In this paper, we are concerned with a positive solution of the non-homogeneous A-Laplacian equation in an open bounded connected domain. We use moving planes method to prove that the domain is a ball and the solution...In this paper, we are concerned with a positive solution of the non-homogeneous A-Laplacian equation in an open bounded connected domain. We use moving planes method to prove that the domain is a ball and the solution is radially symmetric.展开更多
Extracting approximate symmetry planes is a challenge due to the difficulty of accurately measuring numerical values. Introducing the approximate symmetry planes of a 3D point set, this paper presents a new method by ...Extracting approximate symmetry planes is a challenge due to the difficulty of accurately measuring numerical values. Introducing the approximate symmetry planes of a 3D point set, this paper presents a new method by gathering normal vectors of potential of the planes, clustering the high probability ones, and then testing and verifying the planes. An experiment showed that the method is effective, robust and universal for extracting the complete approximate planes of symmetry of a random 3D point set.展开更多
Based on the covariant anomaly cancellation method, which is believed to be more refined than the initial approach of Robinson and Wilczek, we discuss Hawking radiation from the plane symmetric black hole. The result ...Based on the covariant anomaly cancellation method, which is believed to be more refined than the initial approach of Robinson and Wilczek, we discuss Hawking radiation from the plane symmetric black hole. The result shows that Hawking radiation from the non-spherical symmetric black holes also can be derived from the viewpoint of anomaly.展开更多
Non-static plane symmetric cosmological solutions are presented in the presence of cosmic strings in the scalar-tensor theory of gravitation formulated by Sen ＆ Dunn. It is shown that string cosmological models repre...Non-static plane symmetric cosmological solutions are presented in the presence of cosmic strings in the scalar-tensor theory of gravitation formulated by Sen ＆ Dunn. It is shown that string cosmological models representing geometric strings （ρ = λ） and massive strings （ρ ＋λ = 0） do exist in this theory. Further, it is found that the Takabayasi string, i.e. ρ= （1 ＋ζ）λ, does not exist. Some physical and geometrical features of these models are discussed.展开更多
This paper deals with the radial symmetry of positive solutions to the nonlocal problem(-Δ)_(γ)~su=b(x)f(u)in B_(1){0},u=h in R~N B_(1),where b:B_1→R is locally Holder continuous,radially symmetric and decreasing i...This paper deals with the radial symmetry of positive solutions to the nonlocal problem(-Δ)_(γ)~su=b(x)f(u)in B_(1){0},u=h in R~N B_(1),where b:B_1→R is locally Holder continuous,radially symmetric and decreasing in the|x|direction,F:R→R is a Lipschitz function,h:B_1→R is radially symmetric,decreasing with respect to|x|in R^(N)/B_(1),B_(1) is the unit ball centered at the origin,and(-Δ)_γ~s is the weighted fractional Laplacian with s∈(0,1),γ∈[0,2s)defined by(-△)^(s)_(γ)u(x)=CN,slimδ→0+∫R^(N)/B_(δ)(x)u(x)-u(y)/|x-y|N+2s|y|^(r)dy.We consider the radial symmetry of isolated singular positive solutions to the nonlocal problem in whole space(-Δ)_(γ)^(s)u(x)=b(x)f(u)in R^(N)\{0},under suitable additional assumptions on b and f.Our symmetry results are derived by the method of moving planes,where the main difficulty comes from the weighted fractional Laplacian.Our results could be applied to get a sharp asymptotic for semilinear problems with the fractional Hardy operators(-Δ)^(s)u+μ/(|x|^(2s))u=b(x)f(u)in B_(1)\{0},u=h in R^(N)\B_(1),under suitable additional assumptions on b,f and h.展开更多
In this paper, by using the method of moving planes, we are concerned with the symmetry and monotonicity of positive solutions for the fractional Hartree equation.
