A quantitative second order Sobolev regularity for (inhmogeneous) normalized $p(\cdot )$-Laplace equations

Let $\mathrm{\Omega }$ be a domain of ${\mathbb{R}}^n$ with $n\ge 2$ and $p(\cdot )$ be a local Lipschitz funcion in $\mathrm{\Omega }$ with $1<p(x)<\mathrm{\infty }$ in $\mathrm{\Omega }$. We build up an interior quantitative second order Sobolev regularity for the normalized $p(\cdot )$-Laplace equation $-{\mathrm{\Delta }}_{p(\cdot )}^{N}u=0$ in $\mathrm{\Omega }$ as well as the corresponding inhomogeneous equation $-{\mathrm{\Delta }}_{p(\cdot )}^{N}u=f$ in $\mathrm{\Omega }$ with $f\in C^0(\mathrm{\Omega })$. In particular, given any viscosity solution $u$ to $-{\mathrm{\Delta }}_{p(\cdot )}^{N}u=0$ in $\mathrm{\Omega }$, we prove the following: (i) in dimension $n=2$, for any subdomain $U⋐\mathrm{\Omega }$ and any $\beta \ge 0$, one has $\mathrm{\mid }Du{\mathrm{\mid }}^{\beta }Du\in L^{2+\delta }(U)$ locally with a quantitative upper bound, and moreover, the map $(x_1,x_2)\to \mathrm{\mid }Du{\mathrm{\mid }}^{\beta }(u_{x_1},-u_{x_2})$ is quasiregular in $U$ in the sense that |D[|Du|^βDu]|^2\leq -C\det D[|Du|^βDu] \quad a.e. in $U$. (ii) in dimension $n\ge 3$, for any subdomain $U⋐\mathrm{\Omega }$ with ${\inf }_{U}p(x)>1$ and ${\sup }_{U}p(x)<3+\frac{2}{n-2}$, one has $D^{2}u\in L^{2+\delta }(U)$ locally with a quantitative upper bound, and also with a pointwise upper bound |D^2u|^2\le -C\sum_{1\leq i<j\le n}[u_{x_ix_j}u_{x_jx_i}-u_{x_ix_i}u_{x_jx_j}] \quad a.e. in $U$. Here constants $\delta >0$ and $C\ge 1$ are independent of $u$. These extend the related results obtaind by Adamowicz-Hästö \cite{AH2010} when $n=2$ and $\beta =0$.