Two-phase transition problems for fully nonlinear parabolic equations of second order Emmanouil Milakis 35R3535K55free boundary problemsregularityfully nonlinear equationsnon-cylindrical domains In this paper we study an extension of a regularity theory presented by I. Athanasopoulos, L. Caffarelli and S. Salsa in "Caloric functions in Lipschitz domains and the regularity of solutions to phase transition problems" (Ann. of Math. (2) 143 Number 3 (1996), 413--434) and in "Regularity of the free boundary in parabolic phase-transition problems" (Acta Math. 176 Number 2 (1996), 245--282), to some fully nonlinear parabolic equations of second order. We investigate a two-phase free boundary problem in which a fully nonlinear parabolic equation is verified by the solution in the positive and the negative domain. We prove that the solution is Lipschitz up to the Lipschitz free boundary and that Lipschitz free boundaries are $C^1$. Indiana University Mathematics Journal 2005 text pdf 10.1512/iumj.2005.54.2623 10.1512/iumj.2005.54.2623 en Indiana Univ. Math. J. 54 (2005) 1751 - 1768 state-of-the-art mathematics http://iumj.org/access/