We consider the extremal pointset configuration problem of maximizing a kernel-based energy subject to the geometric constraints that the points are contained in a fixed set, the pairwise distances are bounded below, and that every closed ball of fixed radius contains at least one point. We also formulate an extremal density problem, whose solution provides an upper bound for the pointset configuration problem in the limit as the number of points tends to infinity. Existence of solutions to both problems is established and the relationship between the parameters in the two problems is studied. Several examples are studied in detail, including the density problem for the $d$-dimensional ball and sphere, where the solution can be computed exactly using rearrangement inequalities. We develop a computational method for the density problem that is very similar to the Merriman-Bence-Osher (MBO) diffusion-generated method. The method is proven to be increasing for all non-stationary iterations and is applied to study more examples.

中文翻译：

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一个最大能量点集配置问题

我们考虑在几何约束条件下最大化基于内核的能量的极值点集配置问题，即点包含在固定集合中，成对距离在下方有界，并且每个固定半径的闭合球至少包含一个点。我们还制定了一个极值密度问题，它的解决方案为点集配置问题提供了一个上限，因为点的数量趋于无穷大。建立了两个问题的解的存在性，并研究了两个问题中参数之间的关系。详细研究了几个示例，包括 $d$ 维球和球体的密度问题，其中可以使用重排不等式精确计算解。我们为密度问题开发了一种计算方法，该方法与 Merriman-Bence-Osher (MBO) 扩散生成方法非常相似。该方法被证明对于所有非平稳迭代都是递增的，并被应用于研究更多的例子。