- journal articles metadata
European Mathematical Society Publishing House
2024-03-29 11:26:43
20
https://www.ems-ph.org/meta/jmeta-stream.php?jrn=IFB&vol=19&update_since=2024-03-29
Interfaces and Free Boundaries
Interfaces Free Bound.
IFB
1463-9963
1463-9971
Partial differential equations
Numerical analysis
Fluid mechanics
Biology and other natural sciences
10.4171/IFB
http://www.ems-ph.org/doi/10.4171/IFB
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European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society
19
2017
1
Finite horizon model predictive control of electrowetting on dielectric with pinning
Harbir
Antil
George Mason University, FAIRFAX, UNITED STATES
Michael
Hintermüller
Weierstrass-Institut, BERLIN, GERMANY
Ricardo
Nochetto
University of Maryland, COLLEGE PARK, UNITED STATES
Thomas
Surowiec
Philipps-Universität Marburg, MARBURG, GERMANY
Donat
Wegner
Humboldt-Universität zu Berlin, BERLIN, GERMANY
Electrowetting on dielectric, EWOD, contact line pinning, surface tension, sharp interface, optimal control of free boundary problems, mathematical program with equilibrium constraints, MPEC, PDE-constrained optimization, barycenter matching, trajectory tracking
A time-discrete spatially-continuous electrowetting on dielectric (EWOD) model with contact line pinning is considered as the state system in an optimal control framework. The pinning model is based on a complementarity condition. In addition to the physical variables describing velocity, pressure, and voltage, the solid-liquid-air interface, i.e., the contact line, arises as a geometric variable that evolves in time. Due to the complementarity condition, the resulting optimal control of a free boundary problem is thus a mathematical program with equilibrium constraints (MPEC) in function space. In order to cope with the geometric variable, a finite horizon model predictive control approach is proposed. Dual stationarity conditions are derived by applying a regularization procedure, exploiting techniques from PDE-constrained optimization, and then passing to the limit in the regularization parameters. Moreover, a function-space-based numerical procedure is developed by following the theoretical limit argument used in the derivation of the dual stationarity conditions. The performance of the algorithm is demonstrated by several examples; including barycenter matching and trajectory tracking.
Calculus of variations and optimal control; optimization
Fluid mechanics
Operations research, mathematical programming
1
30
10.4171/IFB/375
http://www.ems-ph.org/doi/10.4171/IFB/375
Zero width limit of the heat equation on moving thin domains
Tatsu-Hiko
Miura
University of Tokyo, TOKYO, JAPAN
Heat equation, moving thin domains, evolving surfaces
We study the behavior of a variational solution to the Neumann type problem of the heat equation on a moving thin domain $\Omega_{\varepsilon}(t)$ that converges to an evolving surface $\Gamma (t)$ as the width of $\Omega_\varepsilon(t)$ goes to zero. We show that, under suitable assumptions, the average in the normal direction of $\Gamma(t)$ of a variational solution to the heat equation converges weakly in a function space on $\Gamma(t)$ as the width of $\Omega_\varepsilon(t)$ goes to zero, and that the limit is a unique variational solution to a limit equation on $\Gamma(t)$, which is a new type of linear diffusion equation involving the mean curvature and the normal velocity of $\Gamma(t)$. We also estimate the difference between variational solutions to the heat equation on $\Omega_\varepsilon(t)$ and the limit equation on $\Gamma(t)$.
Partial differential equations
31
77
10.4171/IFB/376
http://www.ems-ph.org/doi/10.4171/IFB/376
A weak formulation for a rate-independent delamination evolution with inertial and viscosity effects subjected to unilateral constraint
Riccardo
Scala
Universidade de Lisboa, LISBOA, PORTUGAL
Second order parabolic equation, viscoelasticity, energetic formulation, delamination, adhesion, rate-independent system, unilateral constraint
We consider a system of two viscoelastic bodies attached on one side by an adhesive where a delamination process occurs. We study the dynamic of the system for small strains, subjected to external forces, suitable boundary conditions, and an unilateral constraint on the jump of the displacement at the interface between the bodies. The constraint arises in a graph inclusion, while the delamination coefficient evolves in a rate-independent way. We prove the existence of a weak solution to the corresponding system of PDEs.
