论文标题
Möbius-Plateau Energy下的固定曲线
Stationary curves under the Möbius-Plateau energy
论文作者
论文摘要
弹性边界能量的高原问题是最近的理论和应用的。但是，必须做出强有力的假设，以避免能量最小化期间边界曲线的自我干扰。我们引入了一类高原问题，以涉及自我抑制能量的边界，以消除能量最小化问题的自我接触。对于自我抑制的能量，我们选择了奥哈拉（O'Hara）引入的莫比乌斯能量，因为Freedman等人所显示的无数规律性。我们首先证明了这个Möbius-Plateau问题的存在定理，在封闭的Lipschitz曲线类别的类别是由浸入式碟片上跨越给定的不可还原结型的。然后，我们将注意力转移到螺旋条带的Möbius-Plateau变化上，这些变化基于边界螺旋的半径符号，它们被归类为“螺钉样”或“带状”。通过分析Euler-Lagrange方程，我们表明类似螺钉的溶液是丰富的，而丝带样溶液对其参数施加了强大的限制：它们必须具有高频（等效地，低螺距），与频率相比较薄，并且保持接近轴。
Plateau problems with elastic boundary energies have been of recent theoretical and applied interest. However, strong assumptions have to be made to avoid self-intersections of the boundary curve during energy minimization. We introduce a class of Plateau problems for boundaries with self-repulsive energies that obviates self-contact in energy minimization problems. For the self-repulsive energy, we choose the Möbius Energy introduced by O'Hara due to its myriad regularity properties shown by Freedman et al. We first prove an existence theorem for this Möbius-Plateau problem in the class of closed Lipschitz curves of a given irreducible knot-type spanned by immersed discs. We then turn our attention to Möbius-Plateau variations of helicoidal strips, which are classified as "screw-like" or "ribbon-like" based on the signs of the radii of the boundary helices. By analyzing the Euler-Lagrange equations, we show that screw-like solutions are plentiful, whilst ribbon-like solutions impose strong constraints on their parameters: they must have high frequency (equivalently, low pitch), thin width in comparison to the frequency, and remain close to the axis.