The first fundamental problems in the infinite plane with cracks and boundary values cyclically symmetric are considered. They are reduced to singular integral equations on a single crack, which would considerably sim...The first fundamental problems in the infinite plane with cracks and boundary values cyclically symmetric are considered. They are reduced to singular integral equations on a single crack, which would considerably simplify the process of method of solution for such problems. Some special cases are illustrated.展开更多
In this article, we consider the fractional Laplacian equation {（-△）α/2u=k（x）f（u）,x∈Rn＋, u=0, x Rn＋, where 0 〈α 〈 2,En＋：= {x = （x1,x2,… ,xn）｜xn〉 0}. When K is strictly decreasing with respect to ...In this article, we consider the fractional Laplacian equation {（-△）α/2u=k（x）f（u）,x∈Rn＋, u=0, x Rn＋, where 0 〈α 〈 2,En＋：= {x = （x1,x2,… ,xn）｜xn〉 0}. When K is strictly decreasing with respect to ｜x＇｜, the symmetry of positive solutions is proved, where x＇ = （x1, x2,…, xn-1） ∈Rn- 1. When K is strictly increasing with respect to xn or only depend on xn, the nonexistence of positive solutions is obtained.展开更多
A high density of {110} shear planes has been observed in the R-phase (Al5CuLi3)following phase transformation from the AlCuLi icosahedral Phase. The Present paper shows that the atomic arrangement in the vicinity of ...A high density of {110} shear planes has been observed in the R-phase (Al5CuLi3)following phase transformation from the AlCuLi icosahedral Phase. The Present paper shows that the atomic arrangement in the vicinity of the shear planes can be described as a two-dimensional periodic array of atom clusters with 5-fold symmetry. This result is obtained by projecting along the [1. 618...01] direction of the atomic positions from four adjacent lattice planes that cross the shear plane. The projection reveals that the shear plane consists of pentagonal arrangements of double triacontahedra,each pentagon incorporating 606 atoms. The calculated diffraction pattern from the pentagon has approximate 10-fold symmetry characteristic of a quasicrystal.展开更多
It is shown that the Pinney equation, Ermakov systems, and their higher-order generalizations describeself-similar solutions of plane curve motions in centro-affine and affine geometries.
In this paper, we use the moving planes method to prove that the domain Ω and the solution u are Steiner symmetric if u is a positive solution of the overdetermined boundary value problem in Ω.
Group-invariant solutions to certain plane curve motions in Euclidean and centro-affine geometries areobtained. The behavior of some solutions is also presented.
In this paper,we first establish narrow region principle and decay at infinity theorems to extend the direct method of moving planes for general fractional p-Laplacian systems.By virtue of this method,we investigate t...In this paper,we first establish narrow region principle and decay at infinity theorems to extend the direct method of moving planes for general fractional p-Laplacian systems.By virtue of this method,we investigate the qualitative properties of positive solutions for the following Schrodinger system with fractional p-Laplacian{(-△)^(s)_(p)u+au^(p-1)=f(u,v),(-△)^(t)_(p)v+bv(p-1)=g(u,v),where 0<s,t<1 and 2<p<∞.We obtain the radial symmetry in the unit ball or the whole space R^(N)(N≥2),the monotonicity in the parabolic domain and the nonexistence on the half space for positive solutions to the above system under some suitable conditions on f and g,respectively.展开更多
The paper generalizes the direct method of moving planes to the Logarithmic Laplacian system.Firstly,some key ingredients of the method are discussed,for example,Narrow region principle and Decay at infinity.Then,the ...The paper generalizes the direct method of moving planes to the Logarithmic Laplacian system.Firstly,some key ingredients of the method are discussed,for example,Narrow region principle and Decay at infinity.Then,the radial symmetry of the solution of the Logarithmic Laplacian system is obtained.展开更多
文摘In this paper, we are concerned with a positive solution of the non-homogeneous A-Laplacian equation in an open bounded connected domain. We use moving planes method to prove that the domain is a ball and the solution is radially symmetric.
文摘Extracting approximate symmetry planes is a challenge due to the difficulty of accurately measuring numerical values. Introducing the approximate symmetry planes of a 3D point set, this paper presents a new method by gathering normal vectors of potential of the planes, clustering the high probability ones, and then testing and verifying the planes. An experiment showed that the method is effective, robust and universal for extracting the complete approximate planes of symmetry of a random 3D point set.
基金supported by National Natural Science Foundation of China under Grant No. 10773008
文摘Based on the covariant anomaly cancellation method, which is believed to be more refined than the initial approach of Robinson and Wilczek, we discuss Hawking radiation from the plane symmetric black hole. The result shows that Hawking radiation from the non-spherical symmetric black holes also can be derived from the viewpoint of anomaly.
文摘Non-static plane symmetric cosmological solutions are presented in the presence of cosmic strings in the scalar-tensor theory of gravitation formulated by Sen ＆ Dunn. It is shown that string cosmological models representing geometric strings （ρ = λ） and massive strings （ρ ＋λ = 0） do exist in this theory. Further, it is found that the Takabayasi string, i.e. ρ= （1 ＋ζ）λ, does not exist. Some physical and geometrical features of these models are discussed.
