论文标题
$ GSBV^p $和强烈各向异性的Mumford-Shah功能的各向异性庞加尼不平等
An anisotropic Poincaré inequality in $GSBV^p$ and the limit of strongly anisotropic Mumford-Shah functionals
论文作者
论文摘要
我们表明，在三维空间中，$ GSBV^p $中的功能在$ 2 $ 3 $的$ 3 $方向上的函数接近一个特殊集合以外的一个变量的函数。提供了卷和周长在这两个方向的两个方向上的边界。作为关键工具，我们证明了$ w^{1，p} $中功能的函数的近似结果。为此，我们提出了一个二维可计数的球结构，允许仔细删除功能的跳跃。作为直接应用，我们向一维模型展示了各向异性三维Mumford-Shah模型的$γ$ - 连接。
We show that functions in $GSBV^p$ in three-dimensional space with small variation in $2$ of $3$ directions are close to a function of one variable outside an exceptional set. Bounds on the volume and the perimeter in these two directions of the exceptional sets are provided. As a key tool we prove an approximation result for such functions by functions in $W^{1,p}$. For this we present a two-dimensional countable ball construction that allows to carefully remove the jumps of the function. As a direct application, we show $Γ$-convergence of an anisotropic three-dimensional Mumford-Shah model to a one-dimensional model.
