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A level set approach for multi-layered interface systems Interfaces Free Bound. (IF 0.718) Pub Date : 2020-12-08 Hiroyoshi Mitake; Hirokazu Ninomiya; Kenta Todoroki
In this paper, a multi-layered interface system is introduced, which is formally derived by a singular limit of a weakly coupled system of the Allen–Cahn type equation. By using the level set approach, this system is written as a quasi-monotone degenerate parabolic system with discontinuous functions. The well-posedness of viscosity solutions is shown, and the singularity of particular viscosity solutions
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Regularity of solutions of a fractional porous medium equation Interfaces Free Bound. (IF 0.718) Pub Date : 2020-12-08 Cyril Imbert; Rana Tarhini; François Vigneron
This article is concerned with a porous medium equation whose pressure law is both nonlinear and nonlocal, namely \[ \partial_t u = { \nabla \cdot} \left(u \nabla(-\Delta)^{\frac{\alpha}{2}-1}u^{m-1} \right)\] where $u:\mathbb{R}_+\times \mathbb{R}^N \to \mathbb{R}_+$, for $0 < \alpha < 2$ and $m\geq2$. We prove that the $L^1\cap L^\infty$ weak solutions constructed by Biler, Imbert and Karch (2015)
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A convergent algorithm for forced mean curvature flow driven by diffusion on the surface Interfaces Free Bound. (IF 0.718) Pub Date : 2020-12-08 Balázs Kovács; Buyang Li; Christian Lubich
The evolution of a closed two-dimensional surface driven by both mean curvature flow and a reaction–diffusion process on the surface is formulated as a system that couples the velocity law not only to the surface partial differential equation but also to the evolution equations for the normal vector and the mean curvature on the surface. Two algorithms are considered for the obtained system. Both methods
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Minimisers of a fractional seminorm and nonlocal minimal surfaces Interfaces Free Bound. (IF 0.718) Pub Date : 2020-12-08 Claudia Bucur; Serena Dipierro; Luca Lombardini; Enrico Valdinoci
The recent literature has intensively studied two classes of nonlocal variational problems, namely the ones related to the minimisation of energy functionals that act on functions in suitable Sobolev–Gagliardo spaces, and the ones related to the minimisation of fractional perimeters that act on measurable sets of the Euclidean space. In this article, we relate these two types of variational problems
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Existence and regularity theorems of one-dimensional Brakke flows Interfaces Free Bound. (IF 0.718) Pub Date : 2020-12-08 Lami Kim; Yoshihiro Tonegawa
Given a closed countably 1-rectifiable set in $\mathbb R^2$ with locally finite $1$-dimensional Hausdorff measure, we prove that there exists a Brakke flow starting from the given set with the following regularity property. For almost all time, the flow locally consists of a finite number of embedded curves of class~$W^{2,2}$ whose endpoints meet at junctions with angles of either 0, 60 or 120 degrees
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On non stress-free junctions between martensitic plates Interfaces Free Bound. (IF 0.718) Pub Date : 2020-09-01 Francesco Della Porta
The analytical understanding of microstructures arising in martensitic phase transitions relies usually on the study of stress-free interfaces between different variants of martensite. However, in the literature there are experimental observations of non stress-free junctions between martensitic plates, where the compatibility theory fails to be predictive. In this work, we focus on $V_{II}$ junctions
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Asymptotic behaviors of a free boundary raised from corporate bond evaluation with credit rating migration risks Interfaces Free Bound. (IF 0.718) Pub Date : 2020-09-01 Wanying Fu; Xinfu Chen; Jin Liang
In this paper, we study asymptotic behaviors of a free boundary raised from the evaluation of a corporate bond subject to credit rating changes of the underlying company. The credit rating migration is modeled by a free boundary which separates different credit rating regions in a state space. We first formulate the mathematical problem and then establish the well-posedness of the problem and the long
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Quantitative analysis of finite-difference approximations of free-discontinuity problems Interfaces Free Bound. (IF 0.718) Pub Date : 2020-09-01 Annika Bach; Andrea Braides; Caterina Ida Zeppieri
Motivated by applications to image reconstruction, in this paper we analyse a finite-difference discretisation of the Ambrosio–Tortorelli functional. Denoted by $\varepsilon$ the elliptic-approximation parameter and by $\delta$ the discretisation step-size, we fully describe the relative impact of $\varepsilon$ and $\delta$ in terms of $\Gamma$-limits for the corresponding discrete functionals, in
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A convex approach to the Gilbert–Steiner problem Interfaces Free Bound. (IF 0.718) Pub Date : 2020-07-06 Mauro Bonafini; Édouard Oudet
We describe a convex relaxation for the Gilbert–Steiner problem both in Rd and on manifolds, extending the framework proposed in [10], and we discuss its sharpness by means of calibration type arguments. The minimization of the resulting problem is then tackled numerically and we present results for an extensive set of examples. In particular we are able to address the Steiner tree problem on surfaces
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Long-time behaviour of solutions to a singular heat equation with an application to hydrodynamics Interfaces Free Bound. (IF 0.718) Pub Date : 2020-07-06 Georgy Kitavtsev; Roman M. Taranets
In this paper, we extend the results of [8] by proving exponential asymptotic $H^1$-convergence of solutions to a one-dimensional singular heat equation with $L^2$-source term that describe evolution of viscous thin liquid sheets while considered in the Lagrange coordinates. Furthermore, we extend this asymptotic convergence result to the case of a time inhomogeneous source. This study has also independent
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Segregation effects and gap formation in cross-diffusion models Interfaces Free Bound. (IF 0.718) Pub Date : 2020-07-06 Martin Burger; José A. Carrillo; Jan-Frederik Pietschmann; Markus Schmidtchen
In this paper, we extend the results of [8] by proving exponential asymptotic $H^1$-convergence of solutions to a one-dimensional singular heat equation with $L^2$-source term that describe evolution of viscous thin liquid sheets while considered in the Lagrange coordinates. Furthermore, we extend this asymptotic convergence result to the case of a time inhomogeneous source. This study has also independent
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From individual-based mechanical models of multicellular systems to free-boundary problems Interfaces Free Bound. (IF 0.718) Pub Date : 2020-07-06 Tommaso Lorenzi; Philip J. Murray; Mariya Ptashnyk
In this paper we present an individual-based mechanical model that describes the dynamics of two contiguous cell populations with different proliferative and mechanical characteristics. An off-lattice modelling approach is considered whereby: (i) every cell is identified by the position of its centre; (ii) mechanical interactions between cells are described via generic nonlinear force laws; and (iii)