Boundary pointwise regularity and applications to the regularity of free boundaries

Abstract

In this paper, we develop a series of boundary pointwise regularity for Dirichlet problems and oblique derivative problems. As applications, we give direct and simple proofs of the higher regularity of the free boundaries in obstacle-type problems and one phase problems.

Boundary pointwise regularity for fully nonlinear parabolic equations and an application to regularity of free boundaries

“…One feature of the theorem is that u ∈ C 2,α (0) even for ∂Ω ∈ C 1,α (0), which is key to the higher regularity of free boundaries (see Theorem 1.30). This was first shown in [16] and the higher order counterpart was obtained in [17].…”

“…For instance, the interior pointwise C 2,α regularity has been used to prove the optimal C 1,1 regularity for solutions in obstacle-type problems (see [8, Proof of Theorem 1.2]). Based on the boundary pointwise C k,α (k ≥ 1) regularity, one can give a direct and short proof of the higher regularity of free boundaries (see [17] and Theorem 1.30). The Liouville theorem on cones can be deduced with the aid of the boundary pointwise regularity (see [13]).…”

“…In addition, we present that if the derivatives of u vanish at boundary, u possesses higher boundary regularity. It was first observed in [16] and used in [17] to prove the higher regularity of free boundaries in obstacle-type problems for the Poisson's equation. As an extension, we will adopt this idea to give a direct and short proof of the higher regularity of free boundaries for fully nonlinear parabolic equations.…”