The purpose of this paper is to analyze regularity properties of local solutions to free discontinuity problems characterized by the presence of multiple phases. The key feature of the problem is related to the way in which two neighboring phases interact: the contact is penalized at jump points, while no cost is assigned to no-jump interfaces which may occur at the zero level of the corresponding state functions. Our main results state that the phases are open and the jump set (globally considered for all the phases) is essentially closed and Ahlfors regular. The proof relies on a multiphase monotonicity formula and on a sharp collective Sobolev extension result for functions with disjoint supports on a sphere, which may be of independent interest.

本文的目的是分析以存在多相为特征的自由不连续问题的局部解的规律性。该问题的关键特征与两个相邻阶段相互作用的方式有关：接触在跳跃点处受到惩罚，而没有分配成本给可能发生在相应状态函数的零级的无跳跃界面。我们的主要结果表明阶段是开放的，跳跃集（全局考虑所有阶段）基本上是封闭的，Ahlfors 是规则的。该证明依赖于多相单调性公式和尖锐的集体 Sobolev 扩展结果，用于球体上具有不相交支撑的函数，这可能是独立的兴趣。