Finite element approximations to surfaces of prescribed variable mean curvature

We give an algorithm for finding finite element approximations to surfaces of prescribed variable mean curvature, which span a given boundary curve. We work in the parametric setting and prove optimal estimates in the H¹ norm. The estimates are verified computationally.     

On the thread problem for minimal surfaces

For a given one-dimensional fixed boundary $\Gamma$ in and a given constant we consider any one-dimensional free boundary $F$ in subject to the conditions that the length of is equal to , that and form a closed boundary, and that the minimal surface of dimension two being bounded by and minimizes the area among all comparison surfaces being bounded by and some with length equal to . This variational problem is known as the thread problem for minimal surfaces and stems from soap film experiments, in which the fixed boundary parts are pieces of wires and the free boundary parts are threads. The new result of this article will be that has no singular points in , provided the admissible surfaces and boundary parts are supposed to be rectifiable flat chains modulo two. Received February 16, 1995 / Accepted October 20, 1995     

The spiral minimal surfaces and their Legendre and Weierstrass representations

A class of spiral minimal surfaces in E³ is constructed using a symmetry reduction. The reduction leads to a cubic-nonlinear ODE whose phase portrait is described using an auxiliary Riccati's equation and the Warzewski topological principle for its solutions. The new surfaces are invariant with respect to the composition of rotation and dilation. The solutions are obtained in parametric form through the Legendre and the Weierstrass representations, and also their asymptotic behaviour is described.     

The bridge principle for stable minimal surfaces

The author was partially funded by NSF grants DMS85-53231(PYI), DMS-92-07704, and by the IHES.     

On the role of energy convexity in the web function approximation

For a given p > 1 and an open bounded convex set we consider the minimization problem for the functional over Since the energy of the unique minimizer u_p may not be computed explicitly, we restrict the minimization problem to the subspace of web functions, which depend only on the distance from the boundary δΩ. In this case, a representation formula for the unique minimizer v_p is available. Hence the problem of estimating the error one makes when approximating J_p(u_p) by J_p(v_p) arises. When Ω varies among convex bounded sets in the plane, we find an optimal estimate for such error, and we show that it is decreasing and infinitesimal with p. As p → ∞, we also prove that u_p − v_p converges to zero in for all m < ∞. These results reveal that the approximation of minima by means of web functions gains more and more precision as convexity in J_p increases.     

Nonconforming mixed finite element approximation to the stationary Navier-Stokes equations on anisotropic meshes

The main aim of this paper is to study the error estimates of a rectangular nonconforming finite element for the stationary Navier-Stokes equations under anisotropic meshes. That is, the nonconforming rectangular element is taken as approximation space for the velocity and the piecewise constant element for the pressure. The convergence analysis is presented and the optimal error estimates both in a broken H¹-norm for the velocity and in an L²-norm for the pressure are derived on anisotropic meshes.     

On two-dimensional parametric variational problems

Let be a two-dimensional parametric variational integral the Lagrangian F(x,z) of which is positive definite and elliptic, and suppose that is a closed rectifiable Jordan curve in . We then prove that there is a conformally parametrized minimizer of in the class of surfaces of the type of the disk B which are bounded by . An immediate consequence of this theorem is that the Dirichlet integral and the area functional have the same infima, a result whose proof usually requires a Lichtenstein-type mapping theorem or else Morrey's lemma on -conformal mappings. In addition we show that the minimizer of is H?lder continuous in B, and even in if satisfies a chord-arc condition. In Section 1 it is described how our results are related to classical investigations, in particular to the work of Morrey. Without difficulty our approach can be carried over to two-dimensional surfaces of codimension greater than one. Received July 20, 1998 / Accepted October 23, 1998     

Internal approximation of quasi-variational inequalities

Summary In this paper the internal approximation of a quasi-variational inequality is considered. An algorithm of Bensoussan-Lions type is proposed for which the convergence is proved. These results are applied to Signorini problem with friction for which two error estimates and numerical examples are also given.     

Embedded minimal annuli solving an exterior problem

Given a compact, strictly convex body in ³ and a closed Jordan curve ³ satisfying several additional assumptions, the existence of a parametric, annulus type minimal surface is proved, which parametrizes along one boundary component, has a free boundary onX along the other boundary component, and which stays in ³. As a consequence of this and a reasoning developed by W. H. Meeks and S. -T. Yau we find an embedded minimal surface with these properties. Another application is the existence of an embedded minimal surface with a flat end, free boundary onX and controlled topology.This article was processed by the author using the LATEX style filepljourlm from Springer-Verlag.     

A method to exclude branch points of minimal surfaces

The central problem of this paper is to exclude boundary branch points of minimal surfaces. The method consists in showing that the third derivative of the Dirichlet energy is negative along well-chosen paths in admissible Jacobi field directions, if a “Schüffler condition” is satisfied. Received July 21, 1997 / Accepted October 3, 1997     

On the coupling of MIXED-FEM and BEM for an exterior Helmholtz problem in the plane

Summary This paper deals with an elliptic boundary value problem posed in the plane, with variable coefficients, but whose restriction to the exterior of a bounded domain reduces to a Helmholtz equation. We consider a mixed variational formulation in a bounded domain that contains the heterogeneous medium, coupled with a boundary integral method applied to the Helmholtz equation in . We utilize suitable auxiliary problems, duality arguments, and Fredholm alternative to show that the resulting formulation of the problem is well posed. Then, we define a corresponding Galerkin scheme by using rotated Raviart-Thomas subspaces and spectral elements (on the interface). We show that the discrete problem is uniquely solvable and convergent and prove optimal error estimates. Finally we illustrate our analysis with some results from computational experiments.     

