The goal of this work is to analyze a model for the rate-independent evolution of sets with finite perimeter. The evolution of the admissible sets is driven by that of (the complement of) a given time-dependent set, which has to include the admissible sets and hence is to be understood as an external loading. The process is driven by the competition between perimeter minimization and minimization of volume changes. In the mathematical modeling of this process, we distinguish the adhesive case, in which the constraint that the (complement of) the `external load' contains the evolving sets is penalized by a term contributing to the driving energy functional, from the brittle case, enforcing this constraint. The existence of Energetic solutions for the adhesive system is proved by passing to the limit in the associated time-incremental minimization scheme. In the brittle case, this time-discretization procedure gives rise to evolving sets satisfying the stability condition, but it remains an open problem to additionally deduce energy-dissipation balance in the time-continuous limit. This can be obtained under some suitable quantification of data. The properties of the brittle evolution law are illustrated by numerical examples in two space dimensions.

中文翻译：

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集的速率无关演化

这项工作的目的是分析一个具有有限周长的集合的速率无关的演化模型。允许集合的演化由给定时间相关集合（的补充）的演化所驱动，该时间依赖集合必须包括允许集合，因此应理解为外部负载。该过程是由周长最小化和体积变化最小化之间的竞争所驱动的。在此过程的数学建模中，我们将粘合情况与脆性情况区分开，在粘合情况中，“外部载荷”（的补充）包含正在演化的集合的约束受到对驱动能量功能起作用的项的惩罚。强制执行此约束。能量解决方案的存在通过在相关的时间增量最小化方案中达到极限，证明了粘合剂体系的适用性。在脆性情况下，这种时间离散过程产生了满足稳定性条件的演化集，但是在时间连续极限内进一步推导能量耗散平衡仍然是一个悬而未决的问题。这可以在适当的数据量化下获得。通过二维空间中的数值示例说明了脆性演化定律的性质。