Willmore flow of planar networks

Geometric gradient flows for elastic energies of Willmore type play an important role in mathematics and in many applications. The evolution of elastic curves has been studied in detail both for closed as well as for open curves. Although elastic flows for networks also have many interesting features, they have not been studied so far from the point of view of mathematical analysis. So far it was not even clear what are appropriate boundary conditions at junctions. In this paper we give a well-posedness result for Willmore flow of networks in different geometric settings and hence lay a foundation for further mathematical analysis. A main point in the proof is to check whether different proposed boundary conditions lead to a well posed problem. In this context one has to check the Lopatinskii–Shapiro condition in order to apply the Solonnikov theory for linear parabolic systems in Hölder spaces which is needed in a fixed point argument. We also show that the solution we get is unique in a purely geometric sense.     

Unstable Willmore surfaces of revolution subject to natural boundary conditions

In the class of surfaces with fixed boundary, critical points of the Willmore functional are naturally found to be those solutions of the Euler-Lagrange equation where the mean curvature on the boundary vanishes. We consider the case of symmetric surfaces of revolution in the setting where there are two families of stable solutions given by the catenoids. In this paper we demonstrate the existence of a third family of solutions which are unstable critical points of the Willmore functional, and which spatially lie between the upper and lower families of catenoids. Our method does not require any kind of smallness assumption, and allows us to derive some additional interesting qualitative properties of the solutions.     

Discrete Variational Calculus for B-Spline Curves

In this work, we study discrete variational problems, for B-spline curves, which are invariant under translation and rotation. We show this approach has advantages over studying smooth variational problems whose solutions are approximated by B-spline curves. The latter method has been well studied in the literature but leads to high order approximation problems. We are particularly interested in Lagrangians that are invariant under the special Euclidean group for which B-spline approximated curves are well suited. The main application we present here is the curve completion problem in 2D and 3D. Here, the aim is to find various aesthetically pleasing solutions as opposed to a solution of a physical problem. Smooth Lagrangians with special Euclidean symmetries involve curvature, torsion, and arc length. Expressions of these, in the original coordinates, are highly complex. We show that, by contrast, relatively simple discrete Lagrangians offer excellent results for the curve completion problem. The novel methods we develop for the discrete curve completion problem are general, and can be used to solve other discrete variational problems for B-spline curves. Our method completely avoids the difficulties of high order smooth differential invariants.     

A characterization of the Ejiri torus in <Emphasis Type="Italic">S</Emphasis><Superscript>5</Superscript>

We conjecture that a Willmore torus having Willmore functional between 2π ² and 2π ² \(\sqrt 3 \) is either conformally equivalent to the Clifford torus, or conformally equivalent to the Ejiri torus. Ejiri’s torus in S ⁵ is the first example of Willmore surface which is not conformally equivalent to any minimal surface in any real space form. Li and Vrancken classified all Willmore surfaces of tensor product in S ⁿ by reducing them into elastic curves in S ³, and the Ejiri torus appeared as a special example. In this paper, we first prove that among all Willmore tori of tensor product, the Willmore functional of the Ejiri torus in S ⁵ attains the minimum 2π ² \(\sqrt 3 \), which indicates our conjecture holds true for Willmore surfaces of tensor product. Then we show that all Willmore tori of tensor product are unstable when the co-dimension is big enough. We also show that the Ejiri torus is unstable even in S ⁵. Moreover, similar to Li and Vrancken, we classify all constrained Willmore surfaces of tensor product by reducing them with elastic curves in S ³. All constrained Willmore tori obtained this way are also shown to be unstable when the co-dimension is big enough.     

Surfaces of revolution in the Heisenberg group and the spectral generalization of the Willmore functional

We study the generalization of the Willmore functional for surfaces in the three-dimensional Heisenberg group. Its construction is based on the spectral theory of the Dirac operator entering into theWeierstrass representation of surfaces in this group. Using the surfaces of revolution we demonstrate that the generalization resembles the Willmore functional for the surfaces in the Euclidean space in many geometrical aspects. We also observe the relation of these functionals to the isoperimetric problem.     

