一类曲率依赖变分问题解的存在性与正则性
Existence and Regularity for a Curvature Dependent Variational Problem
摘要：
证明了存在使薛定谔算子$-\frac{d^2}{ds^2}+κ^2$的主特征值最小的给定长度的光滑闭曲线.这里，$s$表示弧长，$κ$表示曲率。这些极小化器自动地是平面的、解析的、凸的曲线。直线段，来回穿越，是唯一可能的例外，成为一个更广义的设置。在证明这一点时，我们克服了困难，从缺乏的简洁性和紧凑性的方法相结合。
英文摘要：
It is proved that smooth closed curves of given length minimizing the principal eigenvalue of the Schrödinger operator $-\frac{d^2}{ds^2}+κ^2$ exist. Here $s$ denotes the arclength and $κ$ the curvature. These minimizers are automatically planar, analytic, convex curves. The straight segment, traversed back and forth, is the only possible exception that becomes admissible in a more generalized setting. In proving this, we overcome the difficulty from a lack of coercivity and compactness by a combination of methods.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
