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We study a variational model for soap films in which the films are represented by sets with fixed small volume rather than surfaces. In this problem, a minimizing sequence of completely “wet" films, or sets of finite perimeter spanning a wire frame, may converge to a film containing both wet regions of positive volume and collapsed (dry) surfaces. When collapsing occurs, these limiting objects lie outside the original minimization class and instead are admissible for a relaxed problem. Here we show that the relaxation and the original formulation are equivalent by approximating the collapsed films in the relaxed class by wet films in the original class. [ABSTRACT FROM AUTHOR]

A novel approach to constructing polynomial minimal surfaces (surfaces of zero mean curvature) with isothermal parameterization from Pythagorean triples of complex polynomials is presented, and it is shown that they are Pythagorean normal (PN) surfaces, i.e., their unit normal vectors have a rational dependence on the surface parameters. This construction generalizes a prior approach based on Pythagorean triples of real polynomials, and yields more free shape parameters for surfaces of a specified degree. Moreover, when one of the complex polynomials is just a constant, the minimal surfaces have the Pythagorean–hodograph (PH) preserving property — a planar PH curve in the parameter domain is mapped to a spatial PH curve on the surface. Cubic, quartic and quintic examples of these minimal PN surfaces are presented, including examples of solutions to the Plateau problem, with boundaries generated by planar PH curve segments in the parameter domain. The construction is also generalized to the case of minimal surfaces with non–isothermal parameterizations. Finally, an application to the problem of interpolating three given points in R3 as the corners of a triangular cubic minimal surface patch, such that the three patch sides have prescribed lengths, is addressed.

In this paper, we prove that the product of a paired calibrated set and a set of codimension 1 calibrated by a coflat calibration with small singularity set is Almgren minimal. This is motivated by the attempt to classify all possible singularities for Almgren minimal sets–Plateau's problem in the setting of sets. In particular, a direct application of the above result leads to various types of new singularities for Almgren minimal sets, e.g. the product of any paired calibrated cone (such as the cone over the d − 2 skeleton of the unit cube in R d , d ≥ 4) with homogeneous area minimizing hypercones (such as the Simons cone). [ABSTRACT FROM AUTHOR]

In this paper, we study conformal solitons for the mean curvature flow in hyperbolic space H n + 1 . Working in the upper half-space model, we focus on horo-expanders, which relate to the conformal field - ∂ 0 . We classify cylindrical and rotationally symmetric examples, finding appropriate analogues of grim-reaper cylinders, bowl and winglike solitons. Moreover, we address the Plateau and the Dirichlet problems at infinity. For the latter, we provide the sharp boundary convexity condition to guarantee its solvability and address the case of non-compact boundaries contained between two parallel hyperplanes of ∂ ∞ H n + 1 . We conclude by proving rigidity results for bowl and grim-reaper cylinders. [ABSTRACT FROM AUTHOR]

In this article, we prove the various minimality of the product of a 1-codimensional calibrated manifold and a paired calibrated set. This is motivated by the attempt to classify all possible singularities for Almgren minimal sets – Plateau's problem in the setting of sets. The Almgren minimality was introduced by Almgren to modernize Plateau's problem. It gives a very good description of local behavior for soap films. The natural question of whether the product of any two Almgren minimal sets is still minimal is still open, although it seems obvious in intuition. We prove the Almgren minimality for the product of two large classes of Almgren minimal sets – the class of 1-codimensional calibrated manifolds and the class of paired calibrated sets. The general idea is to properly combine different topological conditions (separation and spanning) under different homology groups, to set up a reasonable topological condition and prove the minimality for the product under this condition, which will imply the Almgren minimality. A main difficulty comes from the codimension – algebraic coherences such as multiplicity, separation and orientation do not exist anymore for codimensions larger than 1. An unexpectedly useful thing in the present paper is the flow of the calibrations. Its most important role among all is helping us to do the decomposition of a competitor with the help of the first projections along the flows. [ABSTRACT FROM AUTHOR]

