In this paper,by choosing some appropriate test functions,we prove the Weyl’s lemma for triharmonic functions based on the new type of mean value formulas.
A direct boundary element method (BEM) has been studied in the paper based on a set of sufficient and necessary boundary integral equations (BIE) for the plane harmonic functions. The new sufficient and necessary BEM ...A direct boundary element method (BEM) has been studied in the paper based on a set of sufficient and necessary boundary integral equations (BIE) for the plane harmonic functions. The new sufficient and necessary BEM leads to accurate results while the conventional insufficient BEM will lead to inaccurate results when the conventional BIE has multiple solutions. Theoretical and numerical analyses show that it is beneficial to use the sufficient and necessary BEM, to avoid hidden dangers due to non-unique solution of the conventional BIE.展开更多
In this paper, applying the theory of complex-functional, not only the spaceharmonic functions in polynomial form. but aIso the spherical functions are obtained.
In this paper we establish the oscillation inequality of harmonic functions and HOlder estimate of the functions in the domain of the Laplacian on connected post critically finite （p.c.f.） self-similar sets.
The main objective is to derive a lower bound from an upper one for harmonic functions in the half space, which extends a result of B. Y. Levin from dimension 2 to dimension n 〉 2. To this end, we first generalize th...The main objective is to derive a lower bound from an upper one for harmonic functions in the half space, which extends a result of B. Y. Levin from dimension 2 to dimension n 〉 2. To this end, we first generalize the Carleman＇s formula for harmonic functions in the half plane to higher dimensional half space, and then establish a Nevanlinna＇s representation for harmonic functions in the half sphere by using HSrmander＇s theorem.展开更多
In order to study the temporal and spatial variation characteristics of the regional ionosphere and the modeling accuracy,the experiment is based on the spherical harmonic function model,using the GPS,Glonass,and Gali...In order to study the temporal and spatial variation characteristics of the regional ionosphere and the modeling accuracy,the experiment is based on the spherical harmonic function model,using the GPS,Glonass,and Galileo dual-frequency observation data from the 305th-334th day of the European CORS network in 2019 to establish a global ionospheric model.By analyzing and evaluating the accuracy of the global ionospheric puncture points,VTEC,and comparing code products,the test results showed that the GPS system has the most dense puncture electricity distribution,the Glonass system is the second,and the Galileo system is the weakest.The values of ionospheric VTEC calculated by GPS,Glonass and Galileo are slightly different,but in terms of trends,they are the same as those of ESA,JPL and UPC.GPS data has the highest accuracy in global ionospheric modeling.GPS,Glonass and Galileo have the same trend,but Glonass data is unstable and fluctuates greatly.展开更多
In this paper we establish sharp H/51der estimates of harmonic functions on a class of connected post critically finite （p.c.f.） self-similar sets, and show that functions in the domain of Laplacian enjoy the same p...In this paper we establish sharp H/51der estimates of harmonic functions on a class of connected post critically finite （p.c.f.） self-similar sets, and show that functions in the domain of Laplacian enjoy the same property. Some weU-known examples, such as the Sierpinski gasket, the unit interval, the level 3 Sierpinski gasket, the hexagasket, the 3-dimensional Sierpinski gasket, and the Vicsek set are also considered.展开更多
A complex-valued harmonic functions that are univalent and sense preserving in the unit disk U can be written in the form f = h ＋ g^-, where h and g are analytic in U. We define and investigate a new class SHPλ（α...A complex-valued harmonic functions that are univalent and sense preserving in the unit disk U can be written in the form f = h ＋ g^-, where h and g are analytic in U. We define and investigate a new class SHPλ（α,β）by generalized Salagean operator of harmonic univalent functions. We give sufficient coefficient conditions for normalized harmonic functions in the class SHPλ（α,β） These conditions are also shown to be necessary when the coefficients are negative. This leads to distortion bounds and extreme points.展开更多
The authors prove the space of harmonic functions with polynomial growth of a fixed rate on a complete noncompact Riemannian manifold with asymptotically nonnegative curvature is finite dimensional.
