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On the sharp interface limit of a phase field model for near spherical two phase biomembranes Interfaces Free Bound. (IF 1.0) Pub Date : 2022-05-13 Charles M. Elliott, Luke Hatcher, Björn Stinner
We consider sharp interface asymptotics for a phase field model motivated by lipid raft formation on near spherical biomembranes involving a coupling between the local mean curvature and the local composition. A reduced diffuse interface energy depending only on the membrane composition is introduced and a $\Gamma$-limit is derived. It is shown that the Euler–Lagrange equations for the limiting functional
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Bifurcations of spherically asymmetric solutions to an evolution equation for curves Interfaces Free Bound. (IF 1.0) Pub Date : 2022-04-26 Takeo Sugai
We show that a certain non-local curvature flow for planar curves has non-trivial self-similar solutions with $n$-fold rotational symmetry, bifurcated from a trivial circular solution. Moreover, we show that the trivial solution is stable with respect to perturbations which keep the geometric center and the enclosed area, and that, for $n$ different from 3, the $n$-fold symmetric solution is stable
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Analysis of a tumor model as a multicomponent deformable porous medium Interfaces Free Bound. (IF 1.0) Pub Date : 2022-03-29 Pavel Krejčí, Elisabetta Rocca, Jürgen Sprekels
We propose a diffuse interface model to describe a tumor as a multicomponent deformable porous medium. We include mechanical effects in the model by coupling the mass balance equations for the tumor species and the nutrient dynamics to a mechanical equilibrium equation with phase-dependent elasticity coefficients. The resulting PDE system couples two Cahn–Hilliard type equations for the tumor phase
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The average-distance problem with an Euler elastica penalization Interfaces Free Bound. (IF 1.0) Pub Date : 2022-03-10 Qiang Du, Xin Yang Lu, Chong Wang
We consider the minimization of an average-distance functional defined on a two-dimensional domain $\Omega$ with an Euler elastica penalization associated with $\partial\Omega$, the boundary of $\Omega$. The average distance is given by $$ \int_{\Omega}\operatorname{dist}^p(x,\partial\Omega)\operatorname{d}x, $$ where $p\geq 1$ is a given parameter and $\operatorname{dist}(x,\partial\Omega)$ is the
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Fully nonlinear free transmission problems with nonhomogeneous degeneracies Interfaces Free Bound. (IF 1.0) Pub Date : 2022-03-09 Cristiana De Filippis
We prove existence and regularity results for free transmission problems governed by fully nonlinear elliptic equations with nonhomogeneous degeneracies.
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Finite element error analysis for a system coupling surface evolution to diffusion on the surface Interfaces Free Bound. (IF 1.0) Pub Date : 2022-02-04 Klaus Deckelnick, Vanessa Styles
We consider a numerical scheme for the approximation of a system that couples the evolution of a two-dimensional hypersurface to a reaction–diffusion equation on the surface. The surfaces are assumed to be graphs and evolve according to forced mean curvature flow. The method uses continuous, piecewise linear finite elements in space and a backward Euler scheme in time. Assuming the existence of a smooth
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Longest minimal length partitions Interfaces Free Bound. (IF 1.0) Pub Date : 2022-01-18 Beniamin Bogosel, Édouard Oudet
This article provides numerical evidence that under volume constraint the ball is the set which maximizes the perimeter of the least-perimeter partition into cells with prescribed areas. We introduce a numerical maximization algorithm which performs multiple optimization steps at each iteration to approximate minimal partitions. Using these partitions we compute perturbations of the domain which increase
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A formula for membrane mediated point particle interactions on near spherical biomembranes Interfaces Free Bound. (IF 1.0) Pub Date : 2021-12-10 Charles M. Elliott, Philip J. Herbert
We consider a model of a biomembrane with attached proteins. The membrane is represented by a near spherical continuous surface and attached proteins are described as discrete rigid structures which attach to the membrane at a finite number of points. The resulting surface minimises a quadratic elastic energy (obtained by a perturbation of the Canham–Helfrich energy) subject to the point constraints
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Free boundary problems for Stokes flow, with applications to the growth of biological tissues Interfaces Free Bound. (IF 1.0) Pub Date : 2021-11-05 John R. King, Chandrasekhar Venkataraman
