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

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)

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

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

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

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

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

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

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

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

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

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)