Singularities of the solution of the laplacian in domains with circular edges

In this paper we generalize the results from [4] to special domains with curved edges. For general elliptic boundary value problems the behavior of the solutions near arbitrary, smooth edges is analyzed by Maz'ja and Rossmann [3]. First following Dauge [1] we derive a regularity theorem for the solution of the Dirichlet problem of the Laplacian with a decomposition into edge singularities of nontensor product form. In this case the regularity of the remainder term in the decomposition corresponds to the one in the two-dimensional case [2]. Following [4] we obtain a refined decomposition where all singularity terms are of tensor product form. We illustrate our results with several examples.     

Regularity of solutions for the Navier-Stokes system of incompressible flows on a polygon

We are concerned with singularities and regularities of solutions for the Navier-Stokes system of incompressible flows on a polygonal domain with a concave vertex. We subtract the corner singularities by the Stokes operator from the solution velocity and pressure functions of the system. It is shown that the stress intensity factors are functions of time variable, belong to a fractional Sobolev space on the time interval and can be expressed in terms of given data. An increased regularity for the remainder is obtained.     

Dirichlet problem for semilinear edge-degenerate elliptic equations with singular potential term

In this paper, we introduce weighted p-Sobolev spaces on manifolds with edge singularities. We give the proof for the corresponding edge type Sobolev inequality, Poincaré inequality and Hardy inequality. As an application of these inequalities, we prove the existence of nontrivial weak solutions for the Dirichlet problem of semilinear elliptic equations with singular potentials on manifolds with edge singularities.     

Finite element method for elliptic problems with edge singularities

We consider tangentially regular solution of the Dirichlet problem for an homogeneous strongly elliptic operator with constant coefficients, on an infinite vertical polyhedral cylinder based on a bounded polygonal domain in the horizontal xy-plane. The usual complex blocks of singularities in the non-tensor product singular decomposition of the solution are made more explicit by a suitable choice of the regularizing kernel. This permits to design a well-posed semi-discrete singular function method (SFM), which differs from the usual SFM in that the dimension of the space of trial and test functions is infinite. Partial Fourier transform in the z-direction (of edges) enables us to overcome the difficulty of an infinite dimension and to obtain optimal orders of convergence in various norms for the semi-discrete solution. Asymptotic error estimates are also proved for the coefficients of singularities. For practical computations, an optimally convergent full-disc! ! ! retization approach, which consists in coupling truncated Fourier series in the z-direction with the SFM in the xy-plane, is implemented. Other good (though not optimal) schemes, which are based on a tensor product form of singularities are investigated. As an illustration of the results, we consider the Laplace operator.     

Regularity of mixed boundary value problems in ℝ3 and boundary element methods on graded meshes

The solution of the three-dimensional mixed boundary value problem for the Laplacian in a polyhedral domain has special singular forms at corners and edges. A ‘tensor-product’ decomposition of those singular forms along the edges is derived. We present a strongly elliptic system of boundary integral equations which is equivalent to the mixed boundary value problem. Regularity results for the solution of this system of integral equations are given which allow us to analyse the influence of graded meshes on the rate of convergence of the corresponding boundary element Galerkin solutions. We show that it suffices to refine the mesh only towards the edges of the surfaces to regain the optimal rate of convergence.     

Polygonal interface problems for the biharmonic operator

We study some boundary value problems on two-dimensional polygonal topological networks, where on each face, the considered operator is the biharmonic operator. The transmission conditions we impose along the edges are inspired by the models introduced by H. Le Dret [13] and Destuynder and Nevers [9]. The boundary conditions on the external edges are the classical ones. This class of problem contains the boundary value problems for the biharmonic equation in a plane polygon (see [3, 11, 12, 18]). Conforming to the classical results cited above, we prove that the weak solution of our problem admits a decomposition into a regular part and a singular part, the latter being a linear combination of singular functions depending on the domain and the considered boundary value problem. Finally, we give the exact formula for the coefficients of these singularities.     

