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

Abstract

Let $\Omega$ be a domain of $\mathbb R^n$ with $n\ge 2$ and $p(\cdot)$ be a local Lipschitz funcion in $\Omega$ with $1<p(x)<\infty$ in $\Omega$. We build up an interior quantitative second order Sobolev regularity for the normalized $p(\cdot)$-Laplace equation $-\Delta^N_{p(\cdot)}u=0$ in $\Omega$ as well as the corresponding inhomogeneous equation $-\Delta^N_{p(\cdot)}u=f$ in $\Omega$ with $f\in C^0(\Omega)$. In particular, given any viscosity solution $u$ to $-\Delta^N_{p(\cdot)}u=0$ in $\Omega$, we prove the following: (i) in dimension $n=2$, for any subdomain $U\Subset\Omega$ and any $\beta\ge 0$, one has $|Du|^\beta Du\in L^{2+\delta}(U)$ locally with a quantitative upper bound, and moreover, the map $(x_1,x_2)\to |Du|^\beta(u_{x_1},-u_{x_2})$ is quasiregular in $U$ in the sense that $$|D[|Du|^\beta Du]|^2\leq -C\det D[|Du|^\beta Du] \quad \mbox{a.e. in $U$}.$$ (ii) in dimension $n\geq3$, for any subdomain $U\Subset\Omega$ with $ \inf_U p(x)>1$ and $\sup_Up(x)<3+\frac2{n-2}$, one has $D^2u\in L^{2+\delta}(U)$ locally with a quantitative upper bound, and also with a pointwise upper bound $$|D^2u|^2\le -C\sum_{1\leq i<j\le n}[u_{x_ix_j}u_{x_jx_i}-u_{x_ix_i}u_{x_jx_j}] \quad \mbox{a.e. in $U$}.$$ Here constants $\delta>0$ and $C\geq 1$ are independent of $u$. These extend the related results obtaind by Adamowicz-H\"ast\"o \cite{AH2010} when $n=2$ and $\beta=0$.