论文标题
分数半米和非局部最小表面的最小化剂
Minimisers of a fractional seminorm and nonlocal minimal surfaces
论文作者
论文摘要
最近的文献已经深入研究了两类的非局部变异问题，即与在适当的Sobolev-Gagliardo空间中起作用的能量功能的最小化相关的问题，这些功能与与分数周围的最小化相关的能量功能，这些空间的最小化作用于可测量的Euclideaneanean空间集合。 在本文中，我们将这两种类型的变分问题联系起来。特别是，我们研究了非本地最小表面与$ w^{s，1} $ seminorm的最小值之间的联系。 特别是，我们表明，当且仅当其水平集是分数周边的最小值时，函数是分数半米的最小化器，并且非局部最小的表面的特征函数是分数seminorm的最小值。我们还为分数半米的最小化剂提供了一个存在结果，这是非局部最小表面的明确的非唯一性示例，以及一个描述了非局部最小表面的完整和空隙模式的阴阳结果。
The recent literature has intensively studied two classes of nonlocal variational problems, namely the ones related to the minimisation of energy functionals that act on functions in suitable Sobolev-Gagliardo spaces, and the ones related to the minimisation of fractional perimeters that act on measurable sets of the Euclidean space. In this article, we relate these two types of variational problems. In particular, we investigate the connection between the nonlocal minimal surfaces and the minimisers of the $W^{s,1}$-seminorm. In particular, we show that a function is a minimiser for the fractional seminorm if and only if its level sets are minimisers for the fractional perimeter, and that the characteristic function of a nonlocal minimal surface is a minimiser for the fractional seminorm; we also provide an existence result for minimisers of the fractional seminorm, an explicit non-uniqueness example for nonlocal minimal surfaces, and a Yin-Yang result describing the full and void patterns of nonlocal minimal surfaces.
