Regularity of the Solutions for Parabolic Two-Phase Free Boundary Problems

Two‐Phase Free Boundary Problems for Parabolic Operators: Smoothness of the Front

“…In this paper we continue to develop the regularity theory of general two-phase free boundary problems for parabolic operators started in [11]. Specifically, let We assume that a ij 2 C 0;˛.…”

“…When L has the general form (1.1) the only known result is in [11], where we prove the Lipschitz continuity of the solution under the hypotheses that the free boundary is flat in space and that a nondegeneracy condition holds. Here we get rid of this nondegeneracy condition and prove directly that the free boundary is C 1 and that viscosity solutions are indeed classical solutions.…”

“…The plan of the paper is the following. In Section 2 we recall some results from [11] about the full monotonicity of u along the direction of a spacial cone x .e n ;  / of axis e n and opening  , its 0monotonicity and its full monotonicity 0 -away from the free boundary, along the direction of a space-time cone t .e n ;  t /, with D 2  t ı D 2 Â. Notice that this gives a control of D t u in terms of D n u, 0 -away from the free boundary.…”

“…Still we have 0 -monotonicity only, so that we need to decrease 0 . This can be done exactly as in [11] (Section 3) through the construction of another family of deformations; this is possible thanks to the progressive improvement of the cones opening.…”

“…In this section we collect for the reader's convenience some well-known Harnack principles and some of the results in [11] more or less continuously quoted in this paper.…”