Complete flat surfaces with two isolated singularities in hyperbolic 3-space

We construct examples of flat surfaces in H³ which are graphs over a two-punctured horosphere and classify complete embedded flat surfaces in H³ with only one end and at most two isolated singularities.     

Ribaucour transformations for flat surfaces in the hyperbolic 3-space

We consider Ribaucour transformations for flat surfaces in the hyperbolic 3-space, H³${\mathbb{H}}^3$. We show that such transformations produce complete, embedded ends of horosphere type and curves of singularities which generically are cuspidal edges. Moreover, we prove that these ends and curves of singularities do not intersect. We apply Ribaucour transformations to rotational flat surfaces in H³${\mathbb{H}}^3$ providing new families of explicitly given flat surfaces H³${\mathbb{H}}^3$ which are determined by several parameters. For special choices of the parameters, we get surfaces that are periodic in one variable and surfaces with any even number or an infinite number of embedded ends of horosphere type.     

Complete linear Weingarten surfaces of Bryant type. A Plateau problem at infinity

In this paper we study a large class of Weingarten surfaces which includes the constant mean curvature one surfaces and flat surfaces in the hyperbolic 3-space. We show that these surfaces can be parametrized by holomorphic data like minimal surfaces in the Euclidean 3-space and we use it to study their completeness. We also establish some existence and uniqueness theorems by studing the Plateau problem at infinity: when is a given curve on the ideal boundary the asymptotic boundary of a complete surface in our family? and, how many embedded solutions are there?

     

Time-like Willmore surfaces in Lorentzian 3-space

Let R13 be the Lorentzian 3-space with inner product (, ). Let Q3 be the conformal compactification of R13, obtained by attaching a light-cone C∞ to R13 in infinity. Then Q3 has a standard conformal Lorentzian structure with the conformal transformation group O(3,2)/{±1}. In this paper, we study local conformal invariants of time-like surfaces in Q3 and dual theorem for Willmore surfaces in Q3. Let M (?) R13 be a time-like surface. Let n be the unit normal and H the mean curvature of the surface M. For any p ∈ M we define S12(p) = {X ∈ R13 (X - c(P),X - c(p)) = 1/H(p)2} with c(p) = P 1/H(p)n(P) ∈ R13. Then S12 (p) is a one-sheet-hyperboloid in R3, which has the same tangent plane and mean curvature as M at the point p. We show that the family {S12(p),p ∈ M} of hyperboloid in R13 defines in general two different enveloping surfaces, one is M itself, another is denoted by M (may be degenerate), and called the associated surface of M. We show that (i) if M is a time-like Willmore surface in Q3 with non-degenerate associated surface M, then M is also a time-like Willmore surface in Q3 satisfying M = M; (ii) if M is a single point, then M is conformally equivalent to a minimal surface in R13.     

On the two-dimensional cauchy problem of a hyperbolic system

In this paper, we prove the global existence of generalized solutions to a two-dimensional Cauchy problem of a hyperbolic system by introducing a new definition of generalized solution. Moreover, the solution may involve delta-wave.     

A variational technique for optimal boundary control in a hyperbolic problem

We investigate the problem of controlling the boundary functions in a one dimensional hyperbolic problem by minimizing the functional including the final state. After proving the existence and uniqueness of the solution to the given optimal control problem, we get the Frechet differential of the functional and give the necessary condition to the optimal solution in the form of the variational inequality via the solution of the adjoint problem. We constitute a minimizing sequence by the method of projection of the gradient and prove its convergence to the optimal solution.     

Stability estimate for the hyperbolic inverse boundary value problem by local Dirichlet-to-Neumann map

In this paper we consider the stability of the inverse problem of determining a function q(x) in a wave equation in a bounded smooth domain in Rn from boundary observations. This information is enclosed in the hyperbolic (dynamic) Dirichlet-to-Neumann map associated to the solutions to the wave equation. We prove in the case of n?2 that q(x) is uniquely determined by the range restricted to a subboundary of the Dirichlet-to-Neumann map whose stability is a type of double logarithm.     

On a refinement of the generalized Catalan numbers for Weyl groups

Let be an irreducible crystallographic root system with Weyl group , coroot lattice and Coxeter number , spanning a Euclidean space , and let be a positive integer. It is known that the set of regions into which the fundamental chamber of is dissected by the hyperplanes in of the form for and is equinumerous to the set of orbits of the action of on the quotient . A bijection between these two sets, as well as a bijection to the set of certain chains of order ideals in the root poset of , are described and are shown to preserve certain natural statistics on these sets. The number of elements of these sets and their corresponding refinements generalize the classical Catalan and Narayana numbers, which occur in the special case and .