On the boundary value problem of the biharmonic operator on domains with angular corners

The paper is concerned with boundary singularities of weak solutions of boundary value problems governed by the biharmonic operator. The presence of angular corner points or points at which the type of boundary condition changes in general causes local singularities in the solution. For that case the general theory of V. A. Kondrat'ev provides a priori estimates in weighted Sobolev norms and asymptotic singular representations for the solution which essentially depend on the zeros of certain transcendental functions. The distribution of these zeros will be analysed in detail for the biharmonic operator under several boundary conditions. This leads to sharp a priori estimates in weighted Sobolev norms where the weight function is characterized by the inner angle of the boundary corner. Such estimates for “negative” Sobolev norms are used to analyse also weakly nonlinear perturbations of the biharmonic operator as, for instance, the von Kármán model in plate bending theory and the stream function formulation of the steady state Navier-Stokes problem. It turns out that here the structure of the corner singularities is essentially the same as in the corresponding linear problem.     

The homological singularities of maps in trace spaces between manifolds

We deal with mappings defined between Riemannian manifolds that belong to a trace space of Sobolev functions. The homological singularities of any such map are represented by a current defined in terms of the boundary of its graph. Under suitable topological assumptions on the domain and target manifolds, we show that the non triviality of the singular current is the only obstruction to the strong density of smooth maps. Moreover, we obtain an upper bound for the minimal integral connection of the singular current that depends on the fractional norm of the mapping.     

Elliptic Interface Problems in Axisymmetric Domains. Part I: Singular Functions of Non - Tensorial Type

We study the regularity of solutions of interface problems for the Poisson equation in axisynunetric domains. The Fourier decomposition of the 3D-problem into a sequence of 2D-variational equations end uniform (with respect to the sequence parameter) a prior; estimates of their solutions are derived. Some non-tensorial singular functions describing the behaviour of the solution near interface edges are given and the smoothness of the stress intensity distribution as well as the tangential regularity are characterized in tenns of Sobolev spaces. In a forthcoming part II of this paper, the results will be applied to error estimates of the so-called Fourier-finite-element method for solving approximately elliptic interface problems in 3D.     

Numerical solution of the Kolmogorov-Feller equation with singularities

A method is proposed for solving the Kolmogorov-Feller integro-differential equation with kernels containing delta function singularities. The method is based on a decomposition of the solution into regular and singular parts.     

Dirichlet Problems in Polyhedral Domains I: Regularity of the Solutions

The solution of the Dirichlet problem relative to an elliptic operator in a polyhedron has a complex singular behaviour near edges and vertices. Here we show that this solution and its conormal derivative have a global regularity in appropriate weighted Sobolev spaces. We also investigate some compact embeddings of these spaces. The present results will be applied in a forthcoming work to the constructive treatment of the problem by optimal convergent finite clement method and boundary element method.     

Computing Viscous Flow in an Elastic Tube

We have developed a numerical method for simulating viscous flow through a compliant closed tube, driven by a pair of fluid source and sink. As is natural for tubular flow simulations, the problem is formulated in axisymmetric cylindrical coordinates, with fluid flow described by the Navier-Stokes equations. Because the tubular walls are assumed to be elastic, when stretched or compressed they exert forces on the fluid. Since these forces are singularly supported along the boundaries, the fluid velocity and pressure fields become unsmooth. To accurately compute the solution, we use the velocity decomposition approach, according to which pressure and velocity are decomposed into a singular part and a remainder part. The singular part satisfies the Stokes equations with singular boundary forces. Because the Stokes solution is unsmooth, it is computed to second-order accuracy using the immersed interface method, which incorporates known jump discontinuities in the solution and derivatives into the finite difference stencils. The remainder part, which satisfies the Navier-Stokes equations with a continuous body force, is regular. The equations describing the remainder part are discretized in time using the semi-Lagrangian approach, and then solved using a pressure-free projection method. Numerical results indicate that the computed overall solution is second-order accurate in space, and the velocity is second-order accurate in time.     

