The main purpose of this paper is to investigate the univalence of normalized polyharmonic mappings with bounded length distortions in the unit disk.We first establish the coefficient estimates for polyharmonic mappin...The main purpose of this paper is to investigate the univalence of normalized polyharmonic mappings with bounded length distortions in the unit disk.We first establish the coefficient estimates for polyharmonic mappings with bounded length distortions.Then,using these results,we establish five Landau-type theorems for subclasses of polyharmonic mappings F and L(F),where F has bounded length distortion and L is a differential operator.展开更多
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
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 paper, we are concerned with the following problem：{（-△）ku=λf（x）｜u｜q-2u＋g（x）｜u｜k＊-2u, x∈Ω, u∈H k0 （Ω）, where Ωis a bounded domain in RN with N ≥2k＋1, 1〈q〈2,λ〉0, f, g are continuous ...In this paper, we are concerned with the following problem：{（-△）ku=λf（x）｜u｜q-2u＋g（x）｜u｜k＊-2u, x∈Ω, u∈H k0 （Ω）, where Ωis a bounded domain in RN with N ≥2k＋1, 1〈q〈2,λ〉0, f, g are continuous functions on Ω which are somewhere positive but which may change sign on Ω. k＊ = N2/N-2k is the critical Sobolev exponent. By extracting the Palais-Smale sequence in the Nehari manifold, the existence of multiple nontrivial solutions to this equation is verified.展开更多
Let 0<α<2,p≥1,m∈ℕ_(+).Consider the positive solution u of the PDE(-△)^(α/2+m)u(x)=u^(p)(x) in R^(n).(0.1) In[1](Transactions of the American Mathematical Society,2021),Cao,Dai and Qin showed that,under the ...Let 0<α<2,p≥1,m∈ℕ_(+).Consider the positive solution u of the PDE(-△)^(α/2+m)u(x)=u^(p)(x) in R^(n).(0.1) In[1](Transactions of the American Mathematical Society,2021),Cao,Dai and Qin showed that,under the condition u∈Lα,(0.1)possesses a super polyharmonic property (-△)^(k+α/2)u≥0 for k=0,1,⋯,m−1.In this paper,we show another kind of super polyharmonic property(−Δ)^(k)u>0 for k=1,⋯,m−1,under the conditions and(−Δ)^(m)u≥0.Both kinds of super polyharmonic properties can lead to an equivalence between(0.1)and the integral equation u(x)=∫_(R^(n))u^(p)(y)/|x-y|^(n-2m-α)dy.One can classify solutions to(0.1)following the work of[2]and[3]by Chen,Li,Ou.展开更多
In this article, a polyharmonic Neumann function in a sector with angle π n （n N） is studied by convolution. Especially, the outward normal derivatives at three corner points are defined properly. We give the recur...In this article, a polyharmonic Neumann function in a sector with angle π n （n N） is studied by convolution. Especially, the outward normal derivatives at three corner points are defined properly. We give the recursive expressions for the polyharmonic Neumann function, obtaining the solution and the condition of solvability for the related polyharmonic Neumann problem.展开更多
This paper is devoted to the following high order elliptic problems under the Navier boundary condition: ?Without assuming the standard subcritical polynomial growth condition ensuring the compactness of a bounded (P....This paper is devoted to the following high order elliptic problems under the Navier boundary condition: ?Without assuming the standard subcritical polynomial growth condition ensuring the compactness of a bounded (P.S.) sequence, we show that the Navier boundary value problem has at least a weak nontrivial solution for all λ>0?by using mountain pass theorem.展开更多
In this article, a class of Dirichlet problem with Lp boundary data for poly-harmonic function in the upper half plane is mainly investigated. By introducing a sequence of kernel functions called higher order Poisson ...In this article, a class of Dirichlet problem with Lp boundary data for poly-harmonic function in the upper half plane is mainly investigated. By introducing a sequence of kernel functions called higher order Poisson kernels and a hierarchy of integral operators called higher order Pompeiu operators, we obtain a main result on integral representation solution as well as the uniqueness of the polyharmonic Dirichlet problem under a certain estimate.展开更多
This paper deals with the function u which satisfies△k_u = 0, where k≥2 is an integer. Such a function u is called a polyharmonic function. The author gives an upper bound of the measure of the nodal set of u, and s...This paper deals with the function u which satisfies△k_u = 0, where k≥2 is an integer. Such a function u is called a polyharmonic function. The author gives an upper bound of the measure of the nodal set of u, and shows some growth property of u.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
We study positive solutions of the following polyharmonic equation with Hardy weights associated to Navier boundary conditions on a half space：where rn is any positive integer satisfying 0 〈 2m 〈 n. We first prove ...We study positive solutions of the following polyharmonic equation with Hardy weights associated to Navier boundary conditions on a half space：where rn is any positive integer satisfying 0 〈 2m 〈 n. We first prove that the positive solutions of （0.1） are super polyharmonic, i.e.,where x＊ = （x1,... ,Xn-1, --Xn） is the reflection of the point x about the plane Rn-1. Then, we use the method of moving planes in integral forms to derive rotational symmetry and monotonicity for the positive solution of （0.3）, in which α can be any real number between 0 and n. By some Pohozaev type identities in integral forms, we prove a Liouville type theorem--the non-existence of positive solutions for （0.1）.展开更多