We formulate, analyse and numerically simulate what are arguably the two simplest Stokes-flow free boundary problems relevant to tissue growth, extending the classical Stokes free boundary problem by incorporating (i) a volumetric source (the nutrient-rich case) and (ii) a volumetric sink, a surface source and surface compression (the nutrient-poor case). Both two- and three-dimensional cases are considered
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On differentiability of the membrane-mediated mechanical interaction energy of discrete–continuum membrane–particle models Interfaces Free Bound. (IF 1.0) Pub Date : 2021-11-05 Carsten Gräser, Tobias Kies
We consider a discrete–continuum model of a biomembrane with embedded particles. While the membrane is represented by a continuous surface, embedded particles are described by rigid discrete objects which are free to move and rotate in lateral direction. For the membrane we consider a linearized Canham–Helfrich energy functional and height and slope boundary conditions imposed on the particle boundaries
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A free boundary problem for binary fluids Interfaces Free Bound. (IF 1.0) Pub Date : 2021-11-05 Roberto Benzi, Michiel Bertsch, Francesco Deangelis
A free boundary problem for the dynamics of a glasslike binary fluid naturally leads to a singular perturbation problem for a strongly degenerate parabolic partial differential equation in 1D. We present a conjecture for an asymptotic formula for the velocity of the free boundary and prove a weak version of the conjecture. The results are based on the analysis of a family of local travelling wave solutions
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On second-order and fourth-order elliptic systems consisting of bulk and surface PDEs: Well-posedness, regularity theory and eigenvalue problems Interfaces Free Bound. (IF 1.0) Pub Date : 2021-11-05 Patrik Knopf, Chun Liu
In this paper, we study second-order and fourth-order elliptic problems which include not only a Poisson equation in the bulk but also an inhomogeneous Laplace–Beltrami equation on the boundary of the domain. The bulk and the surface PDE are coupled by a boundary condition that is either of Dirichlet or Robin type. We point out that both the Dirichlet and the Robin type boundary condition can be handled
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Analysis of the Allen–Cahn–Ohta–Nakazawa model in a ternary system Interfaces Free Bound. (IF 1.0) Pub Date : 2021-11-05 Sookyung Joo, Xiang Xu, Yanxiang Zhao
In this paper we study the global well-posedness of the Allen–Cahn–Ohta–Nakazawa model with two fixed nonlinear volume constraints. Utilizing the gradient flow structure of its free energy, we prove the existence and uniqueness of the solution by following De Giorgi’s minimizing movement scheme in a novel way.
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The multiphase Muskat problem with equal viscosities in two dimensions Interfaces Free Bound. (IF 1.0) Pub Date : 2021-11-03 Jonas Bierler, Bogdan-Vasile Matioc
We study the two-dimensional multiphase Muskat problem describing the motion of three immiscible fluids with equal viscosities in a vertical homogeneous porous medium identified with $\mathbb{R}^2$ under the effect of gravity. We first formulate the governing equations as a strongly coupled evolution problem for the functions that parameterize the sharp interfaces between the fluids. Afterwards we
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$K$-mean convex and $K$-outward minimizing sets Interfaces Free Bound. (IF 1.0) Pub Date : 2021-11-03 Annalisa Cesaroni, Matteo Novaga
We consider the evolution of sets by nonlocal mean curvature and we discuss the preservation along the flow of two geometric properties, which are the mean convexity and the outward minimality. The main tools in our analysis are the level set formulation and the minimizing movement scheme for the nonlocal flow. When the initial set is outward minimizing, we also show the convergence of the (time integrated)
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Pinning of interfaces in a random medium with zero mean Interfaces Free Bound. (IF 1.0) Pub Date : 2021-08-11 Patrick W. Dondl, Martin Jesenko, Michael Scheutzow
We consider two related models for the propagation of a curvature sensitive interface in a time independent random medium. In both cases we suppose that the medium contains obstacles that act on the propagation of the interface with an inhibitory or an acceleratory force. We show that the interface remains bounded for all times even when a small constant external driving force is applied. This phenomenon
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Coarse graining and large-$N$ behavior of the $d$-dimensional $N$-clock model Interfaces Free Bound. (IF 1.0) Pub Date : 2021-08-11 Marco Cicalese, Gianluca Orlando, Matthias Ruf
