A remark on Fefferman's equation in non-smooth domains (Q1352947)

The author studies the nonlinear partial differential equation \(-v(x)\Delta v(x)+C|\nabla v(x)|^2=1\) for \(x\) in a bounded domain \(\Omega\) in \(\mathbb{R}^N\) with the boundary condition \(v(x)\to 0\) when \(x\to\partial\Omega\); here \(C\geq 1\) is a constant. This is a real-variable analogue of the one-dimensional version of \textit{Ch. L. Fefferman's} complex Monge-Ampère equation [Ann. Math., II. Ser. 103, 395-416 (1976; Zbl 0322.32012)]. The author proves that if \(\Omega\) satisfies both an interior and an exterior cone condition, then there is a unique positive, twice continuously differentiable solution \(v\), and for every positive \(\varepsilon\) there exists a positive \(\delta\) such that \(|1-C^{1/2}v(x)/d(x,\partial\Omega)|<\varepsilon\) and \(||\nabla v(x)|-C^{-1/2}|<\varepsilon\) when \(d(x,\partial\Omega)<\delta\); here \(d(x,\partial\Omega)\) denotes the distance from \(x\) to the boundary of \(\Omega\). If the cone condition is replaced by the assumption that \(\Omega\) is nontangentially accessible, then there is a weaker result.