We investigate the relation between energy minimizing maps valued into spheres having topological singularities at given points and optimal networks connecting them (e.g., Steiner trees, Gilbert-Steiner irrigation networks). We show the equivalence of the corresponding variational problems, interpreting in particular the branched optimal transport problem as a homological Plateau problem for rectifiable currents with values in a suitable normed group. This generalizes the pioneering work by Brezis, Coron and Lieb [10]. [ABSTRACT FROM AUTHOR]

In this paper, we will prove a theorem for a functional where we prove that the necessary condition for the extreme of a functional is the variation of the functional to be equal to zero and we will give an example of its application, the problem of the Plateau. [ABSTRACT FROM AUTHOR]

We consider codimension 1 area-minimizing m -dimensional currents T mod an even integer p = 2 Q in a C 2 Riemannian submanifold Σ of Euclidean space. We prove a suitable excess-decay estimate towards the unique tangent cone at every point q ∈ spt (T) ∖ spt p (∂ T) where at least one such tangent cone is Q copies of a single plane. While an analogous decay statement was proved in Minter and Wickramasekera (2024) as a corollary of a more general theory for stable varifolds, in our statement we strive for the optimal dependence of the estimates upon the second fundamental form of Σ. This improvement is in fact crucial in De Lellis et al., (2022) to prove that the singular set of T can be decomposed into a C 1 , α (m − 1) -dimensional submanifold and an additional closed remaining set of Hausdorff dimension at most m − 2. [ABSTRACT FROM AUTHOR]

Soap films hanging from a wire frame are studied in the framework of capillarity theory. Minimizers in the corresponding variational problem are known to consist of positive volume regions with boundaries of constant mean curvature/pressure, possibly connected by "collapsed" minimal surfaces. We prove here that collapsing only occurs if the mean curvature/pressure of the bulky regions is negative, and that, when this last property holds, the whole soap film lies in the convex hull of its boundary wire frame. [ABSTRACT FROM AUTHOR]

In this paper, we will give a C1,β-regularity result on the boundary for two dimensional sliding almost minimal sets in R3. This effect may apply to the regularity of the soap films at the boundary, and may also lead to the existence of a solution to the Plateau problem with sliding boundary conditions proposed by Guy David in the case that the boundary is a 2-dimensional smooth submanifold. [ABSTRACT FROM AUTHOR]

In this paper, we will give a C1,β-regularity result on the boundary for two dimensional sliding almost minimal sets in R3. This effect may apply to the regularity of the soap films at the boundary, and may also lead to the existence of a solution to the Plateau problem with sliding boundary conditions proposed by Guy David in the case that the boundary is a 2-dimensional smooth submanifold. [ABSTRACT FROM AUTHOR]

The propagation criterion is defined in cryptography in order to analyze the security of cryptographic components with respect to differential cryptanalysis. This paper explains a class of plateaued functions satisfying the propagation criterion of degree l$l$ by using the Maiorana–McFarland construction. And a variety of other desirable criteria for functions with cryptographic application could also be satisfied, such as balancedness, high non‐linearity, non‐existence of non‐zero linear structures, good global avalanche characteristics. [ABSTRACT FROM AUTHOR]

The authors generalize the Fenchel theorem for strong spacelike closed curves of index 1 in the 3-dimensional Minkowski space, showing that the total curvature must be less than or equal to 2π. Here the strong spacelike condition means that the tangent vector and the curvature vector span a spacelike 2-plane at each point of the curve γ under consideration. The assumption of index 1 is equivalent to saying that γ winds around some timelike axis with winding number 1. This reversed Fenchel-type inequality is proved by constructing a ruled spacelike surface with the given curve as boundary and applying the Gauss-Bonnet formula. As a by-product, this shows the existence of a maximal surface with γ as the boundary. [ABSTRACT FROM AUTHOR]

