A note on one-dimensional symmetry for Hamilton-Jacobi equations with extremal Pucci operators and application to Bernstein type estimate

We prove a Liouville-type theorem that is one-dimensional symmetry and classification results for non-negative $L^q$-viscosity solutions of the equation \begin{equation*} -\mathcal{M}_{\lambda, \Lambda}^{\pm}(D^2u)\pm |Du|^p=0, x\in \mathbb{R}_+^n, \end{equation*} with boundary condition $u(\tilde{x},0)=M\geq 0, \tilde{x}\in \mathbb{R}^{n-1}$, where $\mathcal{M}_{\lambda, \Lambda}^{\pm}$ are the Pucci's operators with parameters $\lambda, \Lambda \in \mathbb{R}_+$ $0<\lambda\leq \Lambda$ and $p>1$. The results are an extension of the results by Porreta and Ver\'on in arXiv:0805.2533 for the case $p\in (1,2]$ and by o Filippucci, Pucci and Souplet in arXiv:1906.05161 for the case $p>2$, both for the Laplacian case (i.e. $\lambda=\Lambda=1$). As an application in the case $p>2$, we prove a sharp Bernstein estimation for $L^q$-viscosity solutions of the fully nonlinear equation \begin{equation*} -\mathcal{M}_{\lambda, \Lambda}^{\pm}(D^2u)= |Du|^p+f(x), \quad x\in \Omega, \label{ecuacion1} \end{equation*} with boundary condition $u=0$ on $\partial \Omega$, where $\Omega \subset \mathbb{R}^n$.