Oscillating facets (Q266494)

Summary: We study a singular one-dimensional parabolic problem with initial data in the \textit{BV} space, i.e. the energy space, for various boundary data. We pay special attention to Dirichlet conditions, which need not be satisfied in a pointwise manner. The equation we study has two singular slopes, so that in principle solutions to the value problems may have an infinite number of oscillations, which seems surprising for a parabolic problem. We investigate this issue. We also study the facet creation process and the stopping of solutions caused by the evolution of facets. Our estimate of stopping time is based on the comparison principle for viscosity solutions. For this purpose we show that our solutions are viscosity solutions in the sense of \textit{M. H. Giga} et al. [Arch. Ration. Mech. Anal. 211, No. 2, 419--453 (2014; Zbl 1288.35137)].