Some sharp Sobolev regularity for inhomogeneous $\infty$-Laplace equation in plane

Abstract

Suppose $\Omega ⋐{\mathbb{R}}^2$ and $f\in B{V}_{loc}(\Omega )\cap C^0(\Omega )$ with $|f|>0$ in $\Omega$. Let $u\in C^0(\Omega )$ be a viscosity solution to the inhomogeneous $\infty$-Laplace equation $$-{\Delta }_{\infty }u:=-\frac{1}{2}\sum _{i=1}^2(|Du{|}^2{)}_i{u}_i=-\sum _{i,j=1}^2{u}_i{u}_j{u}_{ij}=f\mathrm{in}\Omega .$$ The following are proved in this paper. (i) For $\alpha >3/2$, we have $|Du{|}^{\alpha }\in W_{loc}^{1,2}(\Omega )$, which is (asymptotic) sharp when $\alpha \to 3/2$. Indeed, the function $w(x_1,x_2)=-x_1^{4/3}$ is a viscosity solution to $-{\Delta }_{\infty }w=\frac{4^3}{3^4}$ in ${\mathbb{R}}^2$. For any $p>2$, $|Dw{|}^{\alpha }\notin W_{loc}^{1,p}({\mathbb{R}}^2)$ whenever $\alpha \in (3/2,3-3/p)$. (ii) For $\alpha \in (0,3/2]$ and $p\in [1,3/(3-\alpha ))$, we have $|Du{|}^{\alpha }\in W_{loc}^{1,p}(\Omega )$, which is sharp when $p\to 3/(3-\alpha )$. Indeed, $|Dw{|}^{\alpha }\notin W_{loc}^{1,3/(3-\alpha )}({\mathbb{R}}^2)$. (iii) For $\epsilon >0$, we have $|Du{|}^{-3+\epsilon }\in L_{loc}^1(\Omega )$, which is sharp when $\epsilon \to 0$. Indeed, $|Dw{|}^{-3}\notin L_{loc}^1({\mathbb{R}}^2)$. (iv) For $\alpha >0$, we have $$-(|Du|^α)_iu_i= 2α|Du|^{ α-2}f \ \mbox{ almost everywhere in $Ω$}.$$ Some quantative bounds are also given.