Coupled surface and grain boundary motion: a travelling wave solution

Existence and uniqueness are proven for a travelling wave solution for a problem in which motion by mean curvature is coupled with surface diffusion. This problem pertains to a bicrystal in a “quarter-loop” geometry in which one grain grows at the expense of the other, and the internal grain boundary between the two crystals contacts the exterior surface at a “groove root” or “tri-junction” where various balance laws hold. Far in front and behind the groove root the overall height of the bicrystal is assumed to be unperturbed. Whereas in a previous paper (Acta Mater. 51 (2003) 1981) a partially linearized formulation was considered for which explicit solutions could be found, here we treat the fully nonlinear problem. Employing an angle formulation and a scaled arc-length parameterization, we reduce the problem to the solution of a third order ODE with a jump condition at the origin. Existence is proven if m, the ratio of the exterior surface energy to the surface energy of the grain boundary, is less than about $\approx 0.92$. Uniqueness of these solutions is demonstrated within the class of single-valued solutions. A numerical comparison is made with the solution of the partially linearized formulation found earlier for the sake of illustration.     

Two comparison theorems of bsdes

In this paper, by the equations of Mao [9] and Peng [5], we use the martingale method to establish the comparison theorems of backward stochastic differential equations (BSDEs). We generalize the results of Cao-Yan [1].     

Local existence and uniqueness theorems for a free boundary problem

Free boundary problems are considered, where the tangential and normal components u_t and u_n of an otherwise unknown plane harmonic vector field are prescribed along the unknown boundary curve as a function of the coordinates x, y and the tangent angle θ. The vector field is required to exist either in the interior region G⁺ or in the exterior G^?. In each case the free boundary is characterized by a nonlinear integral equation. A linearised version of this equation is a one-dimensional singular integral equation. Under rather general hypotheses which are easy to check, the properties of the linear equation are described by Noether's theorems. The regularity of the solution is studied and the effect of the nonlinear terms is estimated. A variant of the Nash-Moser implicit-function theorem can be applied. This yields local existence and uniqueness theorems for the free boundary problem in Hölder-classes H^2+μ. The boundary curve depends continuously on the defining data. Finally some examples are given, where the linearised equation can be completely discussed.     

On the comparison of a Kantorovich-type and Moore theorems

It was shown in Zhue and Wolfe (Nonlinear Anal. 15(2):229–232, 1990) that the hypotheses of the affine invariant Moore theorem for solving nonlinear equations using Newton’s method are always valid when those of the Kantorovich theorem due to Deuflhard and Heindl (SIAM J. Numer. Anal. 16:1–10, 1980) hold but not necessarily vice versa. Here we show that this result is not true in general for a weaker version of the Kantorovich theorem shown recently by us in Argyros (Advances in the Efficiency of Computational Methods and Applications, World Scientific, Singapore, 2000; Int. J. Comput. Math. 80:5, 2002) and Argyros and Szidarovszky (The Theory and Applications of Iteration Methods, CRC Press, Boca Raton, 1993).     

A phase-field model of grain boundary motion

We consider a phase-field model of grain structure evolution, which appears in materials sciences. In this paper we study the grain boundary motion model of Kobayashi-Warren-Carter type, which contains a singular diffusivity. The main objective of this paper is to show the existence of solutions in a generalized sense. Moreover, we show the uniqueness of solutions for the model in one-dimensional space. Dedicated to Jürgen Sprekels on the occasion of his 60th birthday     

On boundary uniqueness theorems for a linear elliptic equation with constant coefficients

For the solutions of an elliptic equation with constant coefficients, we prove uniqueness theorems that generalize the classical boundary uniqueness theorems for analytic functions.     

具有脉冲的二阶三点边值问题存在性定理

In this paper, two existence theorems are given concerning the following 3-point boundary value problem of second order differential systems with impulses     

Pseudoconvex fully nonlinear partial differential operators: strong comparison theorems

We introduce a new class of curvature PDOs describing relevant properties of real hypersurfaces of . In our setting, the pseudoconvexity and the Levi form play the same role as the convexity and the real Hessian matrix play in the real Euclidean one. Our curvature operators are second-order fully nonlinear PDOs not elliptic at any point. However, when computed on generalized s-pseudoconvex functions, we shall show that their characteristic form is nonnegative definite with kernel of dimension one. Moreover, we shall show that the missing ellipticity direction can be recovered by taking into account the CR structure of the hypersurfaces. These properties allow us to prove a strong comparison principle, leading to symmetry theorems for domains with constant curvatures and to identification results for domains with comparable curvatures.     

Some comparison theorems for nonnegative splittings of matrices

Summary In a recent paper the author has proposed some theorems on the comparison of the asymptotic rates of convergence of two nonnegative splittings. They extended the corresponding result of Miller and Neumann and implied the earlier theorems of Varga, Beauwens, Csordas and Varga. An open question by Miller and Neumann, which additional and appropriate conditions should be imposed to obtain strict inequality, was also answered. This article continues to investigate the comparison theorems for nonnegative splittings. The new results extend and imply the known theorems by the author, Miller and Neumann.The Project Supported by the Natural Science Foundation of Jiangsu Province Education Commission