This paper introduces a domain decomposition preconditioner for elliptic equations with rough coefficients. The coarse space of the domain decomposition method is constructed via the so-called rough polyharmonic splin...This paper introduces a domain decomposition preconditioner for elliptic equations with rough coefficients. The coarse space of the domain decomposition method is constructed via the so-called rough polyharmonic splines （RPS for short）. As an approximation space of the eUiptic problem, RPS is known to recover the quasi-optimal convergence rate and attain the quasi-optimal localization property. The authors lay out the formulation of the RPS based domain decomposition preconditioner, and numerically verify the performance boost of this method through several examples.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
The remainders and the convergence of cardinal polyharmonic spline interpolation are studied, and the asymptotic behavior of the best approximation by polyharmonic spline and the average K-width of some class of smoot...The remainders and the convergence of cardinal polyharmonic spline interpolation are studied, and the asymptotic behavior of the best approximation by polyharmonic spline and the average K-width of some class of smooth functions defined on the Euclidean space Rn are determined.展开更多
The aim of this article is twofold.One aim is to establish the precise forms of Landau-Bloch type theorems for certain polyharmonic mappings in the unit disk by applying a geometric method.The other is to obtain the p...The aim of this article is twofold.One aim is to establish the precise forms of Landau-Bloch type theorems for certain polyharmonic mappings in the unit disk by applying a geometric method.The other is to obtain the precise values of Bloch constants for certain log-p-harmonic mappings.These results improve upon the corresponding results given in Bai et al.(Complex Anal.Oper.Theory,13(2):321-340,2019).展开更多
Let be a hypercube in Rn. We prove theorems concerning mean-values of harmonic and polyharmonic functions on In(r), which can be considered as natural analogues of the famous Gauss surface and volume mean-value formul...Let be a hypercube in Rn. We prove theorems concerning mean-values of harmonic and polyharmonic functions on In(r), which can be considered as natural analogues of the famous Gauss surface and volume mean-value formulas for harmonic functions on the ball in and their extensions for polyharmonic functions. We also discuss an application of these formulas—the problem of best canonical one-sided L1-approximation by harmonic functions on In(r).展开更多
In this paper we present a numerical method for solving Riemann type problem for the fifth order improperly elliptic equation in complex plane .We reduce this problem to the boundary value problems for properly ellipt...In this paper we present a numerical method for solving Riemann type problem for the fifth order improperly elliptic equation in complex plane .We reduce this problem to the boundary value problems for properly elliptic equations, and then solve those problems by the grid methods.展开更多
基金supported by the Natural Science Foundation of Guangdong Province(2021A1515010058)supported by the Youth Innovation Foundation of Shenzhen Polytechnic University(6024310023K)。
文摘The main purpose of this paper is to investigate the univalence of normalized polyharmonic mappings with bounded length distortions in the unit disk.We first establish the coefficient estimates for polyharmonic mappings with bounded length distortions.Then,using these results,we establish five Landau-type theorems for subclasses of polyharmonic mappings F and L(F),where F has bounded length distortion and L is a differential operator.
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.
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supported by the National Natural Science Foundation of China(11326139,11326145)Tian Yuan Foundation(KJLD12067)+1 种基金Central Specialized Fundation of SCUEC(CZQ13013)the Project of Jiangxi Province Technology Hall(2014BAB211010)
文摘In this paper, we are concerned with the following problem：{（-△）ku=λf（x）｜u｜q-2u＋g（x）｜u｜k＊-2u, x∈Ω, u∈H k0 （Ω）, where Ωis a bounded domain in RN with N ≥2k＋1, 1〈q〈2,λ〉0, f, g are continuous functions on Ω which are somewhere positive but which may change sign on Ω. k＊ = N2/N-2k is the critical Sobolev exponent. By extracting the Palais-Smale sequence in the Nehari manifold, the existence of multiple nontrivial solutions to this equation is verified.
文摘Let 0<α<2,p≥1,m∈ℕ_(+).Consider the positive solution u of the PDE(-△)^(α/2+m)u(x)=u^(p)(x) in R^(n).(0.1) In[1](Transactions of the American Mathematical Society,2021),Cao,Dai and Qin showed that,under the condition u∈Lα,(0.1)possesses a super polyharmonic property (-△)^(k+α/2)u≥0 for k=0,1,⋯,m−1.In this paper,we show another kind of super polyharmonic property(−Δ)^(k)u>0 for k=1,⋯,m−1,under the conditions and(−Δ)^(m)u≥0.Both kinds of super polyharmonic properties can lead to an equivalence between(0.1)and the integral equation u(x)=∫_(R^(n))u^(p)(y)/|x-y|^(n-2m-α)dy.One can classify solutions to(0.1)following the work of[2]and[3]by Chen,Li,Ou.
