Atsushi Nakayasu

On general existence results for one-dimensional singular diffusion equations with spatially inhomogeneous driving force

A general anisotropic curvature flow equation with singular interfacial energy and spatially inhomogeneous driving force is considered for a curve given by the graph of a periodic function. We prove that the initial value problem admits a unique global-in-time viscosity solution for a general periodic continuous initial datum. The notion of a viscosity solution used here is the same as proposed by Giga, Giga and Rybka, who established a comparison principle. We construct the global-in-time solution by careful adaptation of Perron’s method.

Energy solutions to one-dimensional singular parabolic problems with

We study one-dimensional very singular parabolic equations with periodic boundary conditions and initial data in

BV

BV

data are viscosity solutions

, which is the energy space. We show existence of solutions in this energy space and then we prove that they are viscosity solutions in the sense of Giga-Giga.