论文标题
Mumford-shah功能最小化的单调公式，在2D中功能和能量密度的尖锐下限
A monotonicity formula for minimizers of the Mumford-Shah functional in 2d and a sharp lower bound on the energy density
论文作者
论文摘要
我们为在平面域中的Mumford-Shah功能的最小化器建立了一个新的单调公式。我们的公式遵循Bucur-Luckhaus的精神，但与David-Léger熵合作而不是能量。有趣的是，这允许截断阈值。特别是，我们的单调性公式能够在熵的有限间隙方面区分$ c^1 $接口和任何其他类型的奇异性的点。作为推论，我们证明了对Mumford-Shah功能最小化的任何非平滑点周围能量密度的最佳下限。
We establish a new monotonicity formula for minimizers of the Mumford-Shah functional in planar domains. Our formula follows the spirit of Bucur-Luckhaus, but works with the David-Léger entropy instead of the energy. Interestingly, this allows for a sharp truncation threshold. In particular, our monotonicity formula is able to discriminate between points at a $C^1$ interface and any other type of singularity in terms of a finite gap in the entropy. As a corollary, we prove an optimal lower bound on the energy density around any nonsmooth point for minimizers of the Mumford-Shah functional.