Willmore Surfaces of Revolution with Two Prescribed Boundary Circles

We consider the family of smooth embedded surfaces of revolution in ?³ having two concentric circles contained in two parallel planes of ?³ as boundary. Minimizing the Willmore functional within this class of surfaces we prove the existence of smooth axi-symmetric Willmore surfaces having these circles as boundary. When the radii of the circles tend to zero we prove convergence of these solutions to the round sphere.     

Plane curves with curvature depending on distance to a line

Motivated by a problem posed by David A. Singer in 1999 and by the classical Euler elastic curves, we study the plane curves whose curvature is expressed in terms of the (signed) distance to a line. In this way, we provide new characterizations of some well known curves, like the catenary or the grim-reaper. We also find out several interesting families of plane curves (including closed and embedded ones) whose intrinsic equations are expressed in terms of elementary functions or Jacobi elliptic functions and we are able to get arc length parametrizations of them and they are depicted graphically.     

Envelopes and osculates of Willmore surfaces

We view conformal surfaces in the 4-sphere as quaternionic holomorphiccurves in quaternionic projective space. By constructing envelopingand osculating curves, we obtain new holomorphic curves in quaternionicprojective space and thus new conformal surfaces. Applying theseconstructions to Willmore surfaces, we show that the osculatingand enveloping curves of Willmore spheres remain Willmore.     

Numerical solution for the anisotropic Willmore flow of graphs

The Willmore flow is well known problem from the differential geometry. It minimizes the Willmore functional defined as integral of the mean-curvature square over given manifold. For the graph formulation, we derive modification of the Willmore flow with anisotropic mean curvature. We define the weak solution and we prove an energy equality. We approximate the solution numerically by the complementary finite volume method. To show the stability, we re-formulate the resulting scheme in terms of the finite difference method. By using simple framework of the finite difference method (FDM) we show discrete version of the energy equality. The time discretization is done by the method of lines and the resulting system of ODEs is solved by the Runge–Kutta–Merson solver with adaptive integration step. We also show experimental order of convergence as well as results of the numerical experiments, both for several different anisotropies.     

Equivariant constrained Willmore tori in the 3-sphere

In this paper we study equivariant constrained Willmore tori in the 3-sphere. These tori admit a 1-parameter group of Möbius symmetries and are critical points of the Willmore energy under conformal variations. We show that the spectral curve associated to an equivariant torus is given by a double covering of \(\mathbb {C}\) and classify equivariant constrained Willmore tori by the genus \(g\) of their spectral curve. In this case the spectral genus satisfies \(g \le 3\) .     

Mixed impedance transmission problems for vibrating layered elastic bodies

Direct scattering problems for partially coated piecewise homogenous and inhomogeneous layered obstacles in linear elasticity lead to mixed impedance transmission problems for the steady‐state elastic oscillation equations. For a piecewise homogenous isotropic composite body, we employ the potential method and reduce the mixed impedance transmission problem to an equivalent system of boundary pseudodifferential equations. We give a detailed analysis of the corresponding pseudodifferential operators, which live on the interface between the layers and on a proper submanifold of the boundary of the composite elastic body, and establish uniqueness and existence results for the original mixed impedance transmission problem for arbitrary values of the oscillation frequency parameter; this is crucial in the study of inverse elastic scattering problems for partially coated layered obstacles. We also investigate regularity properties of solutions near the collision curves, where the different boundary conditions collide, and establish almost best Hölder smoothness results. Further, we analyze the asymptotic behavior of the stress vector near the collision curve and derive explicit formulas for the stress singularity exponents. The case of Lipschitz surfaces is briefly treated separately. In the case of a composite body containing homogeneous or inhomogeneous finite anisotropic inclusions, we develop an alternative hybrid method based on the so‐called nonlocal approach and reduce the mixed transmission problem to an equivalent functional‐variational equation with a sesquilinear form that ‘lives’ on a bounded part of the layered composite body and its boundary. We show that this sesquilinear form is coercive and that the corresponding variational equation is uniquely solvable. Copyright © 2015 John Wiley & Sons, Ltd.     

