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

Abstract

In this paper, we prove the existence of classical solutions to the Dirichlet problem of a class of quasi-linear elliptic equations on an unbounded cone and a U-type domain in R n (n 2). This problem comes from the study of mean curvature flow or 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.

The Evolutionary Properties on Solitary Solutions of Nonlinear Evolution Equations

“…The evolution procedure of curvature which is given by (6) and (12), respectively, is similar, but it is different to those curves given by (8) and (10). Even though, for self-similar curve group, different points ( , ( , )) possess different evolution procedures of curvature, all evolution results are the same.…”

“…There have a mount of works about curvature motion and curvature motion flow (see [6][7][8][9][10][11]). Particularly, the curve evolution problem played important role in applied sciences.…”

The Dirichlet problem on a strip for the α-translating soliton equation

“…We now give some properties about superfunctions whose proofs are straightforward: in the case of the constant mean curvature equation, we refer [14]; in the context of α-translating solitons, see [12,. 1.…”

The translating soliton equation

“…On the other hand, v 0 = f on ∂Ω m , proving definitively that v 0 ∈ S f . We now give some properties about superfunctions whose proofs are straightforward: in the case of the constant mean curvature equation, we refer [25]; in the context of translating solitons, see [22,]. Lemma 6.5.…”