Second-order boundary estimate for the solution to infinity Laplace equations
Ling Mi · DOAJ (DOAJ: Directory of Open Access Journals) · 2017
In this article, we establish a second-order estimate for the solutions to the infinity Laplace equation $$ -\Delta_{\infty} u=b(x)g(u), \quad u>0, \quad x \in \Omega,\; u|_{\partial \Omega}=0, $$ where $\Omega$ is a bounded domain in $\mathbb{R}^N$, $g\in C^1((0,\infty),(0,\infty))$, $g$ is decreasing on $(0,\infty)$ with $\lim_{s \to 0^+}g(s)=\infty$ and g is normalized regularly varying at zero with index $-\gamma$ ($\gamma>1$), $b \in C({\bar{\Omega}})$ is positive in $\Omega$, may be vanishing on the boundary. Our analysis is based on Karamata regular variation theory.