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Oscillating facets

We study a singular one-dimensional parabolic problem with initial data in the $BV$ space, the energy space, for various boundary data. We pay special attention to Dirichlet conditions, which need not satisfied in a pointwise manner. We study the facet creation process and the extinction of solutions caused by the evolution of facets. Our major tool is the comparison principle provided by the theory of viscosity solutions developed in \cite{miyory}.