  This paper presents asymptotic formulas in the case of the following two
problems for the {\it Pucci's extremal operators} $\mathcal{M}^\pm$. It is
considered the solution $u^\varepsilon(x)$ of $-\varepsilon^2
\mathcal{M}^\pm\left(\nabla ^2 u^\varepsilon\right)+u^\varepsilon=0$ in
$\Omega$ such that $u^\varepsilon=1$ on $\Gamma$. Here, $\Omega\subset
\mathbb{R}^N$ is a domain (not necessarily bounded) and $\Gamma$ is its
boundary. It is also considered $v(x,t)$ the solution of $v_t -
\mathcal{M}^\pm\left(\nabla^2 v\right)=0$ in $\Omega\times (0,\infty)$, $v=1$
on $\Gamma\times(0,\infty)$ and $v=0$ on $\Omega\times \{0\}$. In the spirit of
their previous works, the authors establish the profiles as $\varepsilon$ or
$t\to 0^+$ of the values of $u^\varepsilon(x)$ and $v(x,t)$ as well as of those
of their $q$-means on balls touching $\Gamma$. The results represent a further
step in the extensions of those obtained by Varadhan and by Magnanini-Sakaguchi
in the linear regime.