Partial differential equations
Operator theory
Mechanics of deformable solids
79
107
10.4171/IFB/377
http://www.ems-ph.org/doi/10.4171/IFB/377
Minimising a relaxed Willmore functional for graphs subject to boundary conditions
Klaus
Deckelnick
Otto-von-Guericke-Universität Magdeburg, MAGDEBURG, GERMANY
Hans-Christoph
Grunau
Otto-von-Guericke-Universität Magdeburg, MAGDEBURG, GERMANY
Matthias
Röger
Technische Universität Dortmund, DORTMUND, GERMANY
Graph, Willmore functional, boundary conditions, area estimate, diameter estimate, lower semicontinuity, relaxation, existence of a minimiser
For a bounded smooth domain in the plane and smooth boundary data we consider the minimisation of the Willmore functional for graphs subject to Dirichlet or Navier boundary conditions. For $H^2$-regular graphs we show that bounds for the Willmore energy imply bounds on the surface area and on the height of the graph. We then consider the $L^1$-lower semicontinuous relaxation of the Willmore functional, which is shown to be indeed its largest possible extension, and characterise properties of functions with finite relaxed energy. In particular, we deduce compactness and lower-bound estimates for energy-bounded sequences. The lower bound is given by a functional that describes the contribution by the regular part of the graph and is defined for a suitable subset of $BV(\Omega)$. We further show that finite relaxed Willmore energy implies the attainment of the Dirichlet boundary data in an appropriate sense, and obtain the existence of a minimiser in $L^\infty\cap BV$ for the relaxed energy. Finally, we extend our results to Navier boundary conditions and more general curvature energies of Canham–Helfrich type.
Calculus of variations and optimal control; optimization
Differential geometry
109
140
10.4171/IFB/378
http://www.ems-ph.org/doi/10.4171/IFB/378
2
A new phase field model for inhomogeneous minimal partitions, and applications to droplets dynamics
Elie
Bretin
Université de Lyon, Villeurbanne, France
Simon
Masnou
Université Claude Bernard Lyon 1, Villeurbanne, France
Phase field model, multiphase perimeter, $\Gamma$-convergence, droplets, material sciences, image processing
We propose and analyze in this paper a new derivation of a phase-field model to approximate inhomogeneous multiphase perimeters. It is based on suitable decompositions of perimeters under some embeddability condition which allows not only an explicit derivation of the model from the surface tensions, but also gives rise to a $\Gamma$-convergence result. Moreover, thanks to the nice form of the approximating energy, we can use a simple and robust scheme to simulate its gradient flow. We illustrate the efficiency of our approach with a series of numerical simulations in 2D and 3D, and we address in particular the dynamics of droplets evolving on a fixed solid.
Calculus of variations and optimal control; optimization
Numerical analysis
141
182
10.4171/IFB/379
http://www.ems-ph.org/doi/10.4171/IFB/379
On the shape of the free boundary of variational inequalities with gradient constraints
Mohammad
Safdari
Institute for Research in Fundamental Sciences (IPM), Tehran, Iran
Free boundary, variational inequality, gradient constraint, global regularity
In this article we derive an estimate on the number of local maxima of the free boundary of the minimizer of $$I[v]:=\int_{U}\frac{1}{2}|Dv|^{2}-\eta v\,dx,$$ subject to the pointwise gradient constraint $$(|D_{1}v|^{q}+|D_{2}v|^{q})^{\frac{1}{q}}\le1.$$ This also gives an estimate on the number of connected components of the free boundary. In addition, we further study the free boundary when $U$ is a polygon with some symmetry.
Partial differential equations
183
200
10.4171/IFB/380
http://www.ems-ph.org/doi/10.4171/IFB/380
Weak solutions and regularity of the interface in an inhomogeneous free boundary problem for the $p(x)$-Laplacian
Claudia
Lederman
Universidad de Buenos Aires, Argentina
Noemi
Wolanski
Universidad de Buenos Aires, Argentina
Free boundary problem, variable exponent spaces, regularity of the free boundary, singular perturbation, inhomogeneous problem
In this paper we study a one phase free boundary problem for the $p(x)$-Laplacian with non-zero right hand side. We prove that the free boundary of a weak solution is a $C^{1,\alpha}$ surface in a neighborhood of every „flat" free boundary point. We also obtain further regularity results on the free boundary, under further regularity assumptions on the data. We apply these results to limit functions of an inhomogeneous singular perturbation problem for the $p(x)$-Laplacian that we studied in [25].
Partial differential equations
201
241
10.4171/IFB/381
http://www.ems-ph.org/doi/10.4171/IFB/381
Weak solutions to thin-film equations with contact-line friction
Maria
Chiricotto
Universität Heidelberg, Germany
Lorenzo
Giacomelli
Università di Roma La Sapienza, Italy
Fourth order degenerate parabolic equations, thin film equations, free boundary problems, lubrication theory, moving contact line, droplets
We consider the thin-film equation with a prototypical contact-line condition modeling the effect of frictional forces at the contact line where liquid, solid, and air meet. We show that such condition, relating flux with contact angle, naturally emerges from applying a thermodynamic argument due to Weiqing Ren and Weinan E [Commun. Math. Sci. 9 (2011), 597–606] directly into the framework of lubrication approximation. For the resulting free boundary problem, we prove global existence of weak solutions, as well as global existence and uniqueness of approximating solutions which satisfy the contact line condition pointwise. The analysis crucially relies on new contractivity estimates for the location of the free boundary.
