Optimal Regularity for the parabolic No-Sign Obstacle Problem

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We study the parabolic free boundary problem of obstacle type u-∂ u/∂ t= fχᵤ≠ ₀. Under the condition that f=Hv for some function v with bounded second order spatial derivatives and bounded first order time derivative, we establish the same regularity for the solution u. Both the regularity and the assumptions are optimal. Using this result and assuming that f is Dini continuous, we prove that the free boundary is, near so called low energy points, a C¹ graph. Our result completes the theory for this type of problems for the heat operator.