This article is dedicated to discuss the sliding stability and the uniqueness property for the 2-dimensional minimal cone Y × Y in R 4. This problem is motivated by the classification of singularities for Almgren minimal sets, a model for Plateau's problem in the setting of sets. Minimal cones are blow up limits of Almgren minimal sets, thus the list of all minimal cones gives all possible types of singularities that can occur for minimal sets. As proved in [16] , when several 2-dimensional Almgren (resp. topological) minimal cones are Almgren (resp. topological) sliding stable, and Almgren (resp. topological) unique, the almost orthogonal union of them stays minimal. Hence if several minimal cones admit sliding stability and uniqueness properties, then we can use their almost orthogonal unions to generate new families of minimal cones. One then naturally ask which minimal cones admit these two properties. This list of known 2-dimensional minimal cones in arbitrary ambient dimension is not long, and the stability and uniqueness properties for all the known 2-dimensional minimal cones, except for Y × Y , have already been established in the previous works [18,17]. Among all the known 2-dimensional minimal cones, Y × Y is the only one whose stability and uniqueness properties were left unsolved. This is due to two main reasons: 1) Y × Y is the only known minimal cone which is essentially of codimension larger than 1—that is, we cannot decompose it into transversal unions of minimal cones of codimension 1. 2) Y × Y lives in R 4 , where we know very little about which types of singularities can occur in a minimal set. This makes it difficult to control and estimate the measures of all possible competitors. Due to the above two issues, new ideas are required here for solving the problem. We give affirmative answers to this problem for the stability and uniqueness properties for Y × Y in this paper: we prove that the set Y × Y is both Almgren sliding stable, and Almgren unique; for the topological case, we prove its topological sliding stability and topological uniqueness for the coefficient group Z 2. This result, along with the results in [16,18,17] , allows us to use all the known 2-dimensional minimal cones to generate new 2-dimensional minimal cones by taking almost orthogonal unions. [ABSTRACT FROM AUTHOR]

We give a fairly complete solution to the asymptotic Plateau Problem for area minimizing surfaces in H2×R. In particular, we identify the collection of Jordan curves in ∂∞(H2×R) which bounds an area minimizing surface in H2×R. Furthermore, we study the similar problem for minimal surfaces, and show that the situation is highly different. [ABSTRACT FROM AUTHOR]

Self-shrinkers are important geometric objects in the evolution of mean curvature flows, while the Bernstein theorem is one of the most profound results in minimal surface theory. We prove a Bernstein type result for graphical self-shrinker surfaces with co-dimension 2 in |$\mathbb{R}^4$|⁠. Namely under certain natural conditions on the Jacobian of any smooth map from |$\mathbb{R}^2$| to |$\mathbb{R}^2$| we show that the self-shrinker which is the graph of this map must be affine linear through 0. The proof relies on the derivation of structure equations of graphical self-shrinkers in terms of the parallel form and the existence of some positive functions on self-shrinkers related to these Jacobian conditions. [ABSTRACT FROM AUTHOR]

I review the present status of the problem of initial conditions for inflation and describe several ways to solve this problem for many popular inflationary models, including the recent generation of the models with plateau potentials favored by cosmological observations. [ABSTRACT FROM AUTHOR]

By using a suitable triple cover we show how to possibly model the construction of a minimal surface with positive genus spanning all six edges of a tetrahedron, working in the space of BV functions and interpreting the film as the boundary of a Caccioppoli set in the covering space. After a question raised by R. Hardt in the late 1980's, it seems common opinion that an area-minimizing surface of this sort does not exist for a regular tetrahedron, although a proof of this fact is still missing. In this paper we show that there exists a surface of positive genus spanning the boundary of an elongated tetrahedron and having area strictly less than the area of the conic surface. [ABSTRACT FROM AUTHOR]

We investigate the existence of non-constant uniformly-bounded minimal solutions of the Allen–Cahn equation on a Gromov-hyperbolic group. We show that whenever the Laplace term in the Allen–Cahn equation is small enough, there exist minimal solutions satisfying a large class of prescribed asymptotic behaviours. For a phase field model on a hyperbolic group, such solutions describe phase transitions that asymptotically converge towards prescribed phases, given by asymptotic directions. In the spirit of de Giorgi's conjecture, we then fix an asymptotic behaviour and let the Laplace term go to zero. In the limit we obtain a solution to a corresponding asymptotic Plateau problem by Γ-convergence. [ABSTRACT FROM AUTHOR]