Partial differential equations
Fluid mechanics
243
271
10.4171/IFB/382
http://www.ems-ph.org/doi/10.4171/IFB/382
Convergence of thresholding schemes incorporating bulk effects
Tim
Laux
Max-Planck-Institut für Mathematik in den Naturwissenschaften, Leizpig, Germany
Drew
Swartz
Purdue University, West Lafayette, USA
Mean curvature flow, thresholding, MBO scheme, minimizing movements, volume preserving
In this paper we establish the convergence of three computational algorithms for interface motion in a multi-phase system, which incorporate bulk effects. The algorithms considered fall under the classification of thresholding schemes, in the spirit of the celebrated Merriman–Bence–Osher algorithm for producing an interface moving by mean curvature. The schemes considered here all incorporate either a local force coming from an energy in the bulk, or a non-local force coming from a volume constraint. We first establish the convergence of a scheme proposed by Ruuth-Wetton for approximating volume-preserving mean-curvature flow. Next we study a scheme for the geometric flow generated by surface tension plus bulk energy. Here the limit is motion by mean curvature (MMC) plus forcing term. Last we consider a thresholding scheme for simulating grain growth in a polycrystal surrounded by air, which incorporates boundary effects on the solid-vapor interface. The limiting flow is MMC on the inner grain boundaries, and volume-preserving MMC on the solid-vapor interface.
Numerical analysis
Partial differential equations
273
304
10.4171/IFB/383
http://www.ems-ph.org/doi/10.4171/IFB/383
3
Sharp stability inequalities for planar double bubbles
Marco
Cicalese
Technische Universität München, Germany
Gian Paolo
Leonardi
Università di Modena e Reggio Emilia, Modena, Italy
Francesco
Maggi
The University of Texas at Austin, USA
Isoperimetric problems, partitioning problems, stability, double-bubble
In this paper we address the global stability problem for double-bubbles in the plane. This is accomplished by combining the improved convergence theorem for planar clusters developed in [8] with an ad hoc analysis of the problem, which addresses the delicate interaction between the (possible) dislocation of singularities and the multiple-volumes constraint.
Calculus of variations and optimal control; optimization
305
350
10.4171/IFB/384
http://www.ems-ph.org/doi/10.4171/IFB/384
A free boundary problem with log–term singularity
Olivaine
de Queiroz
Universidade Estadual de Campinas, Brazil
Henrik
Shahgholian
KTH Royal Institute of Technology, Stockholm, Sweden
Free boundary, regularity theory, logarithmic singularity, porosity
We study a minimum problem for a non-differentiable functional whose reaction term does not have scaling properties. Specifically we consider the functional $$\mathcal J(v)=\int_\Omega\left(\frac{|\nabla v|^2}{2} -v^+(\mathrm {log}\: v-1)\right)dx\to \mathrm {min}$$ which should be minimized in some natural admissible class of non-negative functions. Here, $v^+=\max\{0,v\}.$ The Euler–Lagrange equation associated with $\mathcal J$ is $$-\Delta u= \chi_{\{u>0\}}\mathrm {log}\: u,$$ which becomes singular along the free boundary $\partial\{u>0\}.$ Therefore, the regularity results do not follow from classical methods. Besides, the logarithmic forcing term does not have scaling properties, which are very important in the study of free boundary theory. Despite these difficulties, we obtain optimal regularity of a minimizer and show that, close to every free boundary point, they exhibit a super-characteristic growth like $$r^2|\mathrm {log}\: r|.$$ This estimate is crucial in the study of analytic and geometric properties of the free boundary.
Combinatorics
351
369
10.4171/IFB/385
http://www.ems-ph.org/doi/10.4171/IFB/385
Numerical investigation of the free boundary regularity for a degenerate advection-diffusion problem
Léonard
Monsaingeon
Université de Lorraine, Vandœuvre-lès-Nancy, France
Degenerate diffusion, traveling waves, free boundaries, Hamilton–Jacobi equations, numerical investigation
We study the free boundary regularity of the traveling wave solutions to a degenerate advection-diffusion problem of Porous Medium type, whose existence was proved in [24]. We set up a finite difference scheme allowing to compute approximate solutions and capture the free boundaries, and we carry out a numerical investigation of their regularity. Based on some nondegeneracy assumptions supported by solid numerical evidence, we prove the Lipschitz regularity of the free boundaries. Our simulations indicate that this regularity is optimal, and the free boundaries seem to develop Lipschitz corners at least for some values of the nonlinear diffusion exponent. We discuss analytically the existence of corners in the framework of viscosity solutions to certain periodic Hamilton–Jacobi equations, whose validity is again supported by numerical evidence.