We study the asymptotic behavior of the $N$-clock model, a nearest neighbors ferromagnetic spin model on the $d$-dimensional cubic $\varepsilon$-lattice in which the spin field is constrained to take values in a discretization $\mathcal{S}_N$ of the unit circle $\mathbb{S}^{1}$ consisting of $N$ equispaced points. Our $\Gamma$-convergence analysis consists of two steps: we first fix $N$ and let the
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Sharp interface limit of a Stokes/Cahn–Hilliard system. Part I: Convergence result Interfaces Free Bound. (IF 1.0) Pub Date : 2021-08-11 Helmut Abels, Andreas Marquardt
We consider the sharp interface limit of a coupled Stokes/Cahn–Hilliard system in a two-dimensional, bounded and smooth domain, i.e., we consider the limiting behavior of solutions when a parameter $\epsilon>0$ corresponding to the thickness of the diffuse interface tends to zero. We show that for sufficiently short times the solutions to the Stokes/Cahn–Hilliard system converge to solutions of a sharp
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Some remarks on segregation of $k$ species in strongly competing systems Interfaces Free Bound. (IF 1.0) Pub Date : 2021-08-11 Flavia Lanzara, Eugenio Montefusco
Spatial segregation occurs in population dynamics when $k$ species interact in a highly competitive way. As a model for the study of this phenomenon, we consider the competition-diffusion system of $k$ differential equations $$ -\Delta u_i(x)=-\mu u_i (x)\sum_{j\neq i} u_j (x), \quad i=1,\ldots,k $$ in a domain $D$ with appropriate boundary conditions. Any $u_i$ represents a population density and
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On the entropy of parabolic Allen–Cahn equation Interfaces Free Bound. (IF 1.0) Pub Date : 2021-08-11 Ao Sun
We define a local (mean curvature flow) entropy for Radon measures in $\mathbb{R}^n$ or in a compact manifold. Moreover, we prove a monotonicity formula of the entropy of the measures associated with the parabolic Allen–Cahn equations. If the ambient manifold is a compact manifold with non-negative sectional curvature and parallel Ricci curvature, this is a consequence of a new monotonicity formula
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Almost everywhere uniqueness of blow-up limits for the lower dimensional obstacle problem Interfaces Free Bound. (IF 1.0) Pub Date : 2021-07-16 Maria Colombo, Luca Spolaor, Bozhidar Velichkov
We answer a question left open in [4] and [3], by proving that the blow-up of minimizers $u$ of the lower dimensional obstacle problem is unique at generic point of the free boundary.
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A curve shortening equation with time-dependent mobility related to grain boundary motions Interfaces Free Bound. (IF 1.0) Pub Date : 2021-07-16 Masashi Mizuno, Keisuke Takasao
A curve shortening equation related to the evolution of grain boundaries is presented. This equation is derived from the grain boundary energy by applying the maximum dissipation principle. Gradient estimates and large time asymptotic behavior of solutions are considered. In the proof of these results, one key ingredient is a new weighted monotonicity formula that incorporates a time-dependent mobility
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Interface dynamics in a two-phase tumor growth model Interfaces Free Bound. (IF 1.0) Pub Date : 2021-07-16 Inwon C. Kim, Jiajun Tong
We study a tumor growth model in two space dimensions, where proliferation of the tumor cells leads to expansion of the tumor domain and migration of surrounding normal tissues into the exterior vacuum. The model features two moving interfaces separating the tumor, the normal tissue, and the exterior vacuum. We prove local-in-time existence and uniqueness of strong solutions for their evolution starting
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Existence and uniqueness of axially symmetric compressible subsonic jet impinging on an infinite wall Interfaces Free Bound. (IF 1.0) Pub Date : 2021-04-19 Jianfeng Cheng, Lili Du, Qin Zhang
This paper is concerned with the well-posedness theory of the impact of a subsonic axially symmetric jet emerging from a semi-infinitely long nozzle, onto a rigid wall. The fluid motion is described by the steady isentropic Euler system. We showed that there exists a critical value $M_{cr} > 0$, if the given mass flux is less than $M_{cr}$, there exists a unique smooth subsonic axially symmetric jet
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Variational approximation of interface energies and applications Interfaces Free Bound. (IF 1.0) Pub Date : 2021-04-19 Samuel Amstutz, Daniel Gourion, Mohammed Zabiba
Minimal partition problems consist in finding a partition of a domain into a given number of components in order to minimize a geometric criterion. In applicative fields such as image processing or continuum mechanics, it is standard to incorporate in this objective an interface energy that accounts for the lengths of the interfaces between components. The present work is focused on the theoretical
