Title: Two-phase transition problems for fully nonlinear parabolic equations of second order

Authors: Emmanouil Milakis

Issue: Volume 54 (2005), Issue 6, 1751-1768

Abstract: 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 $C^1$.