This paper shows an important relation between the fractal analysis and the boundary properties of harmonic functions.It is proved that the multifractal analysis of a finite measureμonIR^(d)determines the(non-tangent...This paper shows an important relation between the fractal analysis and the boundary properties of harmonic functions.It is proved that the multifractal analysis of a finite measureμonIR^(d)determines the(non-tangential)boundary increasing properties ofPμ,the Poisson integral ofμwhich is harmonic onIR^(d+1)_(+).Some examples are given.展开更多
The unique continuation on quadratic curves for harmonic functions is dis-cussed in this paper.By using complex extension method,the conditional stability of unique continuation along quadratic curves for harmonic fun...The unique continuation on quadratic curves for harmonic functions is dis-cussed in this paper.By using complex extension method,the conditional stability of unique continuation along quadratic curves for harmonic functions is illustrated.The nu-merical algorithm is provided based on collocation method and Tikhonov regularization.The stability estimates on parabolic and hyperbolic curves for harmonic functions are demonstrated by numerical examples respectively.展开更多
A complex-valued harmonic function that is univalent and sense preserving in the unit disk U can be written in the form of f = h + g,where h and g are analytic in U.We define and investigate a new class LH_λ(α,β) b...A complex-valued harmonic function that is univalent and sense preserving in the unit disk U can be written in the form of f = h + g,where h and g are analytic in U.We define and investigate a new class LH_λ(α,β) by generalized Salagean operator of harmonic univalent functions.We give sufficient coefficient conditions for normalized harmonic functions in the class LH_λ(α,β).These conditions are also shown to be necessary when the coefficients are negative.This leads to distortion bounds and extreme points.展开更多
In this paper, we find two auxiliary functions and make use of the maximum principle to study the level sets of harmonic function defined on a convex ring with homogeneous Dirichlet boundary conditions in R2. In highe...In this paper, we find two auxiliary functions and make use of the maximum principle to study the level sets of harmonic function defined on a convex ring with homogeneous Dirichlet boundary conditions in R2. In higher dimensions, we also have a similar result to Jagy＇s.展开更多
In this paper, we first introduce the concept ＂harmonically convex functions＂ in the second sense and establish several Hermite-Hadamard type inequalities for harmonically convex functions in the second sense. Final...In this paper, we first introduce the concept ＂harmonically convex functions＂ in the second sense and establish several Hermite-Hadamard type inequalities for harmonically convex functions in the second sense. Finally, some applications to special mean are shown.展开更多
Let(X,d,μ)be a metric measure space with non-negative Ricci curvature.This paper is concerned with the boundary behavior of harmonic function on the(open)upper half-space X×R_(+).We derive that a function f of b...Let(X,d,μ)be a metric measure space with non-negative Ricci curvature.This paper is concerned with the boundary behavior of harmonic function on the(open)upper half-space X×R_(+).We derive that a function f of bounded mean oscillation(BMO)is the trace of harmonic function u(x,t)on X×R_(+),u(x,0)=f(x),whenever u satisfies the following Carleson measure condition supx_(B),r_(B) ∫_(0)^(r_(B) ) f_(B(x_(B),r_(B))) |t■u(x,t)|^(2)dμ(x)dt/t≤C<∞,where ■=(■_(x),■_(t))denotes the total gradient and B(x_(B),r_(B)) denotes the(open)ball centered at x_(B) with radius r_(B).Conversely,the above condition characterizes all the harmonic functions whose traces are in BMO space.展开更多
We consider the local boundary values of generalized harmonic functions associated with the rank-one Dunkl operator D in the upper half-plane R2+=R×(0,∞),where(Df)(x)=f′0(x)+(λ/x)[f(x)-f(-x)]for givenλ≥0.A C...We consider the local boundary values of generalized harmonic functions associated with the rank-one Dunkl operator D in the upper half-plane R2+=R×(0,∞),where(Df)(x)=f′0(x)+(λ/x)[f(x)-f(-x)]for givenλ≥0.A C2 function u in R2+is said to beλ-harmonic if(D2x+■2y)u=0.For aλ-harmonic function u in R2+and for a subset E of■R2+=R symmetric about y-axis,we prove that the following three assertions are equivalent:(i)u has a finite non-tangential limit at(x,0)for a.e.x∈E;(ii)u is non-tangentially bounded for a.e.x∈E;(iii)(Su)(x)<∞for a.e.x∈E,where S is a Lusin-type area integral associated with the Dunkl operator D.展开更多
The object of this paper is to study the non-tangential increasing properties of positiveharmonic function u in Lipschitz domain by means of Martin representation theory. A necessaryand sufficient condition of the con...The object of this paper is to study the non-tangential increasing properties of positiveharmonic function u in Lipschitz domain by means of Martin representation theory. A necessaryand sufficient condition of the control of growth of u near any fixed boundary point is obtained.It is shown that the non-tangential increasing degree of u near a boundary point is exactlythe local degree of its representation measure with respect to the harmonic measure. Someexamples are given.展开更多
基金Supported by National Natural Science Foundation of China(Grant Nos.11801006 and 12071489).