Sequences of Willmore surfaces

In this paper we develop the theory of Willmore sequences for Willmore surfaces in the 4-sphere. We show that under appropriate conditions this sequence has to terminate. In this case the Willmore surface either is the Twistor projection of a holomorphic curve into or the inversion of a minimal surface with planar ends in . These results give a unified explanation of previous work on the characterization of Willmore spheres and Willmore tori with non-trivial normal bundles by various authors. K. Leschke thanks the Department of Mathematics and Statistics at the University of Massachusetts, Amherst, and the Center for Geometry, Analysis, Numerics and Graphics for their support and hospitality.     

Existence and uniqueness for variational problem from progressive lens design

We study a functional modelling the progressive lens design, which is a combination of Willmore functional and total Gauss curvature. First, we prove the existence for the minimizers of this class of functionals among the class of revolution surfaces rotated by the curves y = f(x) about the x-axis. Then, choosing such a minimiser as background surfaces to approximate the functional by a quadratic functional, we prove the existence and uniqueness of the solution to the Euler-Lagrange equation for the quadratic functionals. Our results not only provide a strictly mathematical proof for numerical methods, but also give a more reasonable and more extensive choice for the background surfaces.     

Explicit formulas,symmetry and symmetry breaking for Willmore surfaces of revolution

In this paper, we prove explicit formulas for all Willmore surfaces of revolution and demonstrate their use in the discussion of the associated Dirichlet boundary value problems. It is shown by an explicit example that symmetric Dirichlet boundary conditions do in general not entail the symmetry of the surface. In addition, we prove a symmetry result for a subclass of Willmore surfaces satisfying symmetric Dirichlet boundary data.     

On the genus of a cyclic curve over a finite field

Cyclic curves, i.e. curves fixed by a cyclic collineation group, play a central role in the investigation of cyclic arcs in Desarguesian projective planes. In this paper, the genus of a cyclic curve arising from a cyclic k-arc of Singer type is computed.     

Foci and Foliations of Real Algebraic Curves

We discuss diverse results whose common thread is the notion of focus of an algebraic curve. In a unified setting, which combines elements of projective geometry, complex analysis and Riemann surface theory, we explain the roles of ordinary and singular foci in results on numerical ranges of matrices, quadrature domains, Schwarzian reflection, and other topics. We introduce the notion of canonical foliation of a real algebraic curve, which places foci into the context of continuous families of plane curves and provides a useful method of visualization of all relevant structures in a planar graphical image. Lecture held by Joel Langer in the Seminario Matematico e Fisico on October 6, 2006 Received: July 2007     

连续介质中杆件系统的振动问题

弹性连续介质中杆件系统的振动问题,是工程中经常遇到的,这是弹性动力学和结构动力学的混合求解问题。按照一般的方法进行求解,似乎有很多困难,且很复杂。本文采用Lagrange乘子法,给出了这种类型平面问题的广义泛函。并通过实例说明本文方法的应用。     

Spectral analysis for periodic solutions of the Cahn�CHilliard equation on {\mathbb{R}}

We consider the spectrum associated with the linear operator obtained when the Cahn–Hilliard equation on \mathbbR{\mathbb{R}} is linearized about a stationary periodic solution. Our analysis is particularly motivated by the study of spinodal decomposition, a phenomenon in which the rapid cooling (quenching) of a homogeneously mixed binary alloy causes separation to occur, resolving the mixture into regions of different crystalline structure, separated by steep transition layers. In this context, a natural problem regards the evolution of solutions initialized by small, random (in some sense) perturbations of the pre-quenching homogeneous state. Solutions initialized in this way appear to evolve transiently toward certain unstable periodic solutions, with the rate of evolution described by the spectrum associated with these periodic solutions. In the current paper, we use Evans function methods and a perturbation argument to locate the spectrum associated with such periodic solutions. We also briefly discuss a heuristic method due to Langer for relating our spectral information to coarsening rates.