This paper studies the fully nonlinear free boundary problem $$ F(D^2u)=1 \ \text{a.e. in }B_1 \cap \Omega \qquad \hbox{and }\qquad |D^2 u| \leq K , \ \text{a.e. in }B_1\setminus\Omega, $$ where $K>0$, and $\Omega$ is an unknown open set. The main result is the optimal regularity for solutions to this problem: namely, we prove that $W^{2,n}$ solutions are locally $C^{1,1}$ inside $B_1$. Under the extra condition that $ {\Omega \supset \{D{u} \neq 0 \}}$, and a uniform thickness assumption on the coincidence set $\{Du=0\}$. It is also shown a local regularity for the free boundary $\partial \Omega \cap B_1$.

For a small parameter $ \varepsilon>0 $ let $ u_\varepsilon $ be the solution of the following elliptic system with a Dirichlet boundary condition $$ -\nabla \cdot A(x) \nabla u_\varepsilon (x) =0 \text{ in } D \ \ \text{ and } \ \ u_\varepsilon(x) = g(x/ \varepsilon) \text{ on } \partial D, $$ where $ D \subset \mathbb{R}^d $ $ (d\geq 2) $ is a bounded domain, $g$ is $ \mathbb{Z}^d $-periodic, and the operator is uniformly elliptic. Let also $ u_0 $ be the solution to the same elliptic system in $D$ but with Dirichlet data equal to the average of $g$ over its unit cell of periodicity. The main result of the paper states that under strict convexity of the domain $D$, and smoothness of the data involved in the problem, for any $ \kappa>0 $ one has the following pointwise bound $$ | u_\varepsilon(x) - u_0(x) | \leq C_\kappa \min\left\{ 1, \frac{\varepsilon^{(d-1)/2}}{(d(x))^{d-1+\kappa} } \right\}, \qquad \forall x\in D, $$ where $ d(x) $ denotes the distance of $ x $ from the boundary of $D$, and the constant $ C_\kappa =C(\kappa, d, A, D,g)>0 $. As a corollary to the pointwise bound, for any $ 1\leq p< \infty $ and any $ \kappa>0 $ we easily obtain $$ || u_\varepsilon - u_0 ||_{L^p(D)} \leq C_\kappa \varepsilon^{ \frac{1}{2p} - \kappa}. $$

The proofs are based on the analysis of singular oscillatory integrals, which enter the discussion through Poisson representation of solutions. There are two competing quantities in this representation; namely, the singularity of the Poisson kernel, and the oscillation of the boundary data. The smoothness of $A$ and $ \partial D $ allows us to obtain a nice (quantitative) control over singularities of the representation kernel and its derivatives. Next, the periodicity and smoothness of the boundary data $ g $, along with the strict convexity of the domain (which ensures non-zero Gauss curvature everywhere on the boundary) provide a lot of cancellations in the integral representing the solutions. One then does a careful trade-off between singularity of the integration kernel, and decay of the integral of the boundary data to get the desired estimate.

It is worthwhile to remark, that with some modification, the proposed approach leads to homogenization when instead of strict convexity of the domain, one requires that at each point of the boundary at least $1\leq m \leq d-1$ of the principal curvatures are non-vanishing (strict convexity corresponds to the case when $ m=d-1 $ ). The pointwise estimate in this case wll be similar, with $ d-1 $ in the exponent replaced by $m$, while the $L^p$-estimates will remain the same.