We prove a general result about the behaviour of minimizing sequences for nonlocal shape functionals satisfying suitable structural assumptions. Typical examples include functions of the eigenvalues of the fractional Laplacian under homogeneous Dirichlet boundary conditions. Exploiting a nonlocal version of Lions' concentration-compactness principle, we prove that either an optimal shape exists, or there exists a minimizing sequence consisting of two "pieces" whose mutual distance tends to infinity. Our work is inspired by similar results obtained by Bucur in the local case.

中文翻译：

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非局部形状优化中的紧凑性和二分法

我们证明了关于满足合适结构假设的非局部形状函数的最小化序列行为的一般结果。典型示例包括齐次 Dirichlet 边界条件下分数阶拉普拉斯算子的特征值函数。利用 Lions 的浓度-紧凑性原理的非局部版本，我们证明要么存在最佳形状，要么存在由相互距离趋于无穷大的两个“块”组成的最小化序列。我们的工作受到 Bucur 在当地案例中获得的类似结果的启发。