In this paper,a generalized Laguerre-spherical harmonic spectral method is proposed for the Cauchy problem of three-dimensional nonlinear Klein-Gordon equation. The goal is to make the numerical solutions to preserve ...In this paper,a generalized Laguerre-spherical harmonic spectral method is proposed for the Cauchy problem of three-dimensional nonlinear Klein-Gordon equation. The goal is to make the numerical solutions to preserve the same conservation as that for the exact solution.The stability and convergence of the proposed scheme are proved.Numerical results demonstrate the efficiency of this approach.We also establish some basic results on the generalized Laguerre-spherical harmonic orthogonal approximation,which play an important role in spectral methods for various problems defined on the whole space and unbounded domains with spherical geometry.展开更多
Zhao （2003a） first established a congruence for any odd prime p〉3, S（1,1,1 ;p）=-2Bp-3 （mod p）, which holds when p=3 evidently. In this paper, we consider finite triple harmonic sum S（α,β, γ,ρ） （modp） is...Zhao （2003a） first established a congruence for any odd prime p〉3, S（1,1,1 ;p）=-2Bp-3 （mod p）, which holds when p=3 evidently. In this paper, we consider finite triple harmonic sum S（α,β, γ,ρ） （modp） is considered for all positive integers α,β, γ. We refer to w=α＋β＋ γ as the weight of the sum, and show that if w is even, S（α,β, γ,ρ）=0 （mod p） for p≥w＋3; if w is odd, S（α,β, γ,ρ）=-rBp-w （mod p） for p≥w, here r is an explicit rational number independent ofp. A congruence of Catalan number is obtained as a special case.展开更多
Abstract： Let Ω belong to R^N be a smooth bounded domain such that 0 ∈ Ω, N ≥ 5, 2^＊ ：2N/N-4 is the critical Sobolev exponent, and f（x） is a given function. By using the variational methods, the paper proves ...Abstract： Let Ω belong to R^N be a smooth bounded domain such that 0 ∈ Ω, N ≥ 5, 2^＊ ：2N/N-4 is the critical Sobolev exponent, and f（x） is a given function. By using the variational methods, the paper proves the existence of solutions for the singular critical in the homogeneous problem △^u-μ u/{x}^4=｜μ｜^2＊-2u＋f（x） with Dirichlet boundary condition on 偏dΩ under some assumptions on f（x） and μ.展开更多
In order to consider the thermal and electrical coherent transport in a mesoscopic conductor under the influence of electron-electron interaction, in this paper, we establish a method in terms of which one can analyti...In order to consider the thermal and electrical coherent transport in a mesoscopic conductor under the influence of electron-electron interaction, in this paper, we establish a method in terms of which one can analytically obtain the Hartree self-consistent potential instead of computing it by the numerical iterative procedure as usual, which is convenient for us to describe the thermal and electric current flow through a mesoscopic conductor. If we study the electron-electron interaction at the Hartree approximation level, the Hartree potential satisfies the Poisson equation and Schroedinger equation, so when we expand the action function S（x） by Planck constant h, the self-consistent potential and the wavefunction can be solved analytically order by order, and the thermal and electrical conductance can thus be obtained readily. However, we just show the quantum corrections up to the second order.展开更多
This paper is concerned with a dissipative nonlinear Schroedinger equation with a harmonic potential. By using some intricate inequalities and the argument of priori estimates, some behaviors of the solution are inves...This paper is concerned with a dissipative nonlinear Schroedinger equation with a harmonic potential. By using some intricate inequalities and the argument of priori estimates, some behaviors of the solution are investigated.展开更多
Roughly speaking a regular Dirichlet subspace of a Dirichlet form is a subspace which is also a regular Dirichlet form on the same state space. In particular, the domain of regular Dirichlet subspace is a closed subsp...Roughly speaking a regular Dirichlet subspace of a Dirichlet form is a subspace which is also a regular Dirichlet form on the same state space. In particular, the domain of regular Dirichlet subspace is a closed subspace of the Hilbert space induced by the domain and a-inner product of original Dirichlet form. We investigate the orthogonal complement of regular Dirichlet subspace for one-dimensional Brownian motion in this paper. Our main results indicate that this orthogonal complement has a very close connection with the a-harmonic equation under Neumann type condition.展开更多
Using an improved homogeneous balance principle and an F-expansion technique, we construct the new exact periodic traveling wave solutions to the(3+1)-dimensional Gross–Pitaevskii equation with repulsive harmonic pot...Using an improved homogeneous balance principle and an F-expansion technique, we construct the new exact periodic traveling wave solutions to the(3+1)-dimensional Gross–Pitaevskii equation with repulsive harmonic potential. In the limit cases, the solitary wave solutions are obtained as well. We also investigate the dynamical evolution of the solitons with a time-dependent complicated potential.展开更多