Stability for hypersurfaces of constant mean curvature with free boundary

The partitioning problem for a smooth convex bodyB ³ consists in to study, among surfaces which divideB in two pieces of prescribed volume, those which are critical points of the area functional.We study stable solutions of the above problem: we obtain several topological and geometrical restrictions for this kind of surfaces. In the case thatB is a Euclidean ball we obtain stronger results.Antonio Ros is partially supported by DGICYT grant PB91-0731 and Enaldo Vergasta is partially supported by CNPq grant 202326/91-8.     

The <Emphasis Type="Italic">p</Emphasis>-version of the boundary element method for weakly singular operators on piecewise plane open surfaces

We study piecewise polynomial approximations in negative order Sobolev norms of singularities which are inherent to Neumann data of elliptic problems of second order in polyhedral domains. The worst case of exterior crack problems in three dimensions is included. As an application, we prove an optimal a priori error estimate for the p-version of the BEM with weakly singular operators on polyhedral Lipschitz surfaces and piecewise plane open screens. The work of A. Bespalov was supported by the London Mathematical Society. The work of N. Heuer was supported by the FONDAP Programme in Applied Mathematics and Fondecyt project no. 1040615, both Chile. Dedicated to Professor Wolfgang L. Wendland on the occasion of his 70th birthday.     

Centroaffine minimal hypersurfaces in ℝ n+1

In this paper the first and the second variation formulas for the area integral of the centroaffine metric of hypersurfaces in ⁿ⁺¹ are calculated, and some interesting examples of stable and unstable centroaffine minimal hypersurfaces are given.Partially supported by the DFG-project Affine Differential Geometry at the TU Berlin.     

Young measure approximation in micromagnetics

Summary. Modeling of micromagnetic phenomena typically faces the minimization of a non-convex problem, which gives rise to highly oscillatory magnetization structures. Mathematically, this necessitates to extend the notion of Lebesgue-type solutions to Young-measure valued solutions. The present work proposes and analyzes a conforming finite element method that is based on an active set strategy to compute efficiently discrete solutions of the generalized minimization problem. Computational experiments are given to show the efficiency of the scheme. Received January 20, 2000 / Published online May 30, 2001     

On orders of approximation of plane curves by parametric cubic splines

The aim of the present paper is to show that the convergence rate of the parametric cubic spline approximation of a plane curve is of order four instead of order three. For the first and second derivatives, the rates are of order three and two, respectively. Finally some numerical examples are given to illustrate the predicted error behaviour.     

Schwarz operators of minimal surfaces spanning polygonal boundary curves

This paper examines the Schwarz operator A and its relatives Ȧ, Ā and Ǡ that are assigned to a minimal surface X which maps consequtive arcs of the boundary of its parameter domain onto the straight lines which are determined by pairs P _j , P _j+1 of two adjacent vertices of some simple closed polygon . In this case X possesses singularities in those boundary points which are mapped onto the vertices of the polygon Γ. Nevertheless it is shown that A and its closure Ā have essentially the same properties as the Schwarz operator assigned to a minimal surface which spans a smooth boundary contour. This result is used by the author to prove in [Jakob, Finiteness of the set of solutions of Plateau’s problem for polygonal boundary curves. I.H.P. Analyse Non-lineaire (in press)] the finiteness of the number of immersed stable minimal surfaces which span an extreme simple closed polygon Γ, and in [Jakob, Local boundedness of the set of solutions of Plateau’s problem for polygonal boundary curves (in press)] even the local boundedness of this number under sufficiently small perturbations of Γ.     

On a lower bound for the extinction time of surfaces moved by mean curvature

This paper considers a generalized evolution of a compact closed (hyper)-surface moved by its mean curvature. By a comparison of a shrinking ball the surface extincts in a finite time. An estimate of the extinction time from above is given by L. C. Evans and J. Spruck. In this paper we give an estimate from blow. In fact we proved that the extinction time is estimated from below by to times the square of the volume of a set enclosed by the initial surface over the initial area of the surface. The constant two is optimal.Partly supported by the Inamori Foundation     

New upper solution bounds of the discrete algebraic Riccati matrix equation

In this note, we present upper matrix bounds for the solution of the discrete algebraic Riccati equation (DARE). Using the matrix bound of Theorem 2.2, we then give several eigenvalue upper bounds for the solution of the DARE and make comparisons with existing results. The advantage of our results over existing upper bounds is that the new upper bounds of Theorem 2.2 and Corollary 2.1 are always calculated if the stabilizing solution of the DARE exists, whilst all existing upper matrix bounds might not be calculated because they have been derived under stronger conditions. Finally, we give numerical examples to demonstrate the effectiveness of the derived results.     

On the approximation of singular integral equations by equations with smooth kernels

Singular integral equations with Cauchy kernel and piecewise-continuous matrix coefficients on open and closed smooth curves are replaced by integral equations with smooth kernels of the form(t–)[(t–) ²–n ² (t) ²]^–1,0, wheren(t), t , is a continuous field of unit vectors non-tangential to . we give necessary and sufficient conditions under which the approximating equations have unique solutions and these solutions converge to the solution of the original equation. For the scalar case and the spaceL ₂() these conditions coincide with the strong ellipticity of the given equation.This work was fulfilled during the first author's visit to the Weierstrass Institute for Applied Analysis and Stochastics, Berlin in October 1993.