A General Class of Free Boundary Problems for Fully Nonlinear Parabolic Equations
نویسندگان
چکیده
In this paper we consider the fully nonlinear parabolic free boundary problem { F (Du)− ∂tu = 1 a.e. in Q1 ∩ Ω |Du|+ |∂tu| ≤ K a.e. in Q1 \ Ω, where K > 0 is a positive constant, and Ω is an (unknown) open set. Our main result is the optimal regularity for solutions to this problem: namely, we prove that W 2,n x ∩W 1,n t solutions are locally C 1,1 x ∩C 0,1 t inside Q1. A key starting point for this result is a new BMO-type estimate which extends to the parabolic setting the main result in [4]. Once optimal regularity for u is obtained, we also show regularity for the free boundary ∂Ω ∩ Q1 under the extra condition that Ω ⊃ {u 6= 0}, and a uniform thickness assumption on the coincidence set {u = 0},
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