基金Supported by the National Natural Science Foundation of China(11171260)the Scientific Research Foundation for the Returned Overseas Chinese Scholars,State Education Ministrythe Research Fund for Revitalization Project of Zhongnan University of Economics and Law
文摘In this article, a polyharmonic Neumann function in a sector with angle π n （n N） is studied by convolution. Especially, the outward normal derivatives at three corner points are defined properly. We give the recursive expressions for the polyharmonic Neumann function, obtaining the solution and the condition of solvability for the related polyharmonic Neumann problem.
文摘This paper is devoted to the following high order elliptic problems under the Navier boundary condition: ?Without assuming the standard subcritical polynomial growth condition ensuring the compactness of a bounded (P.S.) sequence, we show that the Navier boundary value problem has at least a weak nontrivial solution for all λ>0?by using mountain pass theorem.
文摘In this article, a class of Dirichlet problem with Lp boundary data for poly-harmonic function in the upper half plane is mainly investigated. By introducing a sequence of kernel functions called higher order Poisson kernels and a hierarchy of integral operators called higher order Pompeiu operators, we obtain a main result on integral representation solution as well as the uniqueness of the polyharmonic Dirichlet problem under a certain estimate.
基金supported by the Innovation Foundation of Shanghai University (Grant No. A10-0101-08-905)Shang hai Leading Academic Discipline Project (Grant No. J50101)+3 种基金Key Disciplines of Shanghai Municipality (GrantNo. S30104)Zhou was supported by the National Basic Research Program of China (Grant No. 2006CB705700)National Natural Science Foundation of China (Grant No. 60532080)the Key Project of Chinese Ministryof Education (Grant No. 306017)
文摘In this paper we obtain local Lp estimates for the parabolic polyharmonic equations by a straightforward approach.
基金supported by the National Natural Science Foundation of China(Nos.11401307,11501292)
文摘This paper deals with the function u which satisfies△k_u = 0, where k≥2 is an integer. Such a function u is called a polyharmonic function. The author gives an upper bound of the measure of the nodal set of u, and shows some growth property of u.
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We study positive solutions of the following polyharmonic equation with Hardy weights associated to Navier boundary conditions on a half space：where rn is any positive integer satisfying 0 〈 2m 〈 n. We first prove that the positive solutions of （0.1） are super polyharmonic, i.e.,where x＊ = （x1,... ,Xn-1, --Xn） is the reflection of the point x about the plane Rn-1. Then, we use the method of moving planes in integral forms to derive rotational symmetry and monotonicity for the positive solution of （0.3）, in which α can be any real number between 0 and n. By some Pohozaev type identities in integral forms, we prove a Liouville type theorem--the non-existence of positive solutions for （0.1）.
基金supported by the National Natural Science Foundation of China(No.11471214)the One Thousand Plan of China for young scientists
文摘This paper introduces a domain decomposition preconditioner for elliptic equations with rough coefficients. The coarse space of the domain decomposition method is constructed via the so-called rough polyharmonic splines （RPS for short）. As an approximation space of the eUiptic problem, RPS is known to recover the quasi-optimal convergence rate and attain the quasi-optimal localization property. The authors lay out the formulation of the RPS based domain decomposition preconditioner, and numerically verify the performance boost of this method through several examples.
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The remainders and the convergence of cardinal polyharmonic spline interpolation are studied, and the asymptotic behavior of the best approximation by polyharmonic spline and the average K-width of some class of smooth functions defined on the Euclidean space Rn are determined.
基金supported by Guangdong Natural Science Foundation(2018A030313508)。
文摘The aim of this article is twofold.One aim is to establish the precise forms of Landau-Bloch type theorems for certain polyharmonic mappings in the unit disk by applying a geometric method.The other is to obtain the precise values of Bloch constants for certain log-p-harmonic mappings.These results improve upon the corresponding results given in Bai et al.(Complex Anal.Oper.Theory,13(2):321-340,2019).
文摘Let be a hypercube in Rn. We prove theorems concerning mean-values of harmonic and polyharmonic functions on In(r), which can be considered as natural analogues of the famous Gauss surface and volume mean-value formulas for harmonic functions on the ball in and their extensions for polyharmonic functions. We also discuss an application of these formulas—the problem of best canonical one-sided L1-approximation by harmonic functions on In(r).
文摘In this paper we present a numerical method for solving Riemann type problem for the fifth order improperly elliptic equation in complex plane .We reduce this problem to the boundary value problems for properly elliptic equations, and then solve those problems by the grid methods.
