Existence and Regularity For The Generalized Mean Curvature Flow Equations

TL;DR

This paper establishes the existence and regularity of viscosity solutions for generalized mean curvature flow equations using approximation methods, and also explores the flow's asymptotic behavior and solvability of related degenerate elliptic equations.

Contribution

It introduces a new approach to prove existence and regularity of solutions for generalized mean curvature flow equations and addresses the Dirichlet problem for a related degenerate elliptic equation.

Findings

Viscosity solutions exist and are regular for the generalized mean curvature flow.

The asymptotic behavior of the flow is characterized.

The Dirichlet problem for the degenerate elliptic equation is solvable in viscosity sense.

Abstract

By making use of the approximation method, we obtain the existence and regularity of the viscosity solutions for the generalized mean curvature flow. The asymptotic behavior of the flow is also considered. In particular, the Dirichlet problem of the degenerate elliptic equation $$ -|\nabla v|(\mathrm{div}(\frac{\nabla v}{|\nabla v|})+\nu)=0 $$ is solvable in viscosity sense.