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

TL;DR

This paper investigates the Sobolev regularity of solutions to the inhomogeneous infinity Laplace equation in the plane, establishing sharp conditions for the integrability and differentiability of powers of the gradient.

Contribution

It provides sharp regularity results for the gradient of solutions, including precise thresholds for Sobolev space membership, extending understanding of inhomogeneous infinity Laplace equations.

Findings

$|Du|^{eta} otin W^{1,p}_{loc}$ for certain $eta,p$ thresholds

$|Du|^{-3+ ext{small}}$ is integrable, sharp at $eta=-3$

Explicit formula for divergence of powers of $|Du|$ in terms of $f$

Abstract

Suppose $\Omega\Subset \mathbb R^2$ and $f\in BV_{loc}(\Omega)\cap C^0(\Omega)$ with $|f|>0$ in $\Omega$. Let $u\in C^0(\Omega)$ be a viscosity solution to the inhomogeneous $\infty$-Laplace equation $$ -\Delta_{\infty} u :=-\frac12\sum_{i=1}^2(|Du|^2)_iu_i= -\sum_{i,j=1}^2u_iu_ju_{ij} =f \quad {\rm in}\ \Omega. $$ The following are proved in this paper. (i) For $ \alpha > 3/2$, we have $|Du|^{\alpha}\in W^{1,2}_{loc}(\Omega)$, 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 $p> 2$, $|Dw|^\alpha \notin W^{1,p}_{loc}(\mathbb R^2)$ whenever $\alpha\in(3/2,3-3/p)$. (ii) For $ \alpha \in(0, 3/2]$ and $p\in[1, 3/(3-\alpha))$, we have $|Du|^{\alpha}\in W^{1,p}_{loc}(\Omega)$, which is sharp when $p\to 3/(3-\alpha)$. Indeed, $ |Dw|^\alpha \notin W^{1,3/(3-\alpha)}_{loc}(\mathbb R^2)$. (iii) For $ \epsilon > 0$, we have $|Du|^{-3+\epsilon }\in L^1_{loc}(\Omega )$, which is sharp when $\epsilon\to0$. Indeed, $|Dw|^{-3} \notin L^1_{loc}(\mathbb R^2)$. (iv) For $ \alpha > 0$, we have $$-(|Du|^{\alpha})_iu_i= 2\alpha|Du|^{{ \alpha-2}}f \ \mbox{ almost everywhere in $\Omega$}.$$ Some quantative bounds are also given.