论文标题
不规则曲线的弹性能弱
Weak elastic energy of irregular curves
论文作者
论文摘要
提出了在任何空间维度中（不一定是规则的）可整流曲线的弹性能量较弱的概念。我们的$ p $能量是通过放松过程来定义的，其中采用了合适的$ p $ rotation。我们选择的离散$ p $ rotation具有几何味道：多边形被视为平滑曲线的近似值，因此其离散曲率分布成平滑的密度。对于任何大于1的指数$ p $，$ p $ emgy是有限的，并且仅当曲线的弧长参数化具有相同的增长指数的二阶参数化。此外，在这种情况下，能源与标量曲率的$ p $ TH幂的积分的自然扩展一致。最后，讨论了与离散曲线的其他定义进行比较。
A weak notion of elastic energy for (not necessarily regular) rectifiable curves in any space dimension is proposed. Our $p$-energy is defined through a relaxation process, where a suitable $p$-rotation of inscribed polygonals is adopted. The discrete $p$-rotation we choose has a geometric flavor: a polygonal is viewed as an approximation to a smooth curve and hence its discrete curvature is spread out into a smooth density. For any exponent $p$ greater than one, the $p$-energy is finite if and only if the arc-length parameterization of the curve has a second order summability with the same growth exponent. In that case, moreover, the energy agrees with the natural extension of the integral of the $p$-th power of the scalar curvature. Finally, a comparison with other definitions of discrete curvatures is discussed.
