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European Mathematical Society Publishing House
2024-03-28 09:09:31
5
https://www.ems-ph.org/meta/jmeta-stream.php?jrn=IFB&vol=2&iss=3&update_since=2024-03-28
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
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2
2000
3
On the asymptotic behaviour of anisotropic energies arising in the cardiac bidomain model
Luigi
Ambrosio
Scuola Normale Superiore, PISA, ITALY
Piero
Colli Franzone
Università di Pavia, PAVIA, ITALY
Giuseppe
Savaré
Università di Pavia, PAVIA, ITALY
localized minimization problems; anisotropic energies; cardiac bidomain model
We study the [Gamma]-convergence of a family of vectorial integral functionals, which are the sum of a vanishing anisotropic quadratic form in the gradients and a penalizing double-well potential depending only on a linear combination of the components of their argument. This particular feature arises from the study of the so-called `bidomain model' for the cardiac electric field; one of its consequences is that the L1-norm of a minimizing sequence can be unbounded and therefore a lack of coercivity occurs. We characterize the [Gamma]-limit as a surface integral functional, whose integrand is a convex function of the normal and can be computed by solving a localized minimization problem.
Functional analysis
Probability theory and stochastic processes
213
266
10.4171/IFB/19
http://www.ems-ph.org/doi/10.4171/IFB/19
Global existence for a non-local mean curvature flow as a limit of a parabolic-elliptic phase transition model
Danielle
Hilhorst
Université Paris-Sud, ORSAY CX, FRANCE
Elisabeth
Logak
Université de Cergy-Pontoise, CERGY-PONTOISE CEDEX, FRANCE
Reiner
Schätzle
Universität Tübingen, TÜBINGEN, GERMANY
free boundary problems; mean curvature; reaction-diffusion systems; microphase separation; diblock copolymers
We consider a free boundary problem where the velocity depends on the mean curvature and on some non-local term. This problem arises as the singular limit of a reaction-diffusion system which describes the microphase separation of diblock copolymers. The interface may present singularities in finite time. This leads us to consider weak solutions on an arbitrary time interval and to prove the global-in-time convergence of solutions of the reaction-diffusion system.
Functional analysis
Probability theory and stochastic processes
267
282
10.4171/IFB/20
http://www.ems-ph.org/doi/10.4171/IFB/20
Evolution of compressible and incompressible fluids separated by a closed interface
Irina
Denisova
Russian Acadademy of Sciences, ST. PETERSBURG, RUSSIAN FEDERATION
Free boundary problem, Navier-Stokes equations, two immiscible fluids
This work solves the problem governing the simultaneous motion of two viscous liquids of different kinds: compressible and incompressible. The boundary between the fluids is considered as an unknown (free) interface where the surface tension is taken into account. Although the fluids occupy the whole space 3, one of them should have a finite volume. Local (in time) unique solvability of this problem is obtained in the Sobolev-Slobodetski[inodot]spaces of functions. Estimates of the solution of a model problem for the Stokes equations are considered in detail, the interface between the fluids being a plane. The Schauder method is used to study a linear problem with a compact boundary. The passage to the nonlinear problem is made by successive approximations.
Functional analysis
Probability theory and stochastic processes
283
312
10.4171/IFB/21
http://www.ems-ph.org/doi/10.4171/IFB/21
Thermo-kinetically controlled pattern selection
Michael
Frankel
Indiana University Purdue University Indianapolis, INDIANAPOLIS, UNITED STATES
Laura Kathryn
Gross
University of Akron, AKRON, UNITED STATES
Victor
Roytburd
Rensselaer Polytechnic Institute, TROY, UNITED STATES
bifurcation and pattern formation; free boundary problems
Through a combination of asymptotic and numerical approaches we investigate bifurcation and pattern formation for a free boundary model related to a rapid crystallization of amorphous films and to the self-propagating high-temperature synthesis (solid combustion). The unifying feature of these diverse physical phenomena is the existence of a uniformly propagating wave of phase transition whose stability is controlled by the balance between the energy production at the interface and the energy dissipation into the medium. For the propagation on a two-dimensional strip with thermally insulated edges, we develop a multi-scale weakly-nonlinear analysis that results in a system of ordinary differential equations for the slowly varying amplitudes. We identify a nonlinear parameter which is responsible for the pattern selection, and utilize the amplitude system for predicting the evolving patterns. The pattern selection is confirmed by direct numerical simulations on the free boundary problem. Some numerical results on strongly nonlinear regimes are also presented.
Functional analysis
Probability theory and stochastic processes
313
330
10.4171/IFB/22
http://www.ems-ph.org/doi/10.4171/IFB/22
On Maxwellian equilibria of insulated semiconductors
Luis
Caffarelli
University of Texas at Austin, AUSTIN, UNITED STATES
Jean
Dolbeault
Université de Paris Dauphine, PARIS CEDEX 16, FRANCE
Peter
Markowich
Universität Wien, WIEN, AUSTRIA
Christian
Schmeiser
TU Wien, WIEN, AUSTRIA
Semiconductors, equilibrium, free boundary problems, charge neutrality
A semi-linear elliptic integro-differential equation subject to homogeneous Neumann boundary conditions for the equilibrium potential in an insulated semiconductor device is considered. A variational formulation gives existence and uniqueness. The limit as the scaled Debye length tends to zero is analysed. Two different cases occur. If the number of free electrons and holes is sufficiently high, local charge neutrality prevails throughout the device. Otherwise, depletion regions occur, and the limiting potential is the solution of a free boundary problem.
Functional analysis
Probability theory and stochastic processes
331
339
10.4171/IFB/23
http://www.ems-ph.org/doi/10.4171/IFB/23