基金supported by the NSFC(12001252)the Jiangxi Provincial Natural Science Foundation(20232ACB211001)。
文摘This paper deals with the radial symmetry of positive solutions to the nonlocal problem(-Δ)_(γ)~su=b(x)f(u)in B_(1){0},u=h in R~N B_(1),where b:B_1→R is locally Holder continuous,radially symmetric and decreasing in the|x|direction,F:R→R is a Lipschitz function,h:B_1→R is radially symmetric,decreasing with respect to|x|in R^(N)/B_(1),B_(1) is the unit ball centered at the origin,and(-Δ)_γ~s is the weighted fractional Laplacian with s∈(0,1),γ∈[0,2s)defined by(-△)^(s)_(γ)u(x)=CN,slimδ→0+∫R^(N)/B_(δ)(x)u(x)-u(y)/|x-y|N+2s|y|^(r)dy.We consider the radial symmetry of isolated singular positive solutions to the nonlocal problem in whole space(-Δ)_(γ)^(s)u(x)=b(x)f(u)in R^(N)\{0},under suitable additional assumptions on b and f.Our symmetry results are derived by the method of moving planes,where the main difficulty comes from the weighted fractional Laplacian.Our results could be applied to get a sharp asymptotic for semilinear problems with the fractional Hardy operators(-Δ)^(s)u+μ/(|x|^(2s))u=b(x)f(u)in B_(1)\{0},u=h in R^(N)\B_(1),under suitable additional assumptions on b,f and h.
基金supported by NSFC(11761082)Yunnan Province,Young Academic and Technical Leaders Program(2015HB028)
文摘In this paper, by using the method of moving planes, we are concerned with the symmetry and monotonicity of positive solutions for the fractional Hartree equation.
文摘The first fundamental problems in the infinite plane with cracks and boundary values cyclically symmetric are considered. They are reduced to singular integral equations on a single crack, which would considerably simplify the process of method of solution for such problems. Some special cases are illustrated.
基金supported by the Fundamental Research Founds for the Central Universities(3102015ZY069)the Natural Science Basic Research Plan in Shaanxi Province of China(2016M1008)
文摘In this article, we consider the fractional Laplacian equation {（-△）α/2u=k（x）f（u）,x∈Rn＋, u=0, x Rn＋, where 0 〈α 〈 2,En＋：= {x = （x1,x2,… ,xn）｜xn〉 0}. When K is strictly decreasing with respect to ｜x＇｜, the symmetry of positive solutions is proved, where x＇ = （x1, x2,…, xn-1） ∈Rn- 1. When K is strictly increasing with respect to xn or only depend on xn, the nonexistence of positive solutions is obtained.
文摘A high density of {110} shear planes has been observed in the R-phase (Al5CuLi3)following phase transformation from the AlCuLi icosahedral Phase. The Present paper shows that the atomic arrangement in the vicinity of the shear planes can be described as a two-dimensional periodic array of atom clusters with 5-fold symmetry. This result is obtained by projecting along the [1. 618...01] direction of the atomic positions from four adjacent lattice planes that cross the shear plane. The projection reveals that the shear plane consists of pentagonal arrangements of double triacontahedra,each pentagon incorporating 606 atoms. The calculated diffraction pattern from the pentagon has approximate 10-fold symmetry characteristic of a quasicrystal.
文摘It is shown that the Pinney equation, Ermakov systems, and their higher-order generalizations describeself-similar solutions of plane curve motions in centro-affine and affine geometries.
文摘In this paper, we use the moving planes method to prove that the domain Ω and the solution u are Steiner symmetric if u is a positive solution of the overdetermined boundary value problem in Ω.
文摘Group-invariant solutions to certain plane curve motions in Euclidean and centro-affine geometries areobtained. The behavior of some solutions is also presented.
基金Supported by the National Natural Science Foundation of China(12101452,12071229,11771218)。
文摘In this paper,we first establish narrow region principle and decay at infinity theorems to extend the direct method of moving planes for general fractional p-Laplacian systems.By virtue of this method,we investigate the qualitative properties of positive solutions for the following Schrodinger system with fractional p-Laplacian{(-△)^(s)_(p)u+au^(p-1)=f(u,v),(-△)^(t)_(p)v+bv(p-1)=g(u,v),where 0<s,t<1 and 2<p<∞.We obtain the radial symmetry in the unit ball or the whole space R^(N)(N≥2),the monotonicity in the parabolic domain and the nonexistence on the half space for positive solutions to the above system under some suitable conditions on f and g,respectively.
基金Supported by the National Natural Science Foundation of China(11501342,12001344)。
文摘The paper generalizes the direct method of moving planes to the Logarithmic Laplacian system.Firstly,some key ingredients of the method are discussed,for example,Narrow region principle and Decay at infinity.Then,the radial symmetry of the solution of the Logarithmic Laplacian system is obtained.