文摘In this paper,by choosing some appropriate test functions,we prove the Weyl’s lemma for triharmonic functions based on the new type of mean value formulas.
文摘A direct boundary element method (BEM) has been studied in the paper based on a set of sufficient and necessary boundary integral equations (BIE) for the plane harmonic functions. The new sufficient and necessary BEM leads to accurate results while the conventional insufficient BEM will lead to inaccurate results when the conventional BIE has multiple solutions. Theoretical and numerical analyses show that it is beneficial to use the sufficient and necessary BEM, to avoid hidden dangers due to non-unique solution of the conventional BIE.
文摘In this paper, applying the theory of complex-functional, not only the spaceharmonic functions in polynomial form. but aIso the spherical functions are obtained.
基金supported by the National Natural Science Foundation of China(No.11201232)Qing Lan Project of Jiangsu Province
文摘In this paper we establish the oscillation inequality of harmonic functions and HOlder estimate of the functions in the domain of the Laplacian on connected post critically finite （p.c.f.） self-similar sets.
基金Project supported by the Academic Human Resources Development in Institutions of Higher Learning under the Jurisdiction of Beijing Municipality (IHLB201008257)Scientific Research Common Program of Beijing Municipal Commission of Education (KM200810011005)+1 种基金PHR (IHLB 201102)research grant of University of Macao MYRG142(Y1-L2)-FST111-KKI
文摘The main objective is to derive a lower bound from an upper one for harmonic functions in the half space, which extends a result of B. Y. Levin from dimension 2 to dimension n 〉 2. To this end, we first generalize the Carleman＇s formula for harmonic functions in the half plane to higher dimensional half space, and then establish a Nevanlinna＇s representation for harmonic functions in the half sphere by using HSrmander＇s theorem.
基金Key Research and Development Program of Liaoning Province(2020JH2/10100044)National Natural Science Foundation of China(41904037)National Key Basic Research and Development Program(973 Program)(2016YFC0803102)。
文摘In order to study the temporal and spatial variation characteristics of the regional ionosphere and the modeling accuracy,the experiment is based on the spherical harmonic function model,using the GPS,Glonass,and Galileo dual-frequency observation data from the 305th-334th day of the European CORS network in 2019 to establish a global ionospheric model.By analyzing and evaluating the accuracy of the global ionospheric puncture points,VTEC,and comparing code products,the test results showed that the GPS system has the most dense puncture electricity distribution,the Glonass system is the second,and the Galileo system is the weakest.The values of ionospheric VTEC calculated by GPS,Glonass and Galileo are slightly different,but in terms of trends,they are the same as those of ESA,JPL and UPC.GPS data has the highest accuracy in global ionospheric modeling.GPS,Glonass and Galileo have the same trend,but Glonass data is unstable and fluctuates greatly.
基金supported by the grants NSFC11201232, 12KJB110008Qing Lan Project, 13KJB110015, 12YJAZH096the Project-sponsored by SRF for ROCS, SEM
文摘In this paper we establish sharp H/51der estimates of harmonic functions on a class of connected post critically finite （p.c.f.） self-similar sets, and show that functions in the domain of Laplacian enjoy the same property. Some weU-known examples, such as the Sierpinski gasket, the unit interval, the level 3 Sierpinski gasket, the hexagasket, the 3-dimensional Sierpinski gasket, and the Vicsek set are also considered.
基金supported by the National Natural Science Foundation of China(No.11201199)the Scientific Research Foundation of Jinling Institute of Technology(No.Jit-b-201221)Qing Lan Project
文摘In this paper, the authors prove a Schwarz-Pick lemma for bounded complexvalued harmonic functions in the unit ball of Rn.
基金Supported by the Key Scientific Research Fund of Inner Mongolian Educational Bureau (NJ04115)
文摘A complex-valued harmonic functions that are univalent and sense preserving in the unit disk U can be written in the form f = h ＋ g^-, where h and g are analytic in U. We define and investigate a new class SHPλ（α,β）by generalized Salagean operator of harmonic univalent functions. We give sufficient coefficient conditions for normalized harmonic functions in the class SHPλ（α,β） These conditions are also shown to be necessary when the coefficients are negative. This leads to distortion bounds and extreme points.