Partial differential equations
371
391
10.4171/IFB/386
http://www.ems-ph.org/doi/10.4171/IFB/386
Some results on anisotropic fractional mean curvature flows
Antonin
Chambolle
Ecole Polytechnique, Palaiseau, France
Matteo
Novaga
Università di Pisa, Italy
Berardo
Ruffini
Université de Montpellier, France
Fractional mean curvature flow, convexity, variational scheme
We show the consistency of a threshold dynamics type algorithm for the anisotropic motion by fractional mean curvature, in the presence of a time dependent forcing term. Beside the consistency result, we show that convex sets remain convex during the evolution, and the evolution of a bounded convex set is uniquely defined.
Differential geometry
Ordinary differential equations
393
415
10.4171/IFB/387
http://www.ems-ph.org/doi/10.4171/IFB/387
Hysteresis in porous media: Modelling and analysis
Ben
Schweizer
Technische Universität Dortmund, Germany
Porous media, hysteresis, unsaturated flow, gravity fingering
Unsaturated flow through porous media can be modelled by a partial differential equation using saturation $s$ and pressure $p$ as unknowns. Experimental data as well as elementary physical arguments show that the coupling of the two variables must take into account hysteresis. In this survey, we describe the physical origins of porous media hysteresis, present the ideas of its mathematical description, and review the analysis of the resulting hysteresis models.
Fluid mechanics
Partial differential equations
Operator theory
417
447
10.4171/IFB/388
http://www.ems-ph.org/doi/10.4171/IFB/388
On the propagation of a periodic flame front by an Arrhenius kinetic
Nathaël
Alibaud
ENSMM, Besançon and Université de Bourgogne Franche-Comté, Besançon, France
Gawtum
Namah
ENSMM, Besançon and Université de Bourgogne Franche-Comté, Besançon, France
Free boundary problems, front propagation, combustion, Arrhenius law, travelling wave solutions, periodic solutions, homogenization, curvature effects, asymptotic analysis
We consider the propagation of a flame front in a solid periodic medium. The model is governed by a free boundary system in which the front’s velocity depends on the temperature via an Arrhenius kinetic. We show the existence of travelling wave solutions and consider their homogenization as the period tends to zero. The main difficulty lies in the degeneracy of the Arrhenius function which requires an a priori lower bound of the propagation’s speed. We next analyze the curvature effects on the homogenization and obtain a continuum of limiting waves parametrized by the ratio “curvature coefficient/period.” Remarkable features are the monotonicity of the speed with respect to the “curvature regime,” together with the explicit computations of the minimal and maximal speeds. We finally identify the asymptotic expansion of the heterogeneous front’s profile with respect to the period.
Partial differential equations
Classical thermodynamics, heat transfer
449
494
10.4171/IFB/389
http://www.ems-ph.org/doi/10.4171/IFB/389
4
Rigidity and stability of spheres in the Helfrich model
Yann
Bernard
ETH Zentrum, Zürich, Switzerland
Glen
Wheeler
University of Wollongong, Australia
Valentina-Mira
Wheeler
University of Wollongong, Australia
Spherocytosis, biomembranes, Helfrich model, differential geometry
The Helfrich functional, denoted by $\mathcal H^{c_0}$, is a mathematical expression proposed by Helfrich (1973) for the natural free energy carried by an elastic phospholipid bilayer. Helfrich theorises that idealised elastic phospholipid bilayers minimise $\mathcal H^{c_0}$ among all possible configurations. The functional integrates a spontaneous curvature parameter $c_0$ together with the mean curvature of the bilayer and constraints on area and volume, either through an inclusion of osmotic pressure difference and tensile stress or otherwise. Using the mathematical concept of embedded orientable surface to represent the configuration of the bilayer, one might expect to be able to adapt methods from differential geometry and the calculus of variations to perform a fine analysis of bilayer configurations in terms of the parameters that it depends upon. In this article we focus upon the case of spherical red blood cells with a view to better understanding spherocytes and spherocytosis. We provide a complete classification of spherical solutions in terms of the parameters in the Helfrich model. We additionally present some further analysis on the rigidity and stability of spherocytes.