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Gradient flow formulation and second order numerical method for motion by mean curvature and contact line dynamics on rough surface Interfaces Free Bound. (IF 1.0) Pub Date : 2021-04-19 Yuan Gao, Jian-Guo Liu
We study the dynamics of a droplet moving on an inclined rough surface in the absence of inertial and viscous stress effects. In this case, the dynamics of the droplet is a purely geometric motion in terms of the wetting domain and the capillary surface. Using a single graph representation, we interpret this geometric motion as a gradient flow on a manifold. We propose unconditionally stable first/second
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A level set approach for multi-layered interface systems Interfaces Free Bound. (IF 1.0) 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 1.0) 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) are locally
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A convergent algorithm for forced mean curvature flow driven by diffusion on the surface Interfaces Free Bound. (IF 1.0) 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 into a system, which couples the velocity law not only to the surface partial differential equation but also to the evolution equations for the geometric quantities, namely the normal vector and the mean curvature on the surface. Two algorithms are considered
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Minimisers of a fractional seminorm and nonlocal minimal surfaces Interfaces Free Bound. (IF 1.0) 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 1.0) 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 1.0) 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 1.0) Pub Date : 2020-09-01 Wanying Fu, Xinfu Chen, Jin Liang
In this thesis, we study asymptotic behaviors of a free boundary arised from the evaluation of a corporate bond subject to changes of the credit rating 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 we establish the well-posedness of the problem
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Quantitative analysis of finite-difference approximations of free-discontinuity problems Interfaces Free Bound. (IF 1.0) Pub Date : 2020-09-01 Annika Bach, Andrea Braides, Caterina Ida Zeppieri
Motivated by applications to image reconstruction, in this paper we analyse a \emph{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
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Segregation effects and gap formation in cross-diffusion models Interfaces Free Bound. (IF 1.0) 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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Erratum to: A homogenization result in the gradient theory of phase transitions Interfaces Free Bound. (IF 1.0) Pub Date : 2020-07-06 Riccardo Cristoferi,Irene Fonseca,Adrian Hagerty,Cristina Popovici
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A convex approach to the Gilbert–Steiner problem Interfaces Free Bound. (IF 1.0) Pub Date : 2020-07-06 Mauro Bonafini, Édouard Oudet
We describe a convex relaxation for the Gilbert-Steiner problem both in $R^d$ and on manifolds, extending the framework proposed in [9], 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 1.0) Pub Date : 2020-07-06 Georgy Kitavtsev, Roman Taranets
In this paper, we extend the results of [1] 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 1.0) Pub Date : 2020-07-06 Tommaso Lorenzi, Philip 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)
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The vanishing exponent limit for motion by a power of mean curvature Interfaces Free Bound. (IF 1.0) Pub Date : 2020-04-15 Qing Liu
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A bulk-surface reaction-diffusion system for cell polarization Interfaces Free Bound. (IF 1.0) Pub Date : 2020-04-15 Barbara Niethammer, Matthias Röger, Juan Velázquez
We propose a model for cell polarization as a response to an external signal which results in a system of PDEs for different variants of a protein on the cell surface and interior respectively. We study stationary states of this model in certain parameter regimes in which several reaction rates on the membrane as well as the diffusion coefficient within the cell are large. It turns out that in suitable
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Well-posedness for degenerate elliptic PDE arising in optimal learning strategies Interfaces Free Bound. (IF 1.0) Pub Date : 2020-04-15 Tim Laux, J. Miguel Villas-Boas
We derive a comparison principle for a degenerate elliptic partial differential equation without boundary conditions which arises naturally in optimal learning strategies. Our argument is direct and exploits the degeneracy of the differential operator to construct (logarithmically) diverging barriers.
