Problems for elliptic singular equations with a quadratic gradient term

We investigate the homogeneous Dirichlet problem for a class of second-order elliptic partial differential equations with a quadratic gradient term and singular data. In particular, we study the asymptotic behaviour of the solution near the boundary under suitable assumptions on the growth of the coefficients of the equation.     

Problems for elliptic singular equations with a gradient term

We investigate the homogeneous Dirichlet problem for a class of second-order nonlinear elliptic partial differential equations with singular data. In particular, we study the asymptotic behaviour of the solution near the boundary up to the second order under various assumptions on the growth of the coefficients of the equation.     

THE PERMUTATION FORMULA OF SINGULAR INTEGRALS WITH BOCHNER-MARTINELLI KERNEL ON STEIN MANIFOLDS

Using the method of localization, the authors obtain the permutation formula of singular integrals with Bochner-Martinelli kernel for a relative compact domain with C(1) smooth boundary on a Stein manifold. As an application the authors discuss the regularization problem for linear singular integral equations with Bochner-Martinelli kernel and variable coefficients; using permutation formula, the singular integral equation can be reduced to a fredholm equation.     

Generalized boundary value problems for analytic functions with convolutions and its applications

In this paper, we first establish a locality theory for the Noethericity of generalized boundary value problems on the spaces . By means of this theory, of the classical boundary value theory, and of the theory of Fourier analysis, we discuss the necessary and sufficient conditions of the solvability and obtain the general solutions and the Noether conditions for one class of generalized boundary value problems. All cases as regards the index of the coefficients in the equations are considered in detail. Moreover, we apply our theoretical results to the solvability of singular integral equations with variable coefficients. Thus, this paper will be of great significance for the study of improving and developing complex analysis, integral equation, and boundary value theory.     

Existence and asymptotic behavior of solutions to a singular parabolic equation

In this paper, we study the initial-boundary value problem for a class of singular parabolic equations. Under some conditions, we obtain the existence and asymptotic behavior of solutions to the problem by parabolic regularization method and the sub-super solutions method. As a byproduct, we prove the existence of solutions to some problems with gradient terms, which blow up on the boundary.     

A new extension of boundary elements method for classical elliptic partial differential equations with constant coefficients

This article is devoted to an extension of boundary elements method (BEM) for solving elliptic partial differential equations of general type with constant coefficients. As the fundamental solution of these equations was not available in the literature, BEM was not able to handle them, directly. So the dual reciprocity method (DRM) has been applied to tackle these problems. In this work, a fundamental solution for these equations is obtained and a new formulation is derived to solve them. Besides, we show that the rate of convergence of the new scheme is quadratic when singular (boundary and domain) integrals are calculated, accurately. The new scheme is applicable on complex domains, without needing internal nodes, just same as conventional BEM. So the CPU time of the new scheme is much less than that of the DRM. Numerical examples presented in the article show ability and efficiency of the new scheme in solving two‐dimensional nonhomogenous elliptic boundary value problems, clearly. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 2027–2042, 2015     

The unique solvability of certain multiplicative-convolution equations

In the class of distributions of slow (moderate) growth we consider a class of equations with operations of convolution and multiplication on the real axis. This class contains convolution equations, in particular, ordinary differential equations with constant coefficients, equations in finite differences, functional differential equations with constant coefficients and shifts, and pair differential equations. By virtue of the analytic representation theory for distributions of moderate growth (the Hilbert or Cauchy transform) the class of equations under consideration is equivalent to the class of boundary value problems of the Riemann type, where an equation corresponds to a boundary value condition in the sense of distributions of moderate growth. As a research technique we use the Fourier transform, the generalized Fourier transform (the Carleman-Fourier transform), and the theory of convolution equations in the space of distributions of moderate growth.     

Elliptic and parabolic equations with measurable coefficients in weighted Sobolev spaces

We consider both divergence and non-divergence parabolic equations on a half space in weighted Sobolev spaces. All the leading coefficients are assumed to be only measurable in the time and one spatial variable except one coefficient, which is assumed to be only measurable either in the time or the spatial variable. As functions of the other variables the coefficients have small bounded mean oscillation (BMO) semi-norms. The lower-order coefficients are allowed to blow up near the boundary with a certain optimal growth condition. As a corollary, we also obtain the corresponding results for elliptic equations.     

Zur Regularität Selbstadjungierter Randwertaufgaben

In this note we derive a necessary requirement for the self-adjointness of a boundary value problem of ordinary differential equations, which involves only the leading coefficients of the normalized boundary conditions. As application it is shown that every self-adjoint boundary value problem of even order (and thus especially every real self-adjoint boundary value problem) is regular. The proof of this fact depends on the evaluation of a sort of generalized Vandermonde's determinant.     

Classical solutions of quasilinear parabolic systems on two dimensional domains

Using results on abstract evolutions equations and recently obtained results on elliptic operators with discontinuous coefficients including mixed boundary conditions we prove that quasilinear parabolic systems admit a local, classical solution in the space of p–integrable functions, for some p greater than 1, over a bounded two dimensional space domain. The treatment of such equations in a space of integrable functions enables us to define the normal component of the current across the boundary of any Lipschitz subset. As applications we have in mind systems of reaction diffusion equations, e.g. van Roosbroeck’s system.     

