相关论文: Stationary curves under the M\"obius-Plateau energ…
The M\"obius energy is a well-studied knot energy with nice regularity and self-repulsive properties. Stationary curves under the M\"obius energy gradient are of significant theoretical interest as they they can indicate equilibrium states…
In the present paper we investigate generalizations of O'Hara's M\"obius energy on curves \cite{ohara_1991a}, to M\"obius-invariant energies on non-smooth subsets of $\R^n$ of arbitrary dimension and co-dimension. In particular, we show…
We study equilibrium configurations for the Euler-Plateau energy with elastic modulus, which couples an energy functional of Euler-Plateau type with a total curvature term often present in models for the free energy of biomembranes. It is…
The M\"obius energy of a curve is a topic of interest to physical knot theorists, harmonic analysts, and geometric analysts. The Gateaux derivative indicates its variation is dependent on curvature and torsion, leading us to consider the…
Classically, Plateau's problem asks to find a surface of the least area with a given boundary $B$. In this article, we investigate a version of Plateau's problem, where the boundary of an admissible surface is only required to partially…
By introducing a shape manifold as a solution set to solve inverse obstacle scattering problems we allow the reconstruction of general, not necessarily star-shaped curves. The bending energy is used as a stabilizing term in Tikhonov…
The O'Hara energies, introduced by Jun O'Hara in 1991, were proposed to answer the question of what is a "good" figure in a given knot type. A property of the O'Hara energies is that the "better" the figure of a knot is, the less the energy…
In this article we study the gradient flow of the M\"obius energy introduced by O'Hara in 1991. We will show a fundamental $\varepsilon$-regularity result that allows us to bound the infinity norm of all derivatives for some time if the…
We define and prove the existence of unique solutions of an asymptotic Plateau problem for spacelike maximal surfaces in the pseudo-hyperbolic space of signature (2, n): the boundary data is given by loops on the boundary at infinity of the…
A variational model to simultaneously treat Stress-Driven Rearrangement Instabilities, such as boundary discontinuities, internal cracks, external filaments, edge delamination, wetting, and brittle fractures, is introduced. The model is…
Aiming at optimizing the shape of closed embedded curves within prescribed isotopy classes, we use a gradient-based approach to approximate stationary points of the M\"obius energy. The gradients are computed with respect to Sobolev inner…
We prove that if a curve parametrized by arc length is a stationary point of the Moebius energy introduced by Jun O'Hara, then it is smooth whenever the Moebius energy is finite. Our methods, interestingly, only rely on purely analytical…
Motivated by a problem originating in the study of defect structures in nematic liquid crystals, we describe and study a numerical algorithm for the resolution of a Plateau-like problem. The energy contains the area of a two-dimensional…
Current quadratic smoothness energies for curved surfaces either exhibit distortions near the boundary due to zero Neumann boundary conditions, or they do not correctly account for intrinsic curvature, which leads to unnatural-looking…
Curves play a fundamental role across computer graphics, physical simulation, and mathematical visualization, yet most tools for curve design do nothing to prevent crossings or self-intersections. This paper develops efficient algorithms…
We study critical surfaces for a surface energy which contains the squared $L^2$ norm of the difference of the mean curvature $H$ and the spontaneous curvature $c_o$, coupled to the elastic energy of the boundary curve. We investigate the…
We provide an overview of the leading edge problem in this paper. We have used a self-similar function having a dependence on both the self-similar variable $\eta$ and Reynold's number R to covert the momentum and energy equations into a…
This paper is devoted to classical variational problems for planar elastic curves of clamped endpoints, so-called Euler's elastica problem. We investigate a straightening limit that means enlarging the distance of the endpoints, and obtain…
Superconducting states of an anisortopic s-wave superconductor on a M\"obius strip are studied numerically based on the Ginzburg-Landau theory and the Bogoliubov-de Gennes theory. In both, the equations are solved numerically on discitized…
Given finitely many pointed forces in the plane. Suppose that these forces sum up to zero and their net torques also sum up to zero. One can show that there exists a system of springs whose boundary forces exactly counter-balance these…