In this paper, we study the regularity of the free boundaries of the parabolic double obstacle problem for Laplacian and fully nonlinear operator. The result in this paper are generalizations of the theory for the elliptic problem in {LPS} and {LP} to parabolic case and also the theory for the parabolic single obstacle problem in {CPS} 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.

{CPS} Luis A. Caffarelli, Arshak Petrosyan, and Henrik Shahgholian, Regularity of a free boundary in parabolic potential theory, Journal of the American Mathematical Society, 17 (2004), no. 4, 827-869.
{LP} Ki-Ahm Lee, and Jinwan Park, The regularity theory for the double obstacle problem for fully nonlinear operator, arXiv preprint arXiv:1805.02806 (2018).
{LPS} Ki-ahm Lee, Jinwan Park, and Henrik Shahgholian, The regularity theory for the double obstacle problem, Calc. Var. Partial Differential Equations 58 (2019), no. 3, 104.