Minimal Surface Problem Research Articles

Graphical solutions to one-phase free boundary problems

Abstract We study viscosity solutions to the classical one-phase problem and its thin counterpart. In low dimensions, we show that when the free boundary is the graph of a continuous function, the solution is the half-plane solution. This answers, in the salient dimensions, a one-phase free boundary analogue of Bernstein’s problem for minimal surfaces. As an application, we also classify monotone solutions of semilinear equations with a bump-type nonlinearity.

The Calabi–Yau problem for minimal surfaces with Cantor ends

We show that every connected compact or bordered Riemann surface contains a Cantor set whose complement admits a complete conformal minimal immersion in \mathbb{R}^3 with bounded image. The analogous result holds for holomorphic immersions into any complex manifold of dimension at least 2, for holomorphic null immersions into \mathbb{C}^n with n \geq 3 , for holomorphic Legendrian immersions into an arbitrary complex contact manifold, and for superminimal immersions into any selfdual or anti-self-dual Einstein four-manifold.

Perturbing the catenoid: Stability and mechanical properties of nonaxisymmetric minimal surfaces.

Minimal surface problems arise naturally in many soft matter systems whose free energies are dominated by surface or interface energies. Of particular interest are the shapes, stability, and mechanical stresses of minimal surfaces spanning specific geometric boundaries. The "catenoid" is the best-known example where an analytical solution is known which describes the form and stability of a minimal surface held between two parallel, concentric circular frames. Here we extend this problem to nonaxisymmetric, parallel frame shapes of different orientations by developing a perturbation approach around the known catenoid solution. We show that the predictions of the perturbation theory are in good agreement with experiments on soap films and finite element simulations (Surface Evolver). Combining theory, experiment, and simulation, we analyze in depth how the shapes, stability, and mechanical properties of the minimal surfaces depend on the type and orientation of elliptic and three-leaf clover shaped frames. In the limit of perfectly aligned nonaxisymmetric frames, our predictions show excellent agreement with a recent theory established by Alimov etal. [Phys. Fluids 33, 052104 (2021)1070-663110.1063/5.0047461]. Moreover, we put in evidence the intriguing capacity of minimal surfaces between nonaxisymmetric frames to transmit a mechanical torque despite being completely liquid. These forces could be interesting to exploit for mechanical self-assembly of soft matter systems or as highly sensitive force captors.

Structural topology optimization of 3D‐printable microlattices for lightweight materials

AbstractWe present a micro‐macro approach for the structural optimization of graded microlattices that contain unit cells with a triply periodic structure. These individual unit cells are based on minimal surface problems such as the the Schwarz Primitive surface or the Gyroid and exhibit a porous structure that allows for a macroscopic grading of the material fraction of the cells. Both the macro‐scale and the micro‐scale are optimized by means of a Cahn‐Hilliard phase‐field model that is coupled with an elastic problem. The optimization on the macro‐scale is performed with the finite element method (FEM) while the micro‐scale optimization makes use of a spectral method that solves the periodic problem in a fast manner by means of fast Fourier transforms (FFT).

Soap film on two noncircular frames

The classical Plateau problem of finding minimal surfaces supported by two noncircular coaxial rings is studied theoretically and experimentally. Using a fluid dynamics analogy, we generalize the classical catenoid solution for a film on circular rings to the general cases of noncircular rings. Some examples of analytical solutions for elliptical, polygonal, and ovoidal rings are presented. The shapes of a tubular film and a film separated by a lamella at the wrist are obtained in an analytical form. The stability of these films is analyzed and compared with the classical catenoid. The data on critical parameters of all minimal surfaces are collected in the tables that can be used in practical applications. The theory is experimentally validated using soap films on elliptical identical frames. Moreover, the shapes of soap films on two different elliptical frames demonstrate a new feature: a flat separating lamella lying parallel to the rings was never observed in experiments. All lamellae appeared deformed suggesting the existence of a new family of minimal surfaces which does not exist in the case of frames of the same sizes.

The virtual element method for a minimal surface problem

In this paper we consider the Virtual Element discretization of a minimal surface problem, a quasi-linear elliptic partial differential equation modeling the problem of minimizing the area of a surface subject to a prescribed boundary condition. We derive an optimal error estimate and present several numerical tests assessing the validity of the theoretical results.

