TL;DR
This paper investigates the evolution of facets in a one-dimensional parabolic PDE with BV initial data, focusing on facet creation, extinction, and boundary conditions using viscosity solutions.
Contribution
It introduces a novel analysis of facet dynamics and boundary conditions in a singular parabolic problem using viscosity solution techniques.
Findings
Facets are created and extinguished during evolution.
Boundary conditions may not be satisfied pointwise.
Viscosity solutions provide a key comparison principle.
Abstract
We study a singular one-dimensional parabolic problem with initial data in the 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}.
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Taxonomy
TopicsNonlinear Partial Differential Equations · Advanced Mathematical Modeling in Engineering · Differential Equations and Numerical Methods