Lifespan Theorem for simple constrained surface diffusion flows

We consider closed immersed hypersurfaces in R³ and R⁴ evolving by a special class of constrained surface diffusion flows. This class of constrained flows includes the classical surface diffusion flow. In this paper we present a Lifespan Theorem for these flows, which gives a positive lower bound on the time for which a smooth solution exists, and a small upper bound on the total curvature during this time. The hypothesis of the theorem is that the surface is not already singular in terms of concentration of curvature. This turns out to be a deep property of the initial manifold, as the lower bound on maximal time obtained depends precisely upon the concentration of curvature of the initial manifold in L² for M² immersed in R³ and additionally on the concentration in L³ for M³ immersed in R⁴. This is stronger than a previous result on a different class of constrained surface diffusion flows, as here we obtain an improved lower bound on maximal time, a better estimate during this period, and eliminate any assumption on the area of the evolving hypersurface.     

Finite volume method for solving a one-dimensional parabolic inverse problem

In this paper, finite volume method is used to solve a one-dimensional parabolic inverse problem with source term and Neumann boundary conditions for the first time. Some advantages of this approach are developing difference schemes and maintaining certain properties of the physics of the problems, especially for the treatment of the source term and the unknown boundary conditions. Numerical results show that our method is more effective.     

Mesh-dependent stability for finite element approximations of parabolic equations with mass lumping

In this paper, based on some mesh-dependent estimates on the extreme eigenvalues of a general finite element system defined on a simplicial mesh, novel and sharp bounds on the permissible time step size are derived for the mass lumping finite element approximations of parabolic equations. The bounds are dependent not only on the mesh size but also on the mesh shape. These results provide guidance to the stability of numerical solutions of parabolic problems in relation to the unstructured geometric meshing. Numerical experiments on both uniform meshes and adaptive meshes are presented to validate the theoretical analysis.     

Boundedness for parabolic singular integral with rough kernels and its commutators on Triebel-Lizorkin spaces

In this paper, the authors give the boundedness on Triebel-Lizorkin spaces for the parabolic singular integral with rough kernel and its commutator.     

Survival criterion for a population subject to selection and mutations; Application to temporally piecewise constant environments

We study a parabolic Lotka–Volterra type equation that describes the evolution of a population structured by a phenotypic trait, under the effects of mutations and competition for resources modelled by a nonlocal feedback. The limit of small mutations is characterized by a Hamilton–Jacobi equation with constraint that describes the concentration of the population on some traits. This result was already established in Barles and Perthame (2008); Barles et al. (2009); Lorz et al. (2011) in a time-homogeneous environment, when the asymptotic persistence of the population was ensured by assumptions on either the growth rate or the initial data. Here, we relax these assumptions to extend the study to situations where the population may go extinct at the limit. For that purpose, we provide conditions on the initial data for the asymptotic fate of the population. Finally, we show how this study for a time-homogeneous environment allows to consider temporally piecewise constant environments.     

A novel method of camera pose estimation by parabolic motion

Camera pose estimation is a basic and crucial problem in computer vision, accordingly a novel method is proposed for pose estimation based on parabolic motion in our paper. Firstly, the intersection of lines which are the image plane projection of free-falling trajectories in different locations is computed. According to the properties of vanishing point, the intersection is defined as the vanishing point in gravity direction. Secondly, the image plane projected curve of parabolic trajectory is obtained by Sampson Approximation. Finally, the camera pose is estimated by employing the projective geometry properties of vanishing point and vanishing line implicated in the projected parabola, provided that the intrinsic parameters of camera are specified. The absolute Euclidean distance of translation is obtained innovatively with the known frame frequency. Numerical simulation as well as practical experiment in this paper demonstrates the correctness and feasibility of our method, with the known frame frequency, as the experiment show that compared with the traditional checkerboard method the mean errors of rotation axis, rotation angle and translation are respectively 0.017 rad, 0.007 rad and 11.650 mm by our method. It can generally satisfy the accuracy requirements of camera.     

LIPSCHITZ柱形域上具Lp,q边值的抛物方程边值问题

设Ω为Ｒｎ中—Ｌｉｐｓｃｈｉｔｚ区域，ＳＴ为柱形域的边界Ω×（０，Ｔ）本文研究了此区域上的一般二阶常系数抛物方程具Ｌｐｑ边值的抛物方程边值问题．采用Ｒ．Ｂｒｏｗｎ和Ｚ．Ｓｈｅｎ的方法，我们证明了所涉及的双层位势和单层位势算子的可逆性，从而证明了初边值问题的唯一可解性，而且这种解可由位势算子表示．     

楞次定律实验验证——磁炮

本给出了一种验证楞次定律实验现象的良好方法，可以明显观察到线圈中突然通以电流时产生的电磁冲击力。     

Stability properties of an inverse parabolic problem with unknown boundaries

We treat the stability issue for an inverse problem arising from non-destructive evaluation by thermal imaging. We consider the determination of an unknown portion of the boundary of a thermic conducting body by overdetermined boundary data for a parabolic initial-boundary value problem. We obtain that when the unknown part of the boundary is a priori known to be smooth, the data are as regular as possible and all possible measurements are taken into account, the problem is exponentially ill-posed. Then, we prove that a single measurement with some a priori information on the unknown part of the boundary and minimal assumptions on the data, in particular on the thermal conductivity, is enough to have stable determination of the unknown boundary. Given the exponential ill-posedness, the stability estimate obtained is optimal. AMS 2000 Mathematics Subject Classification. Primary 35R30, Secondary 35B60, 33C90