The Dirichlet problem on a strip for the \(\alpha\)-translating soliton equation
The paper studies the existence of classical solutions \(u\in C^2(\Omega)\cap C^0(\partial\Omega)\) to the Dirichlet problem, \[ \text{div}\left(\frac{Du}{\sqrt{1+|Du|^2}}\right)=\left(\frac{1}{\sqrt{1+|Du|^2}}\right)^\alpha \text{ in } \Omega, \] \[u=\phi \text{ on }\partial\Omega.\] where \(\Omega\subset \mathbb{R}^2\) is a smooth domain, \(\alpha\) is a positive constant, and \(\phi\in C^0(\partial\Omega)\). In particular, the author proves the existence of solutions when \(\Omega\) is a strip of \(\mathbb{R}^2\), using the Perron method and a family of grin reapers as barriers.
Dirichlet problem, Perron method, translating soliton, grim reapers
scientific article; zbMATH DE number 6994059
scientific article