A remark on the finite time singularity of the heat flow for harmonic maps

In this paper we study the finite time singularities for the solution of the heat flow for harmonic maps. We derive a gradient estimate for the solution across a finite time singularity. In particular, we find that the solution is asymptotically radial around the isolated singular point in space at a finite singular time. It would be more desirable to understand whether the solution is continuous in space at a finite singular time.Received: 15 March 2001, Accepted: 16 June 2002, Published online: 17 December 2002     

Solution of axisymmetric Maxwell equations

In this article, we study the static and time‐dependent Maxwell equations in axisymmetric geometry. Using the mathematical tools introduced in (Math. Meth. Appl. Sci. 2002; 25 : 49), we investigate the decoupled problems induced in a meridian half‐plane, and the splitting of the solution in a regular part and a singular part, the former being in the Sobolev space H¹ component‐wise. It is proven that the singular parts are related to singularities of Laplace‐like or wave‐like operators. We infer from these characterizations: (i) the finite dimension of the space of singular fields; (ii) global space and space–time regularity results for the electromagnetic field. This paper is the continuation of (Modél. Math. Anal. Numér. 1998; 32 : 359, Math. Meth. Appl. Sci. 2002; 25 : 49). Copyright © 2003 John Wiley & Sons, Ltd.     

An edge driven wavelet frame model for image restoration

Wavelet frame systems are known to be effective in capturing singularities from noisy and degraded images. In this paper, we introduce a new edge driven wavelet frame model for image restoration by approximating images as piecewise smooth functions. With an implicit representation of image singularities sets, the proposed model inflicts different strength of regularization on smooth and singular image regions and edges. The proposed edge driven model is robust to both image approximation and singularity estimation. The implicit formulation also enables an asymptotic analysis of the proposed models and a rigorous connection between the discrete model and a general continuous variational model. Finally, numerical results on image inpainting and deblurring show that the proposed model is compared favorably against several popular image restoration models.     

Timelike Constant Mean Curvature Surfaces with Singularities

We use integrable systems techniques to study the singularities of timelike non-minimal constant mean curvature (CMC) surfaces in the Lorentz–Minkowski 3-space. The singularities arise at the boundary of the Birkhoff big cell of the loop group involved. We examine the behavior of the surfaces at the big cell boundary, generalize the definition of CMC surfaces to include those with finite, generic singularities, and show how to construct surfaces with prescribed singularities by solving a singular geometric Cauchy problem. The solution shows that the generic singularities of the generalized surfaces are cuspidal edges, swallowtails, and cuspidal cross caps.     

Three Dimensional Interface Problems for Elliptic Equations

The author studies the structure of solutions to the interface problems for second order linear elliptic partial differential equations in three space dimension.The set of singular points consists of some singular lines and some isolated singular points.It is proved that near a singular line or a singular point,each weak solution can be decomposed into two parts,a singular part and a regular part.The singular parts are some finite sum of particular solutions to some simpler equations,and the regular parts are bounded in some norms,which are slightly weaker than that in the Sobolev space H~2.     

Mixed boundary value problems of semilinear elliptic PDEs and BSDEs with singular coefficients

In this paper, we prove that there exists a unique weak (Sobolev) solution to the mixed boundary value problem for a general class of semilinear second order elliptic partial differential equations with singular coefficients. Our approach is probabilistic. The theory of Dirichlet forms and backward stochastic differential equations with singular coefficients and infinite horizon plays a crucial role.     

Scalar problem of plane wave diffraction by a system of nonintersecting screens and inhomogeneous bodies

The scalar problem of plane wave diffraction by a system of bodies and infinitely thin screens is considered in a quasi-classical formulation. The solution is sought in the classical sense but is defined not in the entire space ?³ but rather everywhere except for the screen edges. The original boundary value problem for the Helmholtz equation is reduced to a system of weakly singular integral equations in the regions occupied by the bodies and on the screen surfaces. The equivalence of the integral and differential formulations is proven, and the solvability of the system in the Sobolev spaces is established. The integral equations are approximately solved by the Bubnov-Galerkin method. The convergence of the method is proved, its software implementation is described, and numerical results are presented.     

关于二元样条空间下的三角剖分的分解

本文指出，在一定条件下，对于一个二元样条空间，所考虑的三种剖分中的某些胞腔和网线可以消去，而前后两个三角剖分下样条空间的结构有着紧密的联系，从而可以用简单划分下的空间结构表示复杂剖分下的空间结构。该分解剖分的步骤可以递推的进行，尤其对Ｓ＾１ｓ。据此，本文还分析了剖分对Ｓ＾１２的奇异性并给出一组奇异的剖分。