Minimal surfaces on mirror-symmetric frames: a fluid dynamics analogy

Chaplygin’s hodograph method of classical fluid mechanics is applied to explicitly solve the Plateau problem of finding minimal surfaces. The minimal surfaces are formed between two mirror-symmetric polygonal frames having a common axis of symmetry. Two classes of minimal surfaces are found: the class of regular surfaces continuously connecting the supporting frames forming a tube with complex shape; and the class of singular surfaces which have a partitioning film closing the tube in between. As an illustration of the general solution, minimal surfaces supported by triangular frames are fully described. The theory is experimentally validated using soap films. The general solution is compared with the known particular solutions obtained by the Weierstrass inverse method.

An adaptive nonlinear finite element analysis of minimal surface problem based on element energy projection technique

PurposeThis paper aims to propose a new adaptive strategy for two-dimensional (2D) nonlinear finite element (FE) analysis of the minimal surface problem (MSP) based on the element energy projection (EEP) technique.Design/methodology/approachBy linearizing nonlinear problems into a series of linear problems via the Newton method, the EEP technique, which is an effective and reliable point-wise super-convergent displacement recovery strategy for linear FE analysis, can be directly incorporated into the solution procedure. Accordingly, a posteriori error estimate in maximum norm was established and an adaptive 2D nonlinear FE strategy of h-version mesh refinement was developed.FindingsThree classical known surfaces, including a singularity problem, were analysed. Moreover, an example whose analytic solution is unavailable was considered and a comparison was made between present results and those computed by the MATLAB PDE toolbox. The results show that the adaptively-generated meshes reflect the difficulties inherent in the problems and the proposed adaptive analysis can produce FE solutions satisfying the user-preset error tolerance in maximum norm with a fair adaptive convergence rate.Originality/valueThe EEP technique for linear FE analysis was extended to the nonlinear procedure of MSP and can be expected to apply to other 2D nonlinear problems. The employment of the maximum norm makes point-wisely error control on the sought surfaces possible and makes the proposed method distinguished from other adaptive FE analyses.

How a minimal surface leaves a thin obstacle

We prove the optimal regularity and a detailed analysis of the free boundary of the solutions to the thin obstacle problem for nonparametric minimal surfaces with flat obstacles.

Renormalization Group Decoder for a Four-Dimensional Toric Code

We describe a computationally-efficient heuristic algorithm based on a renormalization-group procedure which aims at solving the problem of finding minimal surface given its boundary (curve) in any hypercubic lattice of dimension $D>2$. We use this algorithm to correct errors occurring in a four-dimensional variant of the toric code, having open as opposed to periodic boundaries. For a phenomenological error model which includes measurement errors we use a five-dimensional version of our algorithm, achieving a threshold of $4.35\pm0.1\%$. For this error model, this is the highest known threshold of any topological code. Without measurement errors, a four-dimensional version of our algorithm can be used and we find a threshold of $7.3\pm0.1\%$. For the gate-based depolarizing error model we find a threshold of $0.31\pm0.01\%$ which is below the threshold found for the two-dimensional toric code.

Dirichlet boundary values on Euclidean balls with infinitely many solutions for the minimal surface system

We make systematic developments on Lawson–Osserman constructions relating to the Dirichlet problem (over unit disks) for minimal surfaces of high codimension in their 1977' Acta paper. In particular, we show the existence of boundary functions for which infinitely many analytic solutions and at least one nonsmooth Lipschitz solution exist simultaneously. This newly-discovered amusing phenomenon enriches the understanding on the Lawson–Osserman philosophy.

Approximation of minimal surfaces with free boundaries

In this paper we develop a penalty method to approximate solutions of the free boundary problem for minimal surfaces. To this end we study the problem of finding minimizers of a functional F_{\lambda} which is defined as the sum of the Dirichlet integral and an appropriate penalty term weighted by a parameter \lambda . We prove existence of a solution for \lambda large enough as well as convergence to a solution of the free boundary problem as \lambda tends to infinity. Additionally regularity at the boundary of these solutions is shown, which is crucial for deriving numerical error estimates. Since every solution is harmonic, the analysis is largely simplified by considering boundary values only and using harmonic extensions. In a subsequent paper we develop a fully discrete finite element procedure for approximating solutions to this problem and prove an error estimate which includes an order of convergence with respect to the grid size.

