Related papers: The planar Plateau's problem via capillarity
We show a method to solve the problem of the brachistochrone as well as other variational problems with the help of the soap films that are formed between two suitable surfaces. We also show the interesting connection between some…
In this paper, we study a second order variational problem for locally convex hypersurfaces, which is the affine invariant analogue of the classical Plateau problem for minimal surfaces. We prove existence, regularity and uniqueness results…
The Kirchhoff-Plateau problem concerns the equilibrium shapes of a system in which a flexible filament in the form of a closed loop is spanned by a liquid film, with the filament being modeled as a Kirchhoff rod and the action of the…
By employing the method of moving planes in a novel way we extend some classical symmetry and rigidity results for smooth minimal surfaces to surfaces that have singularities of the sort typically observed in soap films.
Real foams can be viewed as a geometrically well-organized dispersion of more or less spherical bubbles in a liquid. When the foam is so drained that the liquid content significantly decreases, the bubbles become polyhedral-like and the…
Towards identifying the number of minimal surfaces sharing the same boundary from the geometry of the boundary, we propose a numerical scheme with high speed and high accuracy. Our numerical scheme is based on the method of fundamental…
A soap film is actually a thin solid fluid bounded by two surfaces of opposite orientation. It is natural to model the film using one polyhedron for each side. Two problems are to get the polyhedra for both sides to be in the same place…
We present some old and recent regularity results concerning minimal and almost minimal sets in domains of the Euclidean space. We concentrate on a sliding variant of Almgren's notion of minimality, which is well suited in the context of…
In this paper, we shall study the Dirichlet problem for the minimal surfaces equation. We prove some results about the boundary behaviour of a solution of this problem. We describe the behaviour of a non-converging sequence of solutions in…
Motivated by the study of the equilibrium equations for a soap film hanging from a wire frame, we prove a compactness theorem for surfaces with asymptotically vanishing mean curvature and fixed or converging boundaries. In particular, we…
Classically, Plateau's problem asks to find a surface of the least area with a given boundary $B$. In this article, we investigate a version of Plateau's problem, where the boundary of an admissible surface is only required to partially…
We consider the steady-state analysis of a pinned elastic plate lying on the free surface of a thin viscous fluid, forced by the motion of a bottom substrate moving at constant speed. A mathematical model incorporating elasticity,…
We study minimal surfaces which arise in wetting and capillarity phenomena. Using conformal coordinates, we reduce the problem to a set of coupled boundary equations for the contact line of the fluid surface, and then derive simple…
We discuss the behaviour of a thin soap film facing a frame element: the pressure in the Plateau border around the frame is lower than the film pressure, and the film thins out over a certain distance lambda(t), due to the formation of a…
We define and prove the existence of unique solutions of an asymptotic Plateau problem for spacelike maximal surfaces in the pseudo-hyperbolic space of signature (2, n): the boundary data is given by loops on the boundary at infinity of the…
We consider minimal hypersurfaces inside the unit ball whose boundary on the sphere is a small perturbation of the link of a minimizing quadratic cone. We show that such minimal surfaces are uniquely determined by their boundary condition.…
We study the existence of solutions to general measure-minimization problems over topological classes that are stable under localized Lipschitz homotopy, including the standard Plateau problem without the need for restrictive assumptions…
The collapse of a catenoidal soap film when the rings supporting it are moved beyond a critical separation is a classic problem in interface motion in which there is a balance between surface tension and the inertia of the surrounding air,…
Capillary forces are involved in a variety of natural phenomena, ranging from droplet breakup to the physics of clouds. The forces from surface tension can also be exploited in industrial application provided the length scales involved are…
Shallow flow or thin liquid film models are used for a wide range of physical and engineering problems. Shallow flow models allow capturing the free surface of the fluid with little effort and reducing the three-dimensional problem to a…