This paper presents a new family of solutions to the singularly perturbed Allen-Cahn equation α~2Δu + u(1- u^2) = 0 in a smooth bounded domain Ω R^3, with Neumann boundary condition and α > 0 a small paramete...This paper presents a new family of solutions to the singularly perturbed Allen-Cahn equation α~2Δu + u(1- u^2) = 0 in a smooth bounded domain Ω R^3, with Neumann boundary condition and α > 0 a small parameter. These solutions have the property that as α→ 0, their level sets collapse onto a bounded portion of a complete embedded minimal surface with finite total curvature intersecting ?Ω orthogonally and that is non-degenerate respect to ?Ω. The authors provide explicit examples of surfaces to which the result applies.展开更多
Minimal surface is extensively employed in many areas. In this paper, we propose a control mesh representation of a class of minimal surfaces, called generalized helicoid minimal surfaces, which contain the right heli...Minimal surface is extensively employed in many areas. In this paper, we propose a control mesh representation of a class of minimal surfaces, called generalized helicoid minimal surfaces, which contain the right helicoid and catenoid as special examples. We firstly construct the Bézier-like basis called AHT Bézier basis in the space spanned by {1, t, sint, cost, sinht, cosht}, t∈[0,α], α∈[0,5π/2]. Then we propose the control mesh representation of the generalized helicoid using the AHT Bézier basis. This kind of representation enables generating the minimal surfaces using the de Casteljau-like algorithm in CAD/CAGD mod- elling systems.展开更多
In this paper,we first introduce a boundary problem for Lagrangian submanifolds,analogous to the problem for free boundary hypersurfaces and capillary hypersurfaces.Then we present several interesting examples of Lagr...In this paper,we first introduce a boundary problem for Lagrangian submanifolds,analogous to the problem for free boundary hypersurfaces and capillary hypersurfaces.Then we present several interesting examples of Lagrangian submanifolds satisfying this boundary condition and we prove a Lagrangian version of the Nitsche(or Hopf)type theorem.Some problems are proposed at the end of this paper.展开更多
基金supported by the Grant 13-00863S of the Grant Agency of the Czech Republicgrants Fondecyt 1150066,Fondo Basal CMM,Millenium+1 种基金Nucleus CAPDE NC130017NSERC accelerator
文摘This paper presents a new family of solutions to the singularly perturbed Allen-Cahn equation α~2Δu + u(1- u^2) = 0 in a smooth bounded domain Ω R^3, with Neumann boundary condition and α > 0 a small parameter. These solutions have the property that as α→ 0, their level sets collapse onto a bounded portion of a complete embedded minimal surface with finite total curvature intersecting ?Ω orthogonally and that is non-degenerate respect to ?Ω. The authors provide explicit examples of surfaces to which the result applies.
基金Project supported by the National Natural Science Foundation of China (No. 60473130) and the National Basic Research Program (973) of China (No. 2004CB318000)
文摘Minimal surface is extensively employed in many areas. In this paper, we propose a control mesh representation of a class of minimal surfaces, called generalized helicoid minimal surfaces, which contain the right helicoid and catenoid as special examples. We firstly construct the Bézier-like basis called AHT Bézier basis in the space spanned by {1, t, sint, cost, sinht, cosht}, t∈[0,α], α∈[0,5π/2]. Then we propose the control mesh representation of the generalized helicoid using the AHT Bézier basis. This kind of representation enables generating the minimal surfaces using the de Casteljau-like algorithm in CAD/CAGD mod- elling systems.
基金supported by SPP 2026:Geometry at Infinity of Deutsche Forschungsgemeinschaft.A part of this work was carried out when the second author visited the University of British Columbia。
文摘In this paper,we first introduce a boundary problem for Lagrangian submanifolds,analogous to the problem for free boundary hypersurfaces and capillary hypersurfaces.Then we present several interesting examples of Lagrangian submanifolds satisfying this boundary condition and we prove a Lagrangian version of the Nitsche(or Hopf)type theorem.Some problems are proposed at the end of this paper.
