Kobayashi–Warren–Carter type systems with nonhomogeneous Dirichlet boundary data for crystalline orientation

Abstract

In this paper we study the Dirichlet problem for the Kobayashi–Warren–Carter system. This system of parabolic PDE’s models the grain boundary motion in a polycrystal with a prescribed orientation at the boundary of the domain. We obtain global existence in time of energy-dissipative solutions. The regularity of the solutions as well as the energy-dissipation property permit us to derive the steady-state problem as the asymptotic in time limit of the system. We finally study the ω-limit set of the solutions; we completely characterize it in the one dimensional case, showing, in particular that orientations in the ω-limit set belong to the space of SBV functions. In the two dimensional case, we give sufficient conditions for existence of radial symmetric piecewise constant solutions.