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Global weak solvability, continuous dependence on data, and large time growth of swelling moving interfaces Interfaces Free Bound. (IF 1.0) Pub Date : 2020-04-15 Kota Kumazaki, Adrian Muntean
We prove a global existence result for weak solutions to a one-dimensional free boundary problem with flux boundary conditions describing swelling along a halfline. Additionally, we show that solutions are not only unique but also depend continuously on data and parameters. The key observation is that the structure of our system of partial differential equations allows us to show that the moving a
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Derivation of a heteroepitaxial thin-film model Interfaces Free Bound. (IF 1.0) Pub Date : 2020-04-15 Elisa Davoli, Paolo Piovano
A variational model for epitaxially-strained thin films on rigid substrates is derived both by {\Gamma}-convergence from a transition-layer setting, and by relaxation from a sharp-interface description available in the literature for regular configurations. The model is characterized by a configurational energy that accounts for both the competing mechanisms responsible for the film shape. On the one
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A nonlocal diffusion problem with a sharp free boundary Interfaces Free Bound. (IF 1.0) Pub Date : 2019-12-18 Carmen Cortázar,Fernando Quirós,Noemi Wolanski
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A tractable mathematical model for tissue growth Interfaces Free Bound. (IF 1.0) Pub Date : 2019-12-18 Joe Eyles, John King, Vanessa Styles
Using formal asymptotic methods we derive a free boundary problem representing one of the simplest mathematical descriptions of the growth and death of a tumour or other biological tissue. The mathematical model takes the form of a closed interface evolving via forced mean curvature flow (together with a `kinetic under-cooling' regularisation) where the forcing depends on the solution of a PDE that
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Phase field modelling of surfactants in multi-phase flow Interfaces Free Bound. (IF 1.0) Pub Date : 2019-12-18 Oliver Dunbar, Kei Fong Lam, Björn Stinner
A diffuse interface model for surfactants in multi-phase flow with three or more fluids is derived. A system of Cahn-Hilliard equations is coupled with a Navier-Stokes system and an advection-diffusion equation for the surfactant ensuring thermodynamic consistency. By an asymptotic analysis the model can be related to a moving boundary problem in the sharp interface limit, which is derived from first
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A lower bound for the void coalescence load in nonlinearly elastic solids Interfaces Free Bound. (IF 1.0) Pub Date : 2019-12-18 Victor Cañulef-Aguilar, Duvan Henao
The problem of the sudden growth and coalescence of voids in elastic media is considered. The Dirichlet energy is minimized among incompressible and invertible Sobolev deformations of a two-dimensional domain having $n$ microvoids of radius $\varepsilon$. The constraint is added that the cavities should reach at least certain minimum areas $v_{1},...,v_{n}$ after the deformation takes place. They can
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Optimal control problem for viscous systems of conservation laws, with geometric parameter, and application to the Shallow-Water equations Interfaces Free Bound. (IF 1.0) Pub Date : 2019-09-24 Sébastien Court,Karl Kunisch,Laurent Pfeiffer
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A homogenization result in the gradient theory of phase transitions Interfaces Free Bound. (IF 1.0) Pub Date : 2019-09-24 Riccardo Cristoferi, Irene Fonseca, Adrian Hagerty, Cristina Popovici
A homogenization problem arising in the gradient theory of ∞uid-∞uid phase transitions is addressed in the vector-valued setting by means of i-convergence.
