- journal articles metadata
European Mathematical Society Publishing House
2024-03-28 12:08:01
5
https://www.ems-ph.org/meta/jmeta-stream.php?jrn=IFB&vol=18&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
subscribers, moving wall 5 years
European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society
18
2016
3
On Hamilton–Jacobi–Bellman equations with convex gradient constraints
Ryan
Hynd
University of Pennsylvania, PHILADELPHIA, UNITED STATES
Henok
Mawi
Howard University, WASHINGTON, UNITED STATES
Fully nonlinear, free boundary problem, Bernstein’s method
We study PDE of the form $\max\{F(D^2u,x)-f(x), H(Du)\}=0$ where $F$ is uniformly elliptic and convex in its first argument, $H$ is convex, $f$ is a given function and $u$ is the unknown. These equations are derived from dynamic programming in a wide class of stochastic singular control problems. In particular, examples of these equations arise in mathematical finance models involving transaction costs, in queuing theory, and spacecraft control problems. The main aspects of this work are to identify conditions under which solutions are uniquely defined and have Lipschitz continuous gradients.
Partial differential equations
Systems theory; control
291
315
10.4171/IFB/365
http://www.ems-ph.org/doi/10.4171/IFB/365
A Hopf–Lax formula for the time evolution of the level-set equation and a new approach to shape sensitivity analysis
Daniel
Kraft
Universität Graz, GRAZ, AUSTRIA
Level-set method, shape optimisation, Hopf–Lax formula, viscosity solutions, nonfattening, shape-sensitivity analysis
The level-set method is used in many different applications to describe the propagation of shapes and domains. When scalar speed fields are used to encode the desired shape evolution, this leads to the classical level-set equation. We present a concise Hopf–Lax representation formula that can be used to characterise the evolved domains at arbitrary times. This result is also applicable for the case of speed fields without a fixed sign, even though the level-set equation has a non-convex Hamiltonian in these situations. The representation formula is based on the same idea that underpins the Fast-Marching Method, and it provides a strong theoretical justification for a generalised Composite Fast-Marching method. Based on our Hopf–Lax formula, we are also able to present new theoretical results. In particular, we show non-fattening of the zero level set in a measure-theoretic sense, derive a very general shapesensitivity calculus that does not require the usual regularity assumptions on the domains, prove optimal Lipschitz constants for the evolved level-set function and discuss the effect of perturbations in both the speed field and the initial geometry.
Calculus of variations and optimal control; optimization
317
353
10.4171/IFB/366
http://www.ems-ph.org/doi/10.4171/IFB/366
Interfaces determined by capillarity and gravity in a two-dimensional porous medium
Maria
Calle
Universidad Carlos III, MADRID-GETAFE, SPAIN
Carlota Maria
Cuesta
Universidad del Pais Vasco, BILBAO, SPAIN
Juan
Velázquez
Universität Bonn, BONN, GERMANY
Capillarity-gravity interface, two-dimensional porous medium, probabilistic asymptotic analysis
We consider a two-dimensional model of a porous medium where circular grains are uniformly distributed in a squared container. We assume that such medium is partially filled with water and that the stationary interface separating the water phase from the air phase is described by the balance of capillarity and gravity. Taking the unity as the average distance between grains, we identify four asymptotic regimes that depend on the Bond number and the size of the container. We analyse, in probabilistic terms, the possible global interfaces that can form in each of these regimes. In summary, we show that in the regimes where gravity dominates the probability of configurations of grains allowing solutions close to the horizontal solution is close to one. Moreover, in such regimes where the size of the container is sufficiently large we can describe deviations from the horizontal in probabilistic terms. On the other hand, when capillarity dominates while the size of the container is sufficiently large, we find that the probability of finding interfaces close to the graph of a given smooth curve without self-intersections is close to one.
Partial differential equations
Probability theory and stochastic processes
Fluid mechanics
355
391
10.4171/IFB/367
http://www.ems-ph.org/doi/10.4171/IFB/367
On a three-dimensional free boundary problem modeling electrostatic MEMS
Philippe
Laurençot
Université de Toulouse, TOULOUSE CEDEX 9, FRANCE
Christoph
Walker
Leibniz-Universität Hannover, HANNOVER, GERMANY
MEMS, free boundary problem, stationary solutions
We consider the dynamics of an electrostatically actuated thin elastic plate being clamped at its boundary above a rigid plate. While the existing literature focuses so far on a two-dimensional geometry, the present model considers a three-dimensional device where the harmonic electrostatic potential varies in the three-dimensional time-dependent region between the plates. The elastic plate deflection evolves according to a fourth-order semilinear parabolic equation which is coupled to the square of the gradient trace of the electrostatic potential on this plate. The strength of the coupling is tuned by a parameter proportional to the square of the applied voltage. We prove that this free boundary problem is locally well-posed in time and that for small values of solutions exist globally in time. We also derive the existence of a branch of asymptotically stable stationary solutions for small values of and non-existence of stationary solutions for large values thereof, the latter being restricted to a disc-shaped plate.
Partial differential equations
393
411
10.4171/IFB/368
http://www.ems-ph.org/doi/10.4171/IFB/368
On the justification of the quasistationary approximation of several parabolic moving boundary problems – Part II
Friedrich
Lippoth
Leibniz Universität Hannover, HANNOVER, GERMANY
Moving boundary problem, maximal regularity, quasistationary approximation, singular limit
We rigorously justify the quasistationary approximations of two moving boundary problems. We work out a systematic procedure to derive a priori estimates that allow to pass to the singular limit. The problems under our consideration are a one-phase osmosis model and the one-phase Stefan problem with Gibbs–Thomson correction and kinetic undercooling.
Partial differential equations
Systems theory; control
413
439
10.4171/IFB/369
http://www.ems-ph.org/doi/10.4171/IFB/369