Globally Lipschitz minimizers for variational problems with linear growth

We study the minimization of convex, variational integrals of linear growth among all functions in the Sobolev spaceW1,1with prescribed boundary values (or its equivalent formulation as a boundary value problem for a degenerately elliptic Euler–Lagrange equation). Due to insufficient compactness properties of these Dirichlet classes, the existence of solutions does not follow in a standard way by the direct method in the calculus of variations and in fact might fail, as it is well-known already for the non-parametric minimal surface problem. Assuming radial structure, we establish a necessary and sufficient condition on the integrand such that the Dirichlet problem is in general solvable, in the sense that a Lipschitz solution exists for any regular domain and all prescribed regular boundary values,viathe construction of appropriate barrier functions in the tradition of Serrin’s paper [J. Serrin,Philos. Trans. R. Soc. Lond., Ser. A264(1969) 413–496].

Asymptotic plateau problem in $${\mathbb H}^2\times {\mathbb R}$$ H 2 × R

We give a fairly complete solution to the asymptotic Plateau Problem for area minimizing surfaces in H2xR. In particular, we identify the collection of Jordan curves in the asymptotic boundary of H2xR, which bounds an area minimizing surface in H2xR. Furthermore, we study the similar problem for minimal surfaces, and show that the situation is highly different.

Universal moduli spaces of Riemann surfaces

We construct a moduli space for Riemann surfaces that is universal in the sense that it represents compact Riemann surfaces of any finite genus. This moduli space is stratifed according to genus, and it carries a metric and a measure that induce a Riemannian metric and a finite volume measure on each stratum. Applications to the Plateau-Douglas problem for minimal surfaces of varying genus and to the partition function of Bosonic string theory are outlined. The construction starts with a universal moduli space of Abelian varieties. This space carries a structure of an infinite dimensional locally symmetric space which is of interest in its own right. The key to our construction of the universal moduli space then is the Torelli map that assigns to every Riemann surface its Jacobian and its extension to the Satake-Baily-Borel compactifications.

Revisiting the redistancing problem using the Hopf–Lax formula

This article presents a fast new numerical method for redistancing objective functions based on the Hopf–Lax formula [1]. The algorithm suggested here is a special case of the previous work in [2] and an extension that applies the Hopf–Lax formula for computing the signed distance to the front. We propose the split Bregman approach to solve the minimization problem as a solution of the eikonal equation obtained from Hopf–Lax formula. Our redistancing procedure is expected to be generalized and widely applied to many fields such as computational fluid dynamics, the minimal surface problem, and elsewhere.

The Björling problem for timelike minimal surfaces in $${\mathbb {R}}^{4}_{1}$$ R 1 4

In this paper, we extend and solve the Bjorling problem for timelike surfaces in the ambient space \({\mathbb {R}}^{4}_{1}\). To do this, we define a Gauss map ideally suited to this setting using the split-complex variable and then we obtain a Weierstrass representation formula. We use this to construct new examples and give applications. In particular, we obtain one-parameter families of timelike surfaces in \({\mathbb {R}}^{4}_{1}\) which are solutions of the timelike Bjorling problem. In addition, we establish symmetry principles for the class of minimal timelike surfaces in \({\mathbb {R}}^4_1\).

A minimal surface criterion for graph partitioning

We consider a geometric approach to graph partitioning based on the graph Beltrami energy, a discrete version of a functional that appears in classical minimal surface problems. More specifically, the optimality criterion is given by the sum of the minimal Beltrami energies of the partition components. Since the Beltrami energy interpolates between the Total Variation and Dirichlet energies, various results for optimal partitions for these two energies can be recovered. We adapt primal-dual convex optimization methods to solve for the minimal Beltrami energy for each component of a given partition. A rearrangement algorithm is proposed to find the graph partition to minimize a relaxed version of the objective. The method is applied to several clustering problems on graphs constructed from manifold discretizations, synthetic data, the MNIST handwritten digit dataset, and image segmentation. The model has a semisupervised extension and provides a natural representative for the clusters as well.

Teichmüller harmonic map flow from cylinders

We define a geometric flow that is designed to change surfaces of cylindrical type spanning two disjoint boundary curves into solutions of the Douglas-Plateau problem of finding minimal surfaces with given boundary curves. We prove that also in this new setting and for arbitrary initial data, solutions of the Teichmuller harmonic map flow exist for all times. Furthermore, for solutions for which a three-point-condition does not degenerate as \(t\rightarrow \infty \), we show convergence along a sequence \(t_i\rightarrow \infty \) to a critical point of the area given either by a minimal cylinder or by two minimal discs spanning the given boundary curves.

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.