A general class of free boundary problems for fully nonlinear parabolic equations

Abstract

In this paper we consider the fully nonlinear parabolic free boundary problemwhere 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 1,1x ∩ 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 = 0}, and a uniform thickness assumption on the coincidence set {u = 0}, A.

An overview of unconstrained free boundary problems

“…As the right-hand side is bounded, by classical theory for fully nonlinear equations one deduces that u ∈ W 2,p (B 1 ) for any p > 1 (actually, D 2 u belongs to BMO, see for instance the appendix in [11]), hence u is twice differentiable a.e. We then note that if we set Ω := {ψ 1 < u < ψ 2 }, u solves…”

“…x ∩ C 0,1 t ) for the unconstrained obstacle problem with fully nonlinear operators has been recently shown in [11,16], and we expect that results valid in the elliptic setting should carry on also to the parabolic case.…”

“…Note that in example 1.1, although the solution is not C 1,1 , their second derivatives belong to BMO as a consequence of classical elliptic regularity (see, for instance, the appendix in [11] for a proof of BMO regularity for elliptic and parabolic fully nonlinear equations). Nevertheless, we do not expect this fact to be true in general for solutions to (1.3), and it would be interesting to understand under which assumptions such BMO regularity holds.…”

Higher order parabolic boundary Harnack inequality in <i>C</i>¹ and <i>C</i>^{<i>k</i>, <i>α</i>} domains

“…When |γ − α| p = β − 1 we have to split the difference into the spatial part and time part and on the time part use (11). When |γ − α| p = β , we directly use (10) to get the same estimate in other cases. Plugging it all in expression (9), we get…”

“…Let us stress, that one often does not have C 1,α initial regularity. Another example is the free boundary problem for fully non-linear parabolic equations, studied in [10]. There they establish the C 1 regularity of the free boundary in both space and time.…”