In this paper, we consider a class of coupled systems of PDEs, denoted by (ACE) ε for ε ≥ 0. For each ε ≥ 0, the system (ACE) ε consists of an Allen-Cahn type equation in a bounded spacial domain Ω, and another Allen-Cahn type equation on the smooth boundary Γ := ∂Ω, and besides, these coupled equations are transmitted via the dynamic boundary conditions. In particular, the equation in Ω is derived from the nonsmooth energy proposed by Visintin in his monography "Models of phase transitions": hence, the diffusion in Ω is provided by a quasilinear form with singularity. The objective of this paper is to build a mathematical method to obtain meaningful L 2 -based solutions to our systems, and to see some robustness of (ACE) ε with respect to ε ≥ 0. On this basis, we will prove two Main Theorems 1 and 2, which will be concerned with the well-posedness of (ACE) ε for each ε ≥ 0, and the continuous dependence of solutions to (ACE) ε for the variations of ε ≥ 0, respectively.
Abstract. In this paper, we propose a weak formulation of the singular diffusion equation subject to the dynamic boundary condition. The weak formulation is based on a reformulation method by an evolution equation including the subdifferential of a governing convex energy. Under suitable assumptions, the principal results of this study are stated in forms of Main Theorems A and B, which are respectively to verify: the adequacy of the weak formulation; the common property between the weak solutions and those in regular problems of standard PDEs.
We combine the total variation flow suitable for crystal modeling and image analysis with the dynamic boundary conditions. We analyze the behavior of facets at the parts of the boundary where these conditions are imposed. We devote particular attention to the radially symmetric data. We observe that the boundary layer detachment actually can happen at concave parts of the boundary.
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