Lorentz-Minkowski空间中的稳态液滴
Stationary liquid drops in Lorentz-Minkowski space
摘要：
本文分析了Minkowski空间中类空液滴沉积在类空平面$Π$上的形状构形。我们假设存在一个指向$Π$的均匀重力场，并且液滴的体积是规定的。我们感兴趣的是，液滴，它们是，相应的力学系统的，能量的临界点，然后我们说，液滴是静止的。在这种情况下，液体-空气界面由以下条件确定：平均曲率是距$Π$的距离的线性函数，并且液滴与板$Π$形成恒定的双曲线接触角。作为第一个结果，我们将证明液滴必须是旋转对称的轴正交于$Π$。然后我们证明了对于给定的接触角与$Π$的对称解的存在性和唯一性。最后，我们将研究液滴根据其大小可以采用的形状。因此，我们将在$Π$上得到其高度、体积和湿表面面积的估计。
英文摘要：
This paper analyzes the configurations of shapes that shows a spacelike liquid drop in Minkowski space deposited over a spacelike plane $Π$. We assume the presence of a uniform gravity field directed toward $Π$ and that the volume of the drop is prescribed. Our interest are the liquid drops that are critical points of the energy of the corresponding mechanical system and we will say then that the liquid drop is stationary. In such case, the liquid-air interface is determined by the condition that the mean curvature is a linear function of distance from $Π$ and that the drop makes a constant hyperbolic angle of contact with the plate $Π$. As first result, we shall prove that the liquid drop must be rotational symmetric with respect to an axis orthogonal to $Π$. Then we prove the existence and uniqueness of symmetric solutions for a given angle of contact with $Π$. Finally, we shall study the shapes that a liquid drop can adopt in terms of its size. So, we shall derive estimates of its height, volume and area of the wetted surface on $Π$.
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