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Convergence of the Allen–Cahn equation to the mean curvature flow with 90o-contact angle in 2D Interfaces Free Bound. (IF 1.0) Pub Date : 2019-09-24 Helmut Abels, Maximilian Moser
We consider the sharp interface limit of the Allen-Cahn equation with homogeneous Neumann boundary condition in a two-dimensional domain $\Omega$, in the situation where an interface has developed and intersects $\partial\Omega$. Here a parameter $\varepsilon>0$ in the equation, which is related to the thickness of the diffuse interface, is sent to zero. The limit problem is given by mean curvature
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Large time behavior of a two phase extension of the porous medium equation Interfaces Free Bound. (IF 1.0) Pub Date : 2019-07-22 Ahmed Ait Hammou Oulhaj, Clément Cancès, Claire Chainais-Hillairet, Philippe Laurençot
We study the large time behavior of the solutions to a two phase extension of the porous medium equation, which models the so-called seawater intrusion problem. The goal is to identify the self-similar solutions that correspond to steady states of a rescaled version of the problem. We fully characterize the unique steady states that are identified as minimizers of a convex energy and shown to be radially
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A general thermodynamical model for adhesive frictional contacts between viscoelastic or poro-viscoelastic bodies at small strains Interfaces Free Bound. (IF 1.0) Pub Date : 2019-07-22 Tomáš Roubíček
A general model covering a large variety of the so-called adhesive or cohesive, possibly also frictional, contact interfaces between visco-elastic bodies with inertia considered in a thermodynamical context is presented. A semi-implicit time discretisation which conserves energy, is numerically stable and convergent, and which advantageously decouples the system is devised. An extension to porous media
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Well-posedness of non-isentropic Euler equations with physical vacuum Interfaces Free Bound. (IF 1.0) Pub Date : 2019-07-22 Yongcai Geng, Yachun Li, Dehua Wang, Runzhang Xu
We consider the local well-posedness of the one-dimensional nonisentropic Euler equations with moving physical vacuum boundary condition. The physical vacuum singularity requires the sound speed to be scaled as the square root of the distance to the vacuum boundary. The main difficulty lies in the fact that the system of hyperbolic conservation laws becomes characteristic and degenerate at the vacuum
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Non-transversal intersection of the free and fixed boundary in the mean-field theory of superconductivity Interfaces Free Bound. (IF 1.0) Pub Date : 2019-07-22 Emanuel Indrei
Non-transversal intersection of the free and fixed boundary is shown to hold and a classification of blow-up solutions is given for obstacle problems generated by fully nonlinear uniformly elliptic operators in two dimensions which appear in the mean-field theory of superconducting vortices.
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Analysis of blow-ups for the double obstacle problem in dimension two Interfaces Free Bound. (IF 1.0) Pub Date : 2019-07-22 Gohar Aleksanyan
In this article we study a normalised double obstacle problem with polynomial obstacles $ p^1\leq p^2$ under the assumption that $ p^1(x)=p^2(x)$ iff $ x=0$. In dimension two we give a complete characterisation of blow-up solutions depending on the coefficients of the polynomials $p^1, p^2$. In particular, we see that there exists a new type of blow-ups, that we call double-cone solutions since the
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Existence, uniqueness and concentration for a system of PDEs involving the Laplace–Beltrami operator Interfaces Free Bound. (IF 1.0) Pub Date : 2019-05-09 Micol Amar,Roberto Gianni
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Relaxation to a planar interface in the Mullins–Sekerka problem Interfaces Free Bound. (IF 1.0) Pub Date : 2019-05-09 Olga Chugreeva, Felix Otto, Maria Westdickenberg
We analyze the convergence rates to a planar interface in the Mullins-Sekerka model by applying a relaxation method based on relationships among distance, energy, and dissipation. The relaxation method was developed by two of the authors in the context of the 1-d Cahn-Hilliard equation and the current work represents an extension to a higher dimensional problem in which the curvature of the interface
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Global stability for solutions to the exponential PDE describing epitaxial growth Interfaces Free Bound. (IF 1.0) Pub Date : 2019-05-09 Jian-Guo Liu, Robert Strain
In this paper we prove the global existence, uniqueness, optimal large time decay rates, and uniform gain of analyticity for the exponential PDE $h_t=\Delta e^{-\Delta h}$ in the whole space $\mathbb{R}^d_x$. We assume the initial data is of medium size in the critical Wiener algebra $\Delta h \in A(\mathbb{R}^d)$. This exponential PDE was derived in (Krug, Dobbs, and Majaniemi in 1995) and more recently
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Numerical solution of a nonlinear eigenvalue problem arising in optimal insulation Interfaces Free Bound. (IF 1.0) Pub Date : 2019-05-09 Sören Bartels, Giuseppe Buttazzo
The optimal insulation of a heat conducting body by a thin film of variable thickness can be formulated as a nondifferentiable, nonlocal eigenvalue problem. The discretization and iterative solution for the reliable computation of corresponding eigenfunctions that determine the optimal layer thickness are addressed. Corresponding numerical experiments confirm the theoretical observation that a symmetry