邊界接觸角給定下最小曲面在無界扇形域上的唯一定理

Abstract

我們考慮在無界扇形域上，給定邊界接觸角的最小曲面方程式。首先，我們對於方程式線性解的存在性，給出一個充要條件。其次，透過「吹」及「吸」兩種過程，我們研究方程式解在原點及無窮遠處的行為。最後，我們論述方程式解在邊界上是線性的，同時證明解是一個平面。

We consider the minimal surface equation in an infinite sector domain with given capillary boundary conditions.First, we give a necessary and sufficient conditions for the existence of the linear solution. Second, we study the behavior of the solutions of the minimal surface equation at the origin and at the infinite by using the blow up and the sip in process. Finally, we claim that the solution is linear on the boundary and conclude that it is a plane.