Mechanics of deformable solids
General
Geometry
495
523
10.4171/IFB/390
http://www.ems-ph.org/doi/10.4171/IFB/390
1
15
2018
Advection-diffusion equations with random coefficients on evolving hypersurfaces
Ana
Djurdjevac
Freie Universität Berlin, Germany
Advection-diffusion, evolving surfaces, uncertainty quantification, random coefficients, existence
We present the analysis of advection-diffusion equations with random coefficients on moving hypersurfaces. We define a weak and a strong material derivative, which account for the spatial movement. Then we define the solution space for these kind of equations, which is the Bochner-type space of random functions defined on a moving domain. We consider both cases, uniform and log-normal distributions of the diffusion coefficient. Under suitable regularity assumptions we prove the existence and uniqueness of weak solutions of the equation under analysis, and also we give some regularity results about the solution.
Partial differential equations
General
525
552
10.4171/IFB/391
http://www.ems-ph.org/doi/10.4171/IFB/391
1
15
2018
A conservative scheme for non-classical solutions to a strongly coupled PDE-ODE problem
Christophe
Chalons
Université de Versailles Saint-Quentin-en-Yvelines, France
Maria Laura
Delle Monache
Inria Grenoble Rhône - Alpes, France and Rutgers University, Camden, USA
Paola
Goatin
Inria Sophia Antipolis – Méditerranée, France
Scalar conservation laws with local moving constraints, traffic flow modeling, PDE-ODE coupling, conservative finite volume schemes
We consider a strongly coupled PDE-ODE system modeling the influence of a slow and large vehicle on road traffic. The model consists of a scalar conservation law describing the main traffic evolution and an ODE accounting for the trajectory of the slower vehicle that depends on the downstream traffic density. The moving constraint is operated by an inequality on the flux, which accounts for the bottleneck created on the road by the presence of the slower vehicle.We introduce a conservative scheme for the constrained hyperbolic PDE and we use a tracking algorithm for the ODE. We perform numerical tests and compute numerically the order of convergence.
Partial differential equations
Operations research, mathematical programming
553
570
10.4171/IFB/392
http://www.ems-ph.org/doi/10.4171/IFB/392
1
15
2018
Varifold solutions of a sharp interface limit of a diffuse interface model for tumor growth
Stefano
Melchionna
Universität Wien, Austria
Elisabetta
Rocca
Università degli Studi di Pavia, Italy
Free boundary problems, diffuse interface models, sharp interface limit, Cahn–Hilliard equation, Darcy law, tumor growth
We discuss the sharp interface limit of a diffuse interface model for a coupled Cahn–Hilliard–Darcy system that models tumor growth when a certain parameter $\epsilon > 0$, related to the interface thickness, tends to zero. In particular, we prove that weak solutions to the related initial boundary value problem tend to varifold solutions of a corresponding sharp interface model when $\epsilon$ goes to zero.
Partial differential equations
Calculus of variations and optimal control; optimization
Biology and other natural sciences
571
590
10.4171/IFB/393
http://www.ems-ph.org/doi/10.4171/IFB/393
1
15
2018
Stability and asymptotic behavior of transonic flows past wedges for the full Euler equations
Gui-Qiang
Chen
Oxford University, UK and Chinese Academy of Sciences, Beijing, China
Jun
Chen
Southern University of Science and Technology, Shenzhen, Guangdong, China
Mikhail
Feldman
University of Wisconsin, Madison, USA
Shock wave, free boundary, wedge problem, steady, supersonic, subsonic, transonic, mixed type, composite type, hyperbolic-elliptic, full Euler equations, physical admissible, existence, stability, asymptotic behavior, decay rate
The existence, uniqueness, and asymptotic behavior of steady transonic flows past a curved wedge, involving transonic shocks, governed by the two-dimensional full Euler equations are established. The stability of both weak and strong transonic shocks under the perturbation of the upstream supersonic flow and the wedge boundary is proved. The problem is formulated as a one-phase free boundary problem, in which the transonic shock is treated as a free boundary. The full Euler equations are decomposed into two algebraic equations and a first-order elliptic system of two equations in Lagrangian coordinates. With careful elliptic estimates by using appropriate weighted Hölder norms, the iteration map is defined and analyzed, and the existence of its fixed point is established by performing the Schauder fixed point argument. The careful analysis of the asymptotic behavior of the solutions reveals particular characters of the full Euler equations.
Partial differential equations
Differential geometry
Fluid mechanics
591
626
10.4171/IFB/394
http://www.ems-ph.org/doi/10.4171/IFB/394
1
15
2018