The semilinear problem given by $$F(D^2 u, u):= \Delta u - f(u)=0,$$ where $f(u)$ is monotone-increasing and has a jump discontinuity for some values of $u$, can be seen as an unconstrained free boundary problem, where free boundary is given by the level surfaces for $u$ where $f(u)$ has a discontinuity. It was shown in [2] that solutions to this problem are $C^{1,1}$. It is tantalizing to analyze the case of fully nonlinear equations $F(D^2 u, u)=0$ with $F'_u \leq 0 $. Notice that in the particular case $F(D^2 u, u)=G(D^2u)-f(u)$ with $G$ convex and $f$ bounded, elliptic regularity gives that $D^2u$ belongs to BMO, so one only needs to understand the ``last step'' from BMO to $L^\infty$. ...

Define a general class of (double) gradient constrained problems for fully nonlinear equations as $$ \max\bigl\{\min \{ -F(D^2u,x) , |\nabla u -a|-h_1\}, |\nabla u-a|-h_2 \bigr\}=0\qquad \text{in $B_1$}, $$ where the equation is in the viscosity sense. Formally, the equation can be written as $$ F(D^2u,x)\chi_{\{|\nabla u-a| \neq h_1 \} \cup \{|\nabla u-a |\neq h_2\} }=0,\qquad h_1 \leq |\nabla u-a| \leq h_2. $$ Dropping the constraints, $h_1 \leq |\nabla u-a| \leq h_2$, one encounters an unconstrained problem. ...

Let $u$ solve $\Delta u+k^2u=0$ in $K$, with $u=0$, $|\nabla u|=1$, on $\partial K$, where the latter is a compact Lipschitz domain in $\mathbb R^3$ Then $K$ is a ball. ...

For $s$ a real number consider \begin{equation} \hbox{div}(A_s(x,u)\nabla u)= \hbox{div} ({\mathbf f}) \quad\text{in }B_1, \end{equation} where \begin{equation} A_s(x,u)= a(x) + b(x)(u^+)^s. \end{equation}

Here $a$ and $b$ satisfy \begin{equation} \lambda\leq a(x), b(x) \leq\frac{1}{\lambda}\quad (x\in\bar{B}_1), \end{equation} with a constant $0< \lambda< 1$. ...