A monotonicity formula for stable solutions to a class of weighted semilinear elliptic equations with ＂negative exponent＂ is established. It is well known that such a monotonieity formula plays an essential role in ...A monotonicity formula for stable solutions to a class of weighted semilinear elliptic equations with ＂negative exponent＂ is established. It is well known that such a monotonieity formula plays an essential role in the study of finite Morse index solutions of equations with ＂positive exponent＂. Unlike the positive exponent case, we will see that both the monotonicity formula and the sub-harmonicity play crucial roles in classifying positive finite Morse index solutions to the equations with negative exponent and obtaining sharp results for their asymptotic behaviors.展开更多
Let B R^n be the unit ball centered at the origin. The authors consider the following biharmonic equation:{?~2u = λ(1 + u)~p in B,u =?u/?ν= 0 on ?B, where p >n+4/ n-4and ν is the outward unit normal vector. It ...Let B R^n be the unit ball centered at the origin. The authors consider the following biharmonic equation:{?~2u = λ(1 + u)~p in B,u =?u/?ν= 0 on ?B, where p >n+4/ n-4and ν is the outward unit normal vector. It is well-known that there exists a λ*> 0 such that the biharmonic equation has a solution for λ∈ (0, λ*) and has a unique weak solution u*with parameter λ = λ*, called the extremal solution. It is proved that u* is singular when n ≥ 13 for p large enough and satisfies u*≤ r^(-4/ (p-1)) - 1 on the unit ball, which actually solve a part of the open problem left in [D`avila, J., Flores, I., Guerra, I., Multiplicity of solutions for a fourth order equation with power-type nonlinearity, Math. Ann., 348(1), 2009, 143–193] .展开更多
The author studies minimal surfaces in 3-dimensional solvable Lie groups with left invariantRiemannian metrics. A Weierstraβ type integral representation formula for minimal surfaces isobtained.
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 is concerned with the boundedness of solutions for second order differential equations x + f(x, t)x + g(x, t) = 0, which are neither dissipative nor conservative, and where the functions f and g are odd in ...This paper is concerned with the boundedness of solutions for second order differential equations x + f(x, t)x + g(x, t) = 0, which are neither dissipative nor conservative, and where the functions f and g are odd in x and even in t, which are 1-periodic in t, and the function g satisfies g(x,t/x+, as|x| - +. Using the KAM theory for reversible systems, the author proves the existence of invariant tori and thus the boundedness of all the solutions and the existence of quasiperiodic solutions and subharmonic solutions.展开更多
The partial regularity of the weak heat flow of harmonic maps from a Riemannian manifold Al with boundary into general compact Riemannian manifold N without boundary is consid-ered. It is shown that the singular set S...The partial regularity of the weak heat flow of harmonic maps from a Riemannian manifold Al with boundary into general compact Riemannian manifold N without boundary is consid-ered. It is shown that the singular set Sing(u) of the weak heat flow satisfies H(Sing(u)) 0, with is = dimensionM. Here is Hausdorff measure with respect to parabolic metric ρ(x,t),(y,s)=max{|x-y|, }.展开更多
基金supported in part by NSF of China N.10871131The Science and Technology Commission of Shanghai Municipality,Grant N.075105118+1 种基金Shanghai Leading Academic Discipline Project N.T0401Fund for E-institute of Shanghai Universities N.E03004.
文摘In this paper,a generalized Laguerre-spherical harmonic spectral method is proposed for the Cauchy problem of three-dimensional nonlinear Klein-Gordon equation. The goal is to make the numerical solutions to preserve the same conservation as that for the exact solution.The stability and convergence of the proposed scheme are proved.Numerical results demonstrate the efficiency of this approach.We also establish some basic results on the generalized Laguerre-spherical harmonic orthogonal approximation,which play an important role in spectral methods for various problems defined on the whole space and unbounded domains with spherical geometry.
基金Project (No. 10371107) supported by the National Natural Science Foundation of China
文摘Zhao （2003a） first established a congruence for any odd prime p〉3, S（1,1,1 ;p）=-2Bp-3 （mod p）, which holds when p=3 evidently. In this paper, we consider finite triple harmonic sum S（α,β, γ,ρ） （modp） is considered for all positive integers α,β, γ. We refer to w=α＋β＋ γ as the weight of the sum, and show that if w is even, S（α,β, γ,ρ）=0 （mod p） for p≥w＋3; if w is odd, S（α,β, γ,ρ）=-rBp-w （mod p） for p≥w, here r is an explicit rational number independent ofp. A congruence of Catalan number is obtained as a special case.
文摘Abstract： Let Ω belong to R^N be a smooth bounded domain such that 0 ∈ Ω, N ≥ 5, 2^＊ ：2N/N-4 is the critical Sobolev exponent, and f（x） is a given function. By using the variational methods, the paper proves the existence of solutions for the singular critical in the homogeneous problem △^u-μ u/{x}^4=｜μ｜^2＊-2u＋f（x） with Dirichlet boundary condition on 偏dΩ under some assumptions on f（x） and μ.