基金Project supported by the National Natural Science Foundation of China (No.10271089).
文摘The authors prove the space of harmonic functions with polynomial growth of a fixed rate on a complete noncompact Riemannian manifold with asymptotically nonnegative curvature is finite dimensional.
文摘This paper shows an important relation between the fractal analysis and the boundary properties of harmonic functions.It is proved that the multifractal analysis of a finite measureμonIR^(d)determines the(non-tangential)boundary increasing properties ofPμ,the Poisson integral ofμwhich is harmonic onIR^(d+1)_(+).Some examples are given.
基金This work was supported by the National Natural Science Foundation of China(No.11971121).
文摘The unique continuation on quadratic curves for harmonic functions is dis-cussed in this paper.By using complex extension method,the conditional stability of unique continuation along quadratic curves for harmonic functions is illustrated.The nu-merical algorithm is provided based on collocation method and Tikhonov regularization.The stability estimates on parabolic and hyperbolic curves for harmonic functions are demonstrated by numerical examples respectively.
基金Foundation item： Supported by the Natural Science Foundation of Inner Mongolia（2009MS0113） Supported by the Higher School Research Foundation of Inner Mongolia（NJzy08150）
文摘A complex-valued harmonic function that is univalent and sense preserving in the unit disk U can be written in the form of f = h + g,where h and g are analytic in U.We define and investigate a new class LH_λ(α,β) by generalized Salagean operator of harmonic univalent functions.We give sufficient coefficient conditions for normalized harmonic functions in the class LH_λ(α,β).These conditions are also shown to be necessary when the coefficients are negative.This leads to distortion bounds and extreme points.
文摘In this paper, we find two auxiliary functions and make use of the maximum principle to study the level sets of harmonic function defined on a convex ring with homogeneous Dirichlet boundary conditions in R2. In higher dimensions, we also have a similar result to Jagy＇s.
基金The Doctoral Programs Foundation(20113401110009)of Education Ministry of ChinaNatural Science Research Project(2012kj11)of Hefei Normal University+1 种基金Universities Natural Science Foundation(KJ2013A220)of Anhui ProvinceResearch Project of Graduates Innovation Fund(2014yjs02)
文摘In this paper, we first introduce the concept ＂harmonically convex functions＂ in the second sense and establish several Hermite-Hadamard type inequalities for harmonically convex functions in the second sense. Finally, some applications to special mean are shown.
文摘Let(X,d,μ)be a metric measure space with non-negative Ricci curvature.This paper is concerned with the boundary behavior of harmonic function on the(open)upper half-space X×R_(+).We derive that a function f of bounded mean oscillation(BMO)is the trace of harmonic function u(x,t)on X×R_(+),u(x,0)=f(x),whenever u satisfies the following Carleson measure condition supx_(B),r_(B) ∫_(0)^(r_(B) ) f_(B(x_(B),r_(B))) |t■u(x,t)|^(2)dμ(x)dt/t≤C<∞,where ■=(■_(x),■_(t))denotes the total gradient and B(x_(B),r_(B)) denotes the(open)ball centered at x_(B) with radius r_(B).Conversely,the above condition characterizes all the harmonic functions whose traces are in BMO space.
基金the National Natural Science Foundation of China(No.11371258).
文摘We consider the local boundary values of generalized harmonic functions associated with the rank-one Dunkl operator D in the upper half-plane R2+=R×(0,∞),where(Df)(x)=f′0(x)+(λ/x)[f(x)-f(-x)]for givenλ≥0.A C2 function u in R2+is said to beλ-harmonic if(D2x+■2y)u=0.For aλ-harmonic function u in R2+and for a subset E of■R2+=R symmetric about y-axis,we prove that the following three assertions are equivalent:(i)u has a finite non-tangential limit at(x,0)for a.e.x∈E;(ii)u is non-tangentially bounded for a.e.x∈E;(iii)(Su)(x)<∞for a.e.x∈E,where S is a Lusin-type area integral associated with the Dunkl operator D.
基金Project supported by the National Natural Science Foundation of China
文摘The object of this paper is to study the non-tangential increasing properties of positiveharmonic function u in Lipschitz domain by means of Martin representation theory. A necessaryand sufficient condition of the control of growth of u near any fixed boundary point is obtained.It is shown that the non-tangential increasing degree of u near a boundary point is exactlythe local degree of its representation measure with respect to the harmonic measure. Someexamples are given.
