限制势中的导电平滴
Conducting flat drops in a confining potential
摘要：
我们研究了一个几何变分问题所产生的模拟二维带电滴的理想导电液体中存在的外部电位。我们的特点是连续包络的能量在一个参数测量的库仑相互作用的相对强度。因此，当势为限制势且库仑排斥强度小于一个临界值时，我们证明了体积约束极小解的存在性和部分正则性。我们还导出了由规则临界点所满足的欧拉-拉格朗日方程，该方程用电容势的正常$\frac12$-导数表示库仑能的一阶变分。
英文摘要：
We study a geometric variational problem arising from modeling two-dimensional charged drops of a perfectly conducting liquid in the presence of an external potential. We characterize the semicontinuous envelope of the energy in terms of a parameter measuring the relative strength of the Coulomb interaction. As a consequence, when the potential is confining and the Coulomb repulsion strength is below a critical value, we show existence and partial regularity of volume-constrained minimizers. We also derive the Euler--Lagrange equation satisfied by regular critical points, expressing the first variation of the Coulombic energy in terms of the normal $\frac12$-derivative of the capacitary potential.
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