发表机构
University of Münster（明斯特大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过紧致平移论证和全纯刚性陈述，完成了平面Mumford-Shah广义全局极小元的分类，并由此证明了平面内部正则性猜想。
AI 中文摘要
我们提出一个紧致平移论证，用于平面Mumford-Shah广义全局极小元的分类。对裂纹的紧致孤立部分进行两个相互垂直的平移，并在位移中独立松弛，使得一个全平面Hodge恒等式饱和。由此得到的法向条件蕴含一个全纯刚性陈述，从而排除该部分。已建立的分类定理随后仅留下常数、纯跳跃、三叉点和裂纹尖端模型。平面内部正则性猜想由该分类与局部正则性之间的已知等价性得出。对广义全局极小元不施加任何齐次性假设。
英文摘要
We prove the Mumford--Shah conjecture, formulated in 1989, by completing the classification of blow-up limits of minimizers in the plane. The remaining case concerns limits whose discontinuity set is noncompact and has connected complement. In this setting, we rule out bounded connected components using second-variation inequalities obtained from combined inner and outer variations associated with two perpendicular translations of a compact subset separated from the rest of the discontinuity set. Established results then imply that the discontinuity set is connected, and the known classification for connected discontinuity sets completes the proof.
