A Two-Phase Parabolic Free Boundary Problem with Coefficients Below the Lipschitz Threshold

Abstract

We study the regularity of a parabolic free boundary problem of two-phase type with coefficients below the Lipschitz threshold. For the Lipschitz coefficient case one can apply a monotonicity formula to prove the optimal \({C_x^{1,1}\cap C_t^{0,1}}\)-regularity of the solution and that the free boundary is, near the so-called branching points, the union of two graphs that are Lipschitz in time and C1 in space. In our case, the same monotonicity formula does not apply in the same way. Instead we use scaling arguments similar to the ones used for the elliptic case in Edquist et al. (Ann Inst Henri Poincaree, Anal Non Lineaire 26(6):2359–2372, 2009) to prove the optimal regularity. However, whenever the spatial gradient does not vanish on the free boundary, we are in the parabolic setting faced with some extra difficulties, that forces us to strain our assumptions slightly.