文摘In order to consider the thermal and electrical coherent transport in a mesoscopic conductor under the influence of electron-electron interaction, in this paper, we establish a method in terms of which one can analytically obtain the Hartree self-consistent potential instead of computing it by the numerical iterative procedure as usual, which is convenient for us to describe the thermal and electric current flow through a mesoscopic conductor. If we study the electron-electron interaction at the Hartree approximation level, the Hartree potential satisfies the Poisson equation and Schroedinger equation, so when we expand the action function S（x） by Planck constant h, the self-consistent potential and the wavefunction can be solved analytically order by order, and the thermal and electrical conductance can thus be obtained readily. However, we just show the quantum corrections up to the second order.
基金the Scientific Research Fund of Sichuan Provincial Education Department (2006A063).
文摘This paper is concerned with a dissipative nonlinear Schroedinger equation with a harmonic potential. By using some intricate inequalities and the argument of priori estimates, some behaviors of the solution are investigated.
文摘Roughly speaking a regular Dirichlet subspace of a Dirichlet form is a subspace which is also a regular Dirichlet form on the same state space. In particular, the domain of regular Dirichlet subspace is a closed subspace of the Hilbert space induced by the domain and a-inner product of original Dirichlet form. We investigate the orthogonal complement of regular Dirichlet subspace for one-dimensional Brownian motion in this paper. Our main results indicate that this orthogonal complement has a very close connection with the a-harmonic equation under Neumann type condition.
基金Supported by National Natural Science Foundation of China under Grant Nos.11375030 and 61304133
文摘Using an improved homogeneous balance principle and an F-expansion technique, we construct the new exact periodic traveling wave solutions to the(3+1)-dimensional Gross–Pitaevskii equation with repulsive harmonic potential. In the limit cases, the solitary wave solutions are obtained as well. We also investigate the dynamical evolution of the solitons with a time-dependent complicated potential.
基金supported by National Natural Science Foundation of China(Grant Nos.11171092 and 11271133)Innovation Scientists and Technicians Troop Construction Projects of Henan Province(Grant No.114200510011)
文摘A monotonicity formula for stable solutions to a class of weighted semilinear elliptic equations with ＂negative exponent＂ is established. It is well known that such a monotonieity formula plays an essential role in the study of finite Morse index solutions of equations with ＂positive exponent＂. Unlike the positive exponent case, we will see that both the monotonicity formula and the sub-harmonicity play crucial roles in classifying positive finite Morse index solutions to the equations with negative exponent and obtaining sharp results for their asymptotic behaviors.
基金supported by the National Natural Science Foundation of China(Nos.11201119,11471099)the International Cultivation of Henan Advanced Talents and the Research Foundation of Henan University(No.yqpy20140043)
文摘Let B R^n be the unit ball centered at the origin. The authors consider the following biharmonic equation:{?~2u = λ(1 + u)~p in B,u =?u/?ν= 0 on ?B, where p >n+4/ n-4and ν is the outward unit normal vector. It is well-known that there exists a λ*> 0 such that the biharmonic equation has a solution for λ∈ (0, λ*) and has a unique weak solution u*with parameter λ = λ*, called the extremal solution. It is proved that u* is singular when n ≥ 13 for p large enough and satisfies u*≤ r^(-4/ (p-1)) - 1 on the unit ball, which actually solve a part of the open problem left in [D`avila, J., Flores, I., Guerra, I., Multiplicity of solutions for a fourth order equation with power-type nonlinearity, Math. Ann., 348(1), 2009, 143–193] .
基金Partially supported by Grant-in-Aid for Encouragement of Young Scientists (No. 12740051), Japan Society for Promotion of Science.
文摘The author studies minimal surfaces in 3-dimensional solvable Lie groups with left invariantRiemannian metrics. A Weierstraβ type integral representation formula for minimal surfaces isobtained.
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）.
文摘This paper is concerned with the boundedness of solutions for second order differential equations x + f(x, t)x + g(x, t) = 0, which are neither dissipative nor conservative, and where the functions f and g are odd in x and even in t, which are 1-periodic in t, and the function g satisfies g(x,t/x+, as|x| - +. Using the KAM theory for reversible systems, the author proves the existence of invariant tori and thus the boundedness of all the solutions and the existence of quasiperiodic solutions and subharmonic solutions.
基金the National Natural Science Foundation of China (No.10071013).
文摘The partial regularity of the weak heat flow of harmonic maps from a Riemannian manifold Al with boundary into general compact Riemannian manifold N without boundary is consid-ered. It is shown that the singular set Sing(u) of the weak heat flow satisfies H(Sing(u)) 0, with is = dimensionM. Here is Hausdorff measure with respect to parabolic metric ρ(x,t),(y,s)=max{|x-y|, }.