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 partiallywetting 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. [ABSTRACT FROM AUTHOR]

In this paper we will compare the Plateau's problem with Čech and singular homological boundary conditions, we also compare these with the size minimizing problem for integral currents with a given boundary. Finally we get the agreement on the infimum values for all these Plateau's problems. [ABSTRACT FROM AUTHOR]

This paper presents a simple proof of Wloc2,2 regularity of Lipschitz uniform local minimizers of vectorial variational problems. The method is based on the idea that inner variations provide constraints on the structure of singularities of local minimizers. [ABSTRACT FROM AUTHOR]

We show that in the setting of proper metric spaces one obtains a solution of the classical 2-dimensional Plateau problem by minimizing the energy, as in the classical case, once a definition of area has been chosen appropriately. We prove the quasi-convexity of this new definition of area. Under the assumption of a quadratic isoperimetric inequality we establish regularity results for energy minimizers and improve Hölder exponents of some area-minimizing discs. [ABSTRACT FROM AUTHOR]

Using the classical approach we show the existence of disc type solutions to the asymptotic Plateau problem in certain Hadamard manifolds which may have arbitrarily strong curvature and volume growth. [ABSTRACT FROM AUTHOR]

We generalize Meeks and Yau's embeddedness result on the solutions of the Plateau problem to constant mean curvature disks. We show that any minimizing H -disk in an H 0 -convex domain is embedded for any H ∈ [ 0 , H 0 ) . In particular, for the unit ball B in R 3 , this implies that for any H ∈ [ 0 , 1 ] , any Jordan curve in ∂ B bounds an embedded H -disk in B . [ABSTRACT FROM AUTHOR]

In this paper, we establish a min-max theory for minimal surfaces using sweepouts of surfaces of genus . We develop a direct variational method similar to the proof of the famous Plateau problem by Douglas [Solution of the problem of Plateau, Trans. Amer. Math. Soc. 33 (1931) 263-321] and Rado [On Plateau's problem, Ann. Math. 31 (1930) 457-469]. As a result, we show that the min-max value for the area functional can be achieved by a bubble tree limit consisting of branched genus- minimal surfaces with nodes, and possibly finitely many branched minimal spheres. We also prove a Colding-Minicozzi type strong convergence theorem similar to the classical mountain pass lemma. Our results extend the min-max theory by Colding-Minicozzi and the author to all genera. [ABSTRACT FROM AUTHOR]

The Boltzmann H-theorem implies that the solution to the Boltzmann equation tends to an equilibrium, that is, a Maxwellian when time tends to infinity. This has been proved in various settings when the initial energy is finite. However, when the initial energy is infinite, the time asymptotic state is no longer described by a Maxwellian, but a self-similar solution obtained by Bobylev--Cercignani. The purpose of this paper is to rigorously justify this for the spatially homogeneous problem with a Maxwellian molecule type cross section without angular cutoff. [ABSTRACT FROM AUTHOR]

We provide a compactness principle which is applicable to different formulations of Plateau's problem in codimension one and which is exclusively based on the theory of Radon measures and elementary comparison arguments. Exploiting some additional techniques in geometric measure theory, we can use this principle to give a different proof of a theorem by Harrison and Pugh and to answer a question raised by Guy David. [ABSTRACT FROM AUTHOR]

Adhesion of bio-inspired microfibre arrays to a rough surface is studied theoretically. The array consists of vertical elastic rods fixed on a rigid backing layer, and the surface is modeled by rigid steps with a normally distributed height. Analytical expressions are obtained for the adhesion forces in both the approach and retraction processes. It is shown that, with the increasing preload, the pull-off force increases at first and then attains a plateau value. The results agree with the previous experiments and are expected helpful in adhesion control of the array in practical applications. [ABSTRACT FROM AUTHOR]