     

On convexity of the functional for inverse problems of hyperbolic equations

In this paper, a strategy to design a functional for inverse problems of hyperbolic equations is proposed. For an inverse source problem, it is shown that the designed functional is globally strictly convex. For an inverse coefficient problem, we can only prove that it is strictly convex near true solution. This strategy can be generalized to other inverse problems, as long as Lipschitz stability is given.     

Inequalities for the generalized trigonometric and hyperbolic functions

The generalized trigonometric functions occur as an eigenfunction of the Dirichlet problem for the one-dimensional p$p$-Laplacian. The generalized hyperbolic functions are defined similarly. Some classical inequalities for trigonometric and hyperbolic functions, such as Mitrinovi?–Adamovi?’s inequality, Lazarevi?’s inequality, Huygens-type inequalities, Wilker-type inequalities, and Cusa–Huygens-type inequalities, are generalized to the case of generalized functions.     

Helicoidal flat surfaces in the 3‐sphere

In this paper, helicoidal flat surfaces in the 3‐dimensional sphere are considered. A complete classification of such surfaces, that generalizes a classification of rotational flat surfaces, is given in terms of the first and second fundamental forms for asymptotic parameters. The result consists in a relation between helicoidal flat surfaces and linear solutions of the corresponding homogeneous wave equation for the angle function.     

三维欧氏Steiner最小树的Delaunay四面体网格混合智能算法

Steiner最小树问题是组合优化中经典的NP难题,在许多实际问题中有着广泛的应用,而三维欧氏Steiner最小树问题是对二维欧氏Steiner最小树问题的推广。由于三维欧氏Steiner树问题的求解非常困难,至今为止的相关成果较为少见。本文针对该问题,利用Delaunay四面体网格剖分技术,提出了一种混合型智能求解方法,不仅可以尽量避免拓扑结构陷入局部最优,且对较大规模的问题求解亦有良好的效果。算法在Matlab环境下编程实现,经实例测试,获得了满意的效果。     

On a first boundary value problem for hyperbolic equations in the plane

In the paper we study a boundary value problem for a hyperbolic equation with two independent variables; this problem is a generalization of the well-known Darboux problem. Translated fromMatematicheskie Zametki, Vol. 65, No. 2, pp. 294–306, February, 1999.     

On the local cauchy problem for quasilinear hyperbolic functional differential systems

Generalized solutions of weakly coupled functional differential systems are investigated. Theorems on the existence, uniqueness and continuous dependence upon initial data are given. The local Cauchy problem is transformed into functional integral equations. The method of bicharacteristics and integral inequalities are used.     

Propagation of singularities in the Cauchy problem for a class of degenerate hyperbolic operators

We consider second order degenerate hyperbolic Cauchy problems, the degeneracy coming either from low regularity (less than Lipschitz continuity) of the coefficients with respect to time, or from weak hyperbolicity. In the weakly hyperbolic case, we assume an intermediate condition between effective hyperbolicity and the Levi condition. We construct the fundamental solution and study the propagation of singularities using an unified approach to these different kinds of degeneracy.     

On a model of phase relaxation for the hyperbolic Stefan problem

In this paper we study a model of phase relaxation for the Stefan problem with the Cattaneo–Maxwell heat flux law. We prove an existence and uniqueness result for the resulting problem and we show that its solution converges to the solution of the Stefan problem as the two relaxation parameters go to zero, provided a relation between these parameters holds.     

On the Cauchy problem for a generalized Boussinesq equation

In this paper, we consider the Cauchy problem for a generalized Boussinesq equation. We show that, under suitable conditions, a global solution for the initial value problem exists. In addition, we derive the sufficient conditions for the blow-up of the solution to the problem.     

The Dirichlet problem at infinity for harmonic map equations arising from constant mean curvature surfaces in the hyperbolic 3-space

The purpose of this paper is to study some uniqueness, existence and regularity properties of the Dirichlet problem at infinity for proper harmonic maps from the hyperbolic m-space to the open unit n-ball with a specific incomplete metric. When m=n=2, harmonic solutions of this Dirichlet problem yield complete constant mean curvature surfaces in the hyperbolic 3-space. Received: 25 January 2001 / Accepted: 23 February 2001 / Published online: 25 June 2001