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

Abstract

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

Theorems & Definitions (22)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Remark 1.1
• Remark 1.2
• Definition 2.1
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3