In the paper we study questions about solvability of some boundary value prob- lems for a non-homogenous poly-harmonic equation. As a boundary operator we consider differentiation operator of fractional order in Mille...In the paper we study questions about solvability of some boundary value prob- lems for a non-homogenous poly-harmonic equation. As a boundary operator we consider differentiation operator of fractional order in Miller-Ross sense. The considered problem is a generalization of well-known Dirichlet and Neumann problems.展开更多
Discontinuous Galerkin methods as a solution technique of second order elliptic problems,have been increasingly exploited by several authors in the past ten years.It is generally claimed the alledged attractive geomet...Discontinuous Galerkin methods as a solution technique of second order elliptic problems,have been increasingly exploited by several authors in the past ten years.It is generally claimed the alledged attractive geometrical flexibility of these methods,although they involve considerable increase of computational effort,as compared to continuous methods.This work is aimed at proposing a combination of DGM and non-conforming finite element methods to solve elliptic m-harmonic equations in a bounded domain of R^(n),for n=2 or n=3,with m≥n+1,as a valid and reasonable alternative to classical finite elements,or even to boundary element methods.展开更多
Let m be a positive integer and B be the unit ball of Rn （n≥2）. We investigate the existence, uniqueness and the asymptotic behavior of a positive continuous solution to the following semilinear polyharmonic bounda...Let m be a positive integer and B be the unit ball of Rn （n≥2）. We investigate the existence, uniqueness and the asymptotic behavior of a positive continuous solution to the following semilinear polyharmonic boundary value problem （-△）mu=a1（x）uα1＋a2（x）uα2 , lim｜x｜→1 u（x） （1-｜x｜）m-1 =0, where α1,α2∈（-1, 1） and a1, a2 are two nonnegative measurable functions on B satisfying some appropriate assumptions related to Karamata regular variation theory.展开更多
In this article, we introduce higher order conjugate Poisson and Poisson kernels, which are higher order analogues of the classical conjugate Poisson and Poisson kernels, as well as the polyharmonic fundamental soluti...In this article, we introduce higher order conjugate Poisson and Poisson kernels, which are higher order analogues of the classical conjugate Poisson and Poisson kernels, as well as the polyharmonic fundamental solutions, and define multi-layer potentials in terms of the Poisson field and the polyharmonic fundamental solutions, in which the former is formed by the higher order conjugate Poisson and the Poisson kernels. Then by the multi-layer potentials, we solve three classes of boundary value problems(i.e., Dirichlet, Neumann and regularity problems) with L^p boundary data for polyharmonic equations in Lipschitz domains and give integral representation(or potential) solutions of these problems.展开更多
In this paper,the two-dimensional slowly rotating highly viscous fluid flow in small cavities is modelled by the triharmonic equation for the streamfunction.The Dirichlet problem for this triharmonic equation is reca...In this paper,the two-dimensional slowly rotating highly viscous fluid flow in small cavities is modelled by the triharmonic equation for the streamfunction.The Dirichlet problem for this triharmonic equation is recast as a set of three boundary integral equations which however,do not have a unique solution for three exceptional geometries of the boundary curve surrounding the planar solution domain.This defect can be removed either by using modified fundamental solutions or by adding two supplementary boundary integral conditions which the solution of the boundary integral equations must satisfy.The analysis is further generalized to polyharmonic equations.展开更多
基金financially supported by a grant from the Ministry of Science and Education of the Republic of Kazakhstan(0819/GF4)
文摘In the paper we study questions about solvability of some boundary value prob- lems for a non-homogenous poly-harmonic equation. As a boundary operator we consider differentiation operator of fractional order in Miller-Ross sense. The considered problem is a generalization of well-known Dirichlet and Neumann problems.
基金They also gratefully acknowledge the financial support provided by CNPq,the Brazilian National Research Council,through grants 307996/2008-5 and 304518/2002-6.
文摘Discontinuous Galerkin methods as a solution technique of second order elliptic problems,have been increasingly exploited by several authors in the past ten years.It is generally claimed the alledged attractive geometrical flexibility of these methods,although they involve considerable increase of computational effort,as compared to continuous methods.This work is aimed at proposing a combination of DGM and non-conforming finite element methods to solve elliptic m-harmonic equations in a bounded domain of R^(n),for n=2 or n=3,with m≥n+1,as a valid and reasonable alternative to classical finite elements,or even to boundary element methods.
文摘Let m be a positive integer and B be the unit ball of Rn （n≥2）. We investigate the existence, uniqueness and the asymptotic behavior of a positive continuous solution to the following semilinear polyharmonic boundary value problem （-△）mu=a1（x）uα1＋a2（x）uα2 , lim｜x｜→1 u（x） （1-｜x｜）m-1 =0, where α1,α2∈（-1, 1） and a1, a2 are two nonnegative measurable functions on B satisfying some appropriate assumptions related to Karamata regular variation theory.
基金National Natural Science Foundation of China (Grant No. 11401254)。
文摘In this article, we introduce higher order conjugate Poisson and Poisson kernels, which are higher order analogues of the classical conjugate Poisson and Poisson kernels, as well as the polyharmonic fundamental solutions, and define multi-layer potentials in terms of the Poisson field and the polyharmonic fundamental solutions, in which the former is formed by the higher order conjugate Poisson and the Poisson kernels. Then by the multi-layer potentials, we solve three classes of boundary value problems(i.e., Dirichlet, Neumann and regularity problems) with L^p boundary data for polyharmonic equations in Lipschitz domains and give integral representation(or potential) solutions of these problems.
文摘In this paper,the two-dimensional slowly rotating highly viscous fluid flow in small cavities is modelled by the triharmonic equation for the streamfunction.The Dirichlet problem for this triharmonic equation is recast as a set of three boundary integral equations which however,do not have a unique solution for three exceptional geometries of the boundary curve surrounding the planar solution domain.This defect can be removed either by using modified fundamental solutions or by adding two supplementary boundary integral conditions which the solution of the boundary integral equations must satisfy.The analysis is further generalized to polyharmonic equations.
