Research on global properties of solutions of geometric variational problems

1.研究了欧氏三维空间中具有体积约束的浸没表面各向异性表面能的临界点。我们假设表面能满足一定的“凸性条件”。在这种情况下，能量称为常系数椭圆参数泛函，临界点是具有恒定各向异性平均曲率的曲面(CAMC曲面)。能量极小化是一个光滑的凸面，称为Wulff形(平移和齐性)。在各向异性表面能量为旋转不变的情况下，我们得到了以下结果：(1)证明了任何嵌入的闭CAMC曲面(直到平移和齐次)都是Wulff形的。(2)研究了支承在两个相隔一定距离的水平面上的旋转不变椭圆参数泛函的毛细曲面。当润湿能为非负值时，证明了稳定解的存在唯一性。我们还确定了解的几何性质。当润湿能为负时，我们得到了稳定解存在唯一性的一些判据。同时，我们也得到了一些关于这个问题的数值结果。我们得到了一种构造表面能量不一定旋转不变的CAMC曲面的例子的新方法。这不仅对CAMC表面的研究具有重要意义，而且对结晶学、数学生物学等其他领域的研究也具有重要意义。各向异性Delaunay曲面是具有常值各向异性平均曲率的旋转曲面。E说明了这种曲面的生成曲线是如何作为一条曲线上的点的轨迹来获得的，这条曲线是沿着一条直线滚动而不滑动的。这推广了Delaunay关于常平均曲率曲面的经典构造。此外，我们还利用各向异性Delaunay曲面的等温自对偶性刻画了各向异性Delaunay曲面。

1. We studied critical points of an anisotropic surface energy for immersed surfaces in the euclidean three-space with a volume constraint. We assume that the surface energy satisfies a certain "convexity condition". In this case, the energy is called a constant coefficient elliptic parametric functional, and the critical points are surfaces with constant anisotropic mean curvature (CAMC surfaces). The energy-minimizer is a smooth convex surface which is called the Wulff shape (up to translation and homothety). In the case where the anisotropic surface energy is rotationally invariant, we obtained the following results.(1) We proved that any embedded closed CAMC surface was (up to translation and homothety) the Wulff shape.(2) We studied capillary surfaces for certain rotationally invariant elliptic parametric functionals supported on two horizontal planes separated by a fixed distance. When the wetting energy is nonnegative, we proved the existence and uniqueness of the stable solution. Also, we determined the geometric property of the solution. When the wetting energy is negative, we obtained some criterions for the existence and uniqueness of the stable solution. Also, we obtained some numerical results on this problem.2. We obtained a new method of constructing examples of CAMC surfaces whose surface energy is not necessarily rotationally invariant. It will be useful not only for the study on CAMC surfaces but also for other fields such as crystallography, mathematical biology, and so on.3. Anisotropic Delaunay surfaces are surfaces of revolution with constant anisotropic mean curvature. e showed how the generating curves of such surfaces could be obtained as the trace of a point attached to a curve which was rolled without slipping along a line. This generalizes the classical construction for constant mean curvature surfaces due to Delaunay. Moreover, we characterize anisotropic Delaunay surfaces by using their isothermic self-duality.