The dirichlet problem for generalized mean curvature flows^*

^*This work has been partially supported by I.A.N. C.N.R., Pavia (Italy)

Abstract

We study the existence and uniqueness of viscosity solutions for the Dirichlet problem associated to generalized mean curvature flow equations under some assumptions on the data and the shape of domain. The data is constant in time and Lipschitz-continuous in space. We first provide the result on a smooth strictly convex domain and then we generalize it to non-strictly convex set whose boundary is piecewise C² with a finite number of corners. At the end an easy example shows that the existence result can not be extended to more general domains without further assumptions on the data. The main idea we use is to contruct viscosity sub-solutions and super-solutions under hypoteses of Perron's Method whose validity for such equations is set in [3]

Keywords:

• viscosity solutions
• fully non-linear PDEs
• Dirichlet problems
• barriers

^*This work has been partially supported by I.A.N. C.N.R., Pavia (Italy)

^*This work has been partially supported by I.A.N. C.N.R., Pavia (Italy)

Notes

^*This work has been partially supported by I.A.N. C.N.R., Pavia (Italy)