In this paper, we study the regularity of the free boundaries of the parabolic double obstacle problem for the heat operator and fully nonlinear operator. The result in this paper are generalizations of the theory for the elliptic problem in Lee et al. (Calc Var Partial Differ Equ 58(3):104, 2019) and Lee and Park (The regularity theory for the double obstacle problem for fully nonlinear operator, , 2018) to parabolic case and also the theory for the parabolic single obstacle problem in Caffarelli et al. (J Am Math Soc 17(4):827–869, 2004) to double obstacle case. New difficulties in the theory which are generated by the characteristic of parabolic PDEs and the existence of the upper obstacle are discussed in detail. Furthermore, the thickness assumptions to have the regularity of the free boundary are carefully considered.

中文翻译：

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抛物线双障碍问题的规律性理论

在本文中，我们研究了热算子和完全非线性算子的抛物线双障碍问题的自由边界的规律性。本文的结果是 Lee 等人的椭圆问题理论的推广。(Calc Var Partial Differ Equ 58(3):104, 2019) 和 Lee 和 Park（全非线性算子的双障碍问题的规律性理论，2018）抛物线情况以及抛物线单障碍问题的理论卡法雷利等人。（J Am Math Soc 17(4):827–869, 2004）双重障碍案例。详细讨论了由于抛物线偏微分方程的特点和上位障碍的存在而产生的理论新难点。此外，仔细考虑了具有自由边界规则性的厚度假设。