- journal articles metadata
European Mathematical Society Publishing House
2024-03-29 15:09:51
6
https://www.ems-ph.org/meta/jmeta-stream.php?jrn=IFB&vol=13&iss=2&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
13
2011
2
Local simplicity, topology, and sets of finite perimeter
David
Caraballo
Georgetown University, WASHINGTON, UNITED STATES
Local simplicity; decomposable; finite perimeter; regularity; reduced boundary; measuretheoretic boundary; total variation; TV-minimization
We introduce a useful and relatively easy to check condition, local simplicity, which provides significantly more structure to sets of finite perimeter, while not being too restrictive. Local simplicity holds for minimizers in a wide variety of variational problems in materials science, biology, image processing, oncology, and other fields. We prove several regularity and structural properties of locally simple sets and their boundaries, including a vital decomposition theorem that in our setting strengthens the conclusion of theorems of H. Federer and L. Ambrosio, V. Caselles, S. Masnou, and J.-M. Morel. We establish strong connections between topology and sets of finite perimeter, so that ordinary notions of openness, closedness, and connectedness may be readily used in the finite perimeter setting. We apply these results to an image reconstruction procedure from image processing, L1 TV-minimization. The density ratio bounds computed to establish local simplicity are themselves of practical importance, as they provide concrete, easy to compute criteria to check simulations against.
Calculus of variations and optimal control; optimization
Computer science
Information and communication, circuits
General
171
189
10.4171/IFB/253
http://www.ems-ph.org/doi/10.4171/IFB/253
On the Hölder regularity of the landscape function
Alessio
Brancolini
Politecnico di Bari, BARI, ITALY
Sergio
Solimini
Politecnico di Bari, BARI, ITALY
Optimal transportation problems; irrigation models; landscape function
We study the Hölder regularity of the landscape function introduced by Santambrogio in [S]. We develop a new technique which both extends Santambrogio’s result to lower Ahlfors regular measures in general dimension h and simplifies its proof.
General
191
222
10.4171/IFB/254
http://www.ems-ph.org/doi/10.4171/IFB/254
Free boundary regularity for a problem with right hand side
Daniela
De Silva
Columbia University, NEW YORK, UNITED STATES
We consider a one-phase free boundary problem with variable coefficients and nonzero right hand side. We prove that flat free boundaries are C1, using a different approach than the classical supconvolution method of Caffarelli. We use this result to deduce that Lipschitz free boundaries are C1, .
General
223
238
10.4171/IFB/255
http://www.ems-ph.org/doi/10.4171/IFB/255
Interface conditions for limits of the Navier–Stokes–Korteweg model
Katharina
Hermsdörfer
Universität Freiburg, FREIBURG, GERMANY
Christiane
Kraus
Angewandte Analysis und Stochastik, BERLIN, GERMANY
Dietmar
Kröner
Universität Freiburg, FREIBURG, GERMANY
liquid-vapour flows; van der Waals–Cahn–Hilliard energy; Navier–Stokes–Korteweg system; singular perturbations
We study the behaviour of the pressure across phase boundaries in liquid-vapour flows. As mathematical model, we consider the static version of the Navier–Stokes–Korteweg model, which belongs to the class of diffuse interface models. From this static equation, a formula for the pressure jump across the phase interface can be derived. If we perform the sharp interface limit, we see that the resulting interface condition for the pressure seems to be inconsistent with classical results of hydrodynamics. Therefore we will present two approaches to recover the results of hydrodynamics in the sharp interface limit at least for special situations.
Fluid mechanics
Partial differential equations
Statistical mechanics, structure of matter
General
239
254
10.4171/IFB/256
http://www.ems-ph.org/doi/10.4171/IFB/256
The effective energy in the Allen–Cahn model with deformation
M.
Šilhavý
Czech Academy of Sciences, PRAGUE 1, CZECH REPUBLIC
phase transitions; diffuse and sharp phase interface; interfacial energy
The sharp interface limit of a diffuse interface theory of phase transitions is considered in static situations. The diffuse interface model is of the Allen--Cahn type with deformation, with a parameter $\varepsilon$ measuring the width of the interface. Equilibrium states of a given elongation and a given interface width are considered and the asymptotics for $\varepsilon\to0$ of the equilibrium energy is determined. The interface energy is defined as the excess energy over the corresponding two phase state with a sharp interface without the interface energy. It is shown that to within the term of order ${\rm o}(\varepsilon)$ the interface energy is equal to $\sigma\varepsilon$ where the coefficient $\sigma$ is given by a new formula that involves the mechanical contribution to the total energy. Also the corresponding equilibrium states are determined and shown to converge to a sharp interface state for $\varepsilon\to0.$
Mechanics of deformable solids
Calculus of variations and optimal control; optimization
General
255
270
10.4171/IFB/257
http://www.ems-ph.org/doi/10.4171/IFB/257
Asymptotic behaviour of a nonlinear parabolic equation with gradient absorption and critical exponent
Razvan Gabriel
Iagar
Universidad Autónoma de Madrid, MADRID, SPAIN
Philippe
Laurençot
Université de Toulouse, TOULOUSE CEDEX 9, FRANCE
Juan Luis
Vázquez
Universidad Autónoma de Madrid, MADRID, SPAIN
Nonlinear parabolic equations; p-Laplacian equation; gradient absorption; asymptotic patterns; Hamilton–Jacobi equation; viscosity solutions
We consider the problem posed for $x\in ℝ^N $ and $t>0$ with nonnegative and compactly supported initial data. We take the exponent $p>2$ which corresponds to slow $p$-Laplacian diffusion. The main feature of the paper is that the exponent $q$ takes the critical value $q=p-1$ which leads to interesting asymptotics. This is due to the fact that in this case both the Hamilton-Jacobi term $|\nabla u|^q$ and the diffusive term $\Delta_p u$ have a similar size for large times. The study performed in this paper shows that a delicate asymptotic equilibrium happens, so that the large-time behaviour of the solutions is described by a rescaled version of a suitable self-similar solution of the Hamilton-Jacobi equation $|\nabla W|^{p-1}=W$, with logarithmic time corrections. The asymptotic rescaled profile is a kind of sandpile with a cusp on top, and it is independent of the space dimension.
Partial differential equations
Calculus of variations and optimal control; optimization
General
271
295
10.4171/IFB/258
http://www.ems-ph.org/doi/10.4171/IFB/258