论文标题
在限制潜力中进行平坦的滴滴
Conducting flat drops in a confining potential
论文作者
论文摘要
我们研究了在存在外部电势的情况下对完美传导液体的二维带电液滴进行建模引起的几何变异问题。我们根据测量库仑相互作用的相对强度的参数来表征能量的半连续包膜。结果，当电势限制并且库仑排斥强度低于临界价值时，我们表明了体积约束最小化器的存在和部分规律性。我们还通过常规临界点得出了Euler-Lagrange方程，从正常的$ \ frac12 $衍生电势来表达了库仑能量的第一个变化。
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.
