On the asymptotic behaviour of nonlocal perimeters

We study a class of integral functionals known as nonlocal perimeters, which, intuitively, express a weighted interaction between a set and its complement. The weight is provided by a positive kernel K, which might be singular. In the first part of the paper, we show that these functionals are indeed perimeters in a generalised sense that has been recently introduced by A. Chambolle et al. [Archiv. Rational Mech. Anal. 218 (2015) 1263–1329]. Also, we establish existence of minimisers for the corresponding Plateau’s problem and, when K is radial and strictly decreasing, we prove that halfspaces are minimisers if we prescribe “flat” boundary conditions. A Γ-convergence result is discussed in the second part of the work. We study the limiting behaviour of the nonlocal perimeters associated with certain rescalings of a given kernel that has faster-than-L1 decay at infinity and we show that the Γ-limit is the classical perimeter, up to a multiplicative constant that we compute explicitly.

Covers, soap films and BV functions

In this paper we review the double covers method with constrained BV functions for solving the classical Plateau’s problem. Next, we carefully analyze some interesting examples of soap films compatible with covers of degree larger than two: in particular, the case of a soap film only partially wetting a space curve, a soap film spanning a cubical frame but having a large tunnel, a soap film that retracts to its boundary, and various soap films spanning an octahedral frame.

Hölder regularity at the boundary of two-dimensional sliding almost minimal sets

In [15], Jean Taylor proved a regularity theorem away from the boundary for Almgren almost minimal sets of dimension 2 in {\mathbb{R}^{3}}. It is quite important for understanding the soap films and the solutions of Plateau’s problem away from boundary. In this paper, we will give a regularity result on the boundary for two-dimensional sliding almost minimal sets in {\mathbb{R}^{3}}.

Existence and soap film regularity of solutions to Plateau’s problem

Plateau’s problem is to find a surface with minimal area spanning a given boundary. Our paper presents a theorem for codimension one surfaces in ${\mathbb{R}^{n}}$ in which the usual homological definition of span is replaced with a novel algebraic-topological notion. In particular, our new definition offers a significant improvement over existing homological definitions in the case that the boundary has multiple connected components. Let M be a connected, oriented compact manifold of dimension ${n-2}$ and ${\mathfrak{S}}$ the collection of compact sets spanning M. Using Hausdorff spherical measure as a notion of “size,” we prove: There exists an ${X_{0}}$ in ${\mathfrak{S}}$ with smallest size. Any such ${X_{0}}$ contains a “core” ${X_{0}^{*}\in\mathfrak{S}}$ with the following properties: It is a subset of the convex hull of M and is a.e. (in the sense of ${(n-1)}$-dimensional Hausdorff measure) a real analytic ${(n-1)}$-dimensional minimal submanifold. If ${n=3}$, then ${X_{0}^{*}}$ has the local structure of a soap film. Furthermore, set theoretic solutions are elevated to current solutions in a space with a rich continuous operator algebra.

Numerical simulation of Plateau’s problem of minimal surfaces using nonvariational finite element method

Purpose – Numerical solution of Plateau’s problem of minimal surface using non-variational finite element method. The paper aims to discuss this issue. Design/methodology/approach – An efficient algorithm is proposed for the computation of minimal surfaces and numerical results are presented. Findings – The solutions obtained here are examined for different cases of non-linearity and are found sufficiently accurate. Originality/value – The manuscript provide the non-variational solution for Plateau’s problem. Thus it has a good value in engineering application.