Regularity of solutions to fully nonlinear elliptic and parabolic free boundary problems

Abstract

We consider fully nonlinear obstacle-type problems of the formwhere Ω is an unknown open set and K > 0. In particular, structural conditions on F are presented which ensure that W 2,n (B1) solutions achieve the optimal C 1,1 (B 1/2 ) regularity when f is Hölder continuous. Moreover, if f is positive on B1, Lipschitz continuous, and {u = 0} ⊂ Ω, then we obtain local C 1 regularity of the free boundary under a uniform thickness assumption on {u = 0}. Lastly, we extend these results to the parabolic setting.

An overview of unconstrained free boundary problems

“…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.…”

“…Indeed, if N 2 denotes the δ/4-neighbourhood of B 1 \ Ω 2 , then u 2 is uniformly C 2,α in B 3/4 \ N 2 because it solves a nice PDE in a δ/2-neighbourhood of this set, with bounded boundary data. In particular, in B 3/4 \ N 2 , u 1 is a solution to a free boundary problem of the type studied in [6,16], hence u 1 is C 1,1 loc there. As u 1 also solves a nice PDE in a δ/2-neighbourhood of N 2 , we conclude that u 1 is uniformly C 1,1 in B 1/2 .…”

The regularity theory for the double obstacle problem

“…We note that for the general case (FB nosign local ), with f ∈ C 0,α and ψ ∈ C 2,α , the optimal regularity of the solutions are obtained by the theory in [IM16a] (see [FS14,IM16a] and Theorem 2.1 of [LPS] for more detail).…”

Non-transversal intersection of the free and fixed boundary in the mean-field theory of superconductivity

“…for g ∈ C ∞ (∂B 1 ) which belong to C 1,α \ C 1,1 for all α ∈ (0, 1) [AW06]. Up to the boundary and interior C 1,1 estimates were obtained in [IM16a] and [IM16b,FS14], respectively, for more general versions of (1.1) (valid also in higher dimensions; see also [LJH] for applications to double obstacle problems).…”