Title: Lattice tilings minimizing nonlocal perimeters Show Chinese title

Title: 格子铺砌最小化非局部周长

Abstract: We prove the existence of periodic tessellations of $\mathbb{R}^N$ minimizing a general nonlocal perimeter functional, defined as the interaction between a set and its complement through a nonnegative kernel, which we assume to be either integrable at the origin, or singular, with a fractional type singularity. We reformulate the optimal partition problem as an isoperimetric problem among fundamental domains associated with discrete subgroups of $\mathbb{R}^N$ , and we provide the existence of a solution by using suitable concentrated compactness type arguments and compactness results for lattices. Finally, we discuss the possible optimality of the hexagonal tessellation in the planar case. Show Chinese abstract

Abstract: 我们证明了$\mathbb{R}^N$的周期性铺砌的存在性，该铺砌最小化了一个一般的非局部周长泛函，该泛函定义为一个集合与其补集之间的相互作用，通过一个非负核来定义，我们假设该核在原点处可积，或者具有分数类型的奇异性。 我们将最优分割问题重新表述为与$\mathbb{R}^N$的离散子群相关的基本域之间的等周问题，并通过使用适当的集中紧性类型论证和格点的紧性结果来提供解的存在性。 最后，我们讨论了平面情况下六边形铺砌的可能最优性。