The Dirichlet problem for the $\alpha$-translating soliton equation on a strip

TL;DR

This paper proves the existence of classical solutions to the Dirichlet problem for the $oldsymbol{ extalpha}$-translating soliton equation in a strip, employing the Perron method with grim reapers as barriers for convex boundary data.

Contribution

It establishes the existence of solutions for the $ extalpha$-translating soliton equation in a strip using novel barrier techniques with grim reapers.

Findings

Existence of classical solutions in a strip for the $ extalpha$-translating soliton equation.

Use of Perron method with grim reapers as barriers.

Solutions for boundary data formed by convex functions.

Abstract

We prove the existence of classical solutions to the Dirichlet problem for the $\alpha$-translating soliton equation defined in a strip of $\r^2$. We use the Perron method where a family of grim reapers are employed as barriers for solving the Dirichlet problem when the boundary data is formed by two copies of a convex function.