Abstract
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.
Highlights
We consider the total variation flow with the dynamic boundary condition, possibly mixed with the Neumann boundary condition, which can be formally written as follows, ut = div ∇u |∇u| τ vt = − · ν∂u ∂ν u(x, 0) = u0(x) for (x, t) ∈ Ω × (0, T ) =: QT ; for (y, t) ∈ Γ × (0, T ) =: ST ; for (y, t) ∈ (∂Ω \ Γ) × (0, T ); for x ∈ Ω; (1.1)v(y, 0) = v0(y) for y ∈ Γ.Here Ω ⊂ RN is a bounded spatial domain of dimension N ∈ N, and when N > 1, the boundary ∂Ω is supposed to be sufficiently smooth
Even though the total variation flow with the Dirichlet boundary conditions was studied by a number of authors, see [4, 13, 41, 42], the details of the boundary behavior were not extensively discussed
We introduce a family of evolution problems with dynamic boundary condition indexed by parameter τ ∈ (0, ∞)
Summary
Even though the total variation flow with the Dirichlet boundary conditions was studied by a number of authors, see [4, 13, 41, 42], the details of the boundary behavior were not extensively discussed In particular this applies to the evolution of facets touching the boundary. The authors showed there that a boundary layer may detach from the solution in the bulk This phenomenon is attributed to the lack of uniform parabolicity of the mean curvature flow for graphs. Our goal in this paper is to study instances of occurrence of the “boundary layer detachment phenomenon” in the case of the total variation flow under the dynamic boundary condition on a part of the boundary called Γ. In this case we pinpoint the situation of the boundary layer detachment
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
