dy}{|x-y|} = λ\quad \hbox{ for all } x\in Σ, \quad\n$$\n其中 $H_Σ$ 表示曲面的平均曲率，$λ\in\mathbb{R}$ 为与约束 $|Ω|=m$ 相关的 Lagrange 乘子。包围体积为 $m$ 的球体的球面始终是解。对于足够小的 $m$，它们是最小化解。由于能量中的两项相互竞争，寻找非最小化解可能具有挑战性。我们发现了一类新的具有大体积的紧致、嵌入解，其几何形状类似于“珍珠项链”，其轴线位于一个大圆上，形状接近于常平均曲率的 Delaunay unduloid 曲面。此类平衡态的存在性并非显而易见，因为正如经典的 Alexandrov 结论所指出的，对于与之密切相关的常平均曲率问题 $H_Σ= λ$，唯一的紧致嵌入解是球面。
英文摘要：
The liquid drop model was introduced by Gamow in 1928 and Bohr-Wheeler in 1938 to model atomic nuclei. The model describes the competition between the surface tension, which keeps the nuclei together, and the Coulomb force, corresponding to repulsion among protons. More precisely, the problem consists of finding a surface $Σ=\partial Ω$ in $\mathbb{R}^3$ that is critical for the energy $$ E(Ω) = {\rm Per\,} (Ω) + \frac 12 \int_Ω\int_Ω\frac {dxdy}{|x-y|} $$ under the volume constraint $|Ω| = m$. The term ${\rm Per\,} (Ω) $ corresponds to the surface area of $Σ$. The associated Euler-Lagrange equation is $$ H_Σ(x) + \int_{Ω} \frac {dy}{|x-y|} = λ\quad \hbox{ for all } x\in Σ, \quad $$ where $H_Σ$ stands for the mean curvature of the surface, and where $λ\in\mathbb{R}$ is the Lagrange multiplier associated to the constraint $|Ω|=m$. Round spheres enclosing balls of volume $m$ are always solutions. They are minimizers for sufficiently small $m$. Since the two terms in the energy compete, finding non-minimizing solutions can be challenging. We find a new class of compact, embedded solutions with large volumes, whose geometry resembles a "pearl necklace" with an axis located on a large circle, with a shape close to a Delaunay's unduloid surface of constant mean curvature. The existence of such equilibria is not at all obvious, since for the closely related constant mean curvature problem $H_Σ= λ$, the only compact embedded solutions are spheres, as stated by the classical Alexandrov result.
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