We give a detailed classification of all rotationally symmetric figures of equilibrium corresponding to rotating liquid masses subject to surface tension. When the rotation rate is zero, these shapes were studied by Delaunay who found six different qualitative types of complete connected interfaces (spheres, cylinders, unduloids, nodoids, catenoids, and planes). We find twenty-six qualitatively different interfaces providing a complete picture of symmetric equilibrium shapes, some of which have been studied by other authors. In particular, combining our work with that of Beer, Chandrasekhar, Gulliver, Smith, and Ross, we conclude that every compact equilibrium is in either a smooth connected one parameter family of spheroids or a smooth connected one parameter family of tori (possibly immersed in either case). [ABSTRACT FROM AUTHOR]

We link covering spaces with the theory of functions of bounded variation, in order to study minimal networks in the plane and Plateau's problem without fixing a priori the topology of solutions. We solve the minimization problem in the class of (possibly vector-valued) BV functions defined on a covering space of the complement of an (n -2)-dimensional compact embedded Lipschitz manifold S without boundary. This approach has several similarities with Brakke's "soap films" covering construction. The main novelty of our method stands in the presence of a suitable constraint on the fibers, which couples together the covering sheets. In the case of networks, the constraint is defined using a suitable subset of transpositions of m elements, m being the number of points of S. The model avoids all issues concerning the presence of the boundary S, which is automatically attained. The constraint is lifted in a natural way to Sobolev spaces, allowing also an approach based on Γ-convergence. [ABSTRACT FROM AUTHOR]

Best-first search algorithms such as A* need to apply tie-breaking strategies in order to decide which node to expand when multiple search nodes have the same evaluation score. We investigate and improve tie-breaking strategies for cost-optimal search using A*. We first experimentally analyze the performance of common tie-breaking strategies that break ties according to the heuristic value of the nodes. We find that the tie-breaking strategy has a significant impact on search algorithm performance when there are 0-cost operators that induce large plateau regions in the search space. Based on this, we develop two new classes of tie-breaking strategies. We first propose a depth diversification strategy which breaks ties according to the distance from the entrance to the plateau, and then show that this new strategy significantly outperforms standard strategies on domains with 0-cost actions. Next, we propose a new framework for interpreting A* search as a series of satisficing searches within plateaus consisting of nodes with the same f-cost. Based on this framework, we investigate a second, new class of tie-breaking strategy, a multi-heuristic tie-breaking strategy which embeds inadmissible, distance-to-go variations of various heuristics within an admissible search. This is shown to further improve the performance in combination with the depth metric. [ABSTRACT FROM AUTHOR]

We show the area-minimality property of all homogeneous area-minimizing hypercones in Euclidean spaces (classified by Lawlor) following Lawson's original idea in his 72' Trans. A.M.S. paper 'The equivariant Plateau problem and interior regularity'. Moreover, each of them enjoys (coflat) calibrations singular only at the origin. [ABSTRACT FROM AUTHOR]

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 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 and the collection of compact sets spanning M. Using Hausdorff spherical measure as a notion of 'size,' we prove: There exists an in with smallest size. Any such contains a 'core' with the following properties: It is a subset of the convex hull of M and is a.e. (in the sense of -dimensional Hausdorff measure) a real analytic -dimensional minimal submanifold. If , then 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. [ABSTRACT FROM AUTHOR]

We consider the numerical solution of nonlinear elliptic boundary value problems with Kansa’s method. We derive analytic formulas for the Jacobian and Hessian of the resulting nonlinear collocation system and exploit them within the framework of the trust-region algorithm. This ansatz is tested on semilinear, quasilinear and fully nonlinear elliptic PDEs (including Plateau’s problem, Hele–Shaw flow and the Monge–Ampère equation) with excellent results. The new approach distinctly outperforms previous ones based on linearization or finite-difference Jacobians. [ABSTRACT FROM AUTHOR]

The main result of this paper is a characterization of the minimal surface hull of a compact set K in ℝ³ by sequences of conformal minimal discs whose boundaries converge to K in the measure theoretic sense, and also by 2-dimensional minimal currents which are limits of Green currents supported by conformal minimal discs. Analogous results are obtained for the null hull of a compact subset of C³. We also prove a null hull analogue of the Alexander-Stolzenberg-Wermer theorem on polynomial hulls of compact sets of finite linear measure, and a polynomial hull version of classical Bochner's tube theorem. [ABSTRACT FROM AUTHOR]