Existence of translating solutions to the flow by powers of mean curvature on unbounded domains

TL;DR

This paper proves the existence of classical solutions for a class of quasi-linear elliptic equations related to mean curvature flow on unbounded domains, using a modified Perron method with sub- and super-solutions.

Contribution

It introduces a new approach to establish solutions for flow by powers of mean curvature on unbounded domains, extending previous methods.

Findings

Existence of solutions on cone and U-type domains

Application of a modified Perron method

Construction of sub- and super-solutions for the problem

Abstract

In this paper, we prove the existence of classical solutions of the Dirichlet problem for a class of quasi-linear elliptic equations on unbounded domains like a cone or a U-type domain. This problem comes from the study of mean curvature flow and its generalization, the flow by powers of mean curvature. Our approach is a modified version of the classical Perron method, where the solutions to the minimal surface equation are used as sub-solutions and a family auxiliary functions are constructed as super-solutions.