The exact order of information-based complexity of weakly singular integral equations with periodic analytic coefficients

The exact order of complexity of weakly singular integral equations including logarithmic singularities with periodic and analytic coefficients is found. This class of equations contains the boundary equations of exterior boundary value problems for the two-dimensional Helmholtz equation. Translated fromMatematicheskie Zarnetki, Vol. 62, No. 5, pp. 643–656, November, 1997. Translated by O. V. Sipacheva     

On A parabolic equation with a singular lower order term

We obtain the existence of the weak Green's functions of parabolic equations with lower order coefficients in the so called parabolic Kato class which is being proposed as a natural generalization of the Kato class in the study of elliptic equations. As a consequence we are able to prove the existence of solutions of some initial boundary value problems. Moreover, based on a lower and an upper bound of the Green's function, we prove a Harnack inequality for the non-negative weak solutions.

     

Backward stochastic differential equations with rank-based data

In this paper, we investigate Markovian backward stochastic differential equations(BSDEs) with the generator and the terminal value that depend on the solutions of stochastic differential equations with rankbased drift coefficients. We study regularity properties of the solutions of this kind of BSDEs and establish their connection with semi-linear backward parabolic partial differential equations in simplex with Neumann boundary condition. As an application, we study the European option pricing problem with capital size based stock prices.     

Continuous dependence results for non-linear Neumann type boundary value problems

We obtain estimates on the continuous dependence on the coefficient for second-order non-linear degenerate Neumann type boundary value problems. Our results extend previous work of Cockburn et al., Jakobsen and Karlsen, and Gripenberg to problems with more general boundary conditions and domains. A new feature here is that we account for the dependence on the boundary conditions. As one application of our continuous dependence results, we derive for the first time the rate of convergence for the vanishing viscosity method for such problems. We also derive new explicit continuous dependence on the coefficients results for problems involving Bellman-Isaacs equations and certain quasilinear equation.     

Singular integral on bounded strictly pseudoconvex domain

Kytmanov and Myslivets gave a special Cauchy principal value of the singular integral on the bounded strictly pseudoconvex domain with smooth boundary. By means of this Cauchy integral principal value, the corresponding singular integral and a composition formula are obtained. This composition formula is quite different from usual ones in form. As an application, the corresponding singular integral equation and the system of singular integral equations are discussed as well.     

Reflected Generalized Backward Doubly SDEs Driven by Lévy Processes and Applications

In this paper, we study reflected generalized backward doubly stochastic differential equations driven by Teugels martingales associated with Lévy process (RGBDSDELs in short) with one continuous barrier. Under uniformly Lipschitz coefficients, we prove an existence and uniqueness result by means of the penalization method and the fixed-point theorem. As an application, this study allows us to give a probabilistic representation for the solutions to a class of reflected stochastic partial differential integral equations (SPDIEs in short) with a nonlinear Neumann boundary condition.     

闭光滑流形上的高阶线性微-积分方程

利用积分变换技巧,作者给出了C~n中闭光滑可定向流形上一个新的Bochner-Martinelli型积分的高阶偏导数的奇异积分的Hadamard主值,获得了高阶奇异积分的Plemelj公式和合成公式,还讨论了相应的变系数线性微分积分方程的正则化,证明其可转化为一类等价的Fredholm方程。并且指出其特征方程当给出一组适当的边值条件时,在L~*中存在唯一解。     

Efficient Implementation of a Spectral-Tau Method for a Class of Two-Point Boundary-Value and Initial-Boundary-Value Problems

A simple and efficient spectral method for solving the second, third order and fourth order elliptic equations with variable coefficients and nonlinear differential equations is presented. It is different from spectral-collocation method which leads to dense, ill-conditioned matrices. The spectral method in this paper solves for the coefficients of the solution in a Chebyshev series, leads to discrete systems with special structured matrices which can be factorized and solved efficiently. We also extend the method to boundary value problems in two space dimensions and solve 2-D separable equation with variable coefficients. As an application, we solve Cahn-Hilliard equation iteratively via first-order implicit time discretization scheme. Ample numerical results indicate that the proposed method is extremely accurate and efficient.     

奇异Hamilton-Jacobi-Bellman方程粘性解的存在唯一性

研究系数在边界点有奇性的一类Hamilt on- Jacobi- Bellman (HJB)方程的粘性解的存在唯一性问题及解的渐近估计,这类问题包括波动系数振荡或爆破的情况.奇异HJB方程在随机最优控制和金融数学等许多领域都有重要的应用,包括金融数学中的随机利率模型.应用粘性上下解理论建立了一类奇异HJB方程的比较原理,给出了粘性解存在唯一性的条件.     

On the rate of convergence of solutions in domain with periodic multilevel oscillating boundary

In this paper we deal with the homogenization problem for the Poisson equation in a singularly perturbed domain with multilevel periodically oscillating boundary. This domain consists of the body, a large number of thin cylinders joining to the body through the thin transmission zone with rapidly oscillating boundary. Inhomogeneous Fourier boundary conditions with perturbed coefficients are set on the boundaries of the thin cylinders and on the boundary of the transmission zone. We prove the homogenization theorems and derive the estimates for the convergence of the solutions. Copyright © 2010 John Wiley & Sons, Ltd.