This paper is based on the relations between projection Riccati equations and Weierstrass elliptic equation, combined with the Groebner bases in the symbolic computation.Then the novel method for constructing the Weie...This paper is based on the relations between projection Riccati equations and Weierstrass elliptic equation, combined with the Groebner bases in the symbolic computation.Then the novel method for constructing the Weierstrass elliptic solutions to the nonlinear evolution equations is given by using the above relations.展开更多
We sharpen and prove a conjecture suggested by Chen and Xie, which states that in Galerkineigenfunction discretization for -Δu = u3 , when the finite-dimensional subspace is taken as the eigensubspace corresponding t...We sharpen and prove a conjecture suggested by Chen and Xie, which states that in Galerkineigenfunction discretization for -Δu = u3 , when the finite-dimensional subspace is taken as the eigensubspace corresponding to an N-fold eigenvalue of -Δ, the discretized problem has at least 3N-1 distinct nonzero solutions. We also present a related result on the multiplicities of eigenvalues of -Δ.展开更多
H61der and gradient estimates for the correctors in the homogenization are presented based on the translation invariance and Li-Vogelius＇s gradient estimate. If the coefficients are piecewise smooth and the homogeniz...H61der and gradient estimates for the correctors in the homogenization are presented based on the translation invariance and Li-Vogelius＇s gradient estimate. If the coefficients are piecewise smooth and the homogenized solution is smooth enough, the interior error of the first-order expansion is O（e） in the HSlder norm; it is O（e） in W1,∞ based on the Avellaneda-Lin＇s gradient estimate when the coefficients are Lipschitz continuous. These estimates can be partly extended to the nonlinear parabolic equations.展开更多
In this paper, the truncated Painleve analysis and the consistent tanh expansion （CTE） method are developed for the （2＋1）-dimensional breaking soliton equation. As a result, the soliton-cnoidal wave interaction s...In this paper, the truncated Painleve analysis and the consistent tanh expansion （CTE） method are developed for the （2＋1）-dimensional breaking soliton equation. As a result, the soliton-cnoidal wave interaction solution of the equation is explicitly given, which is dimcult to be found by other traditional methods. When the value of the Jacobi elliptic function modulus rn = 1, the soliton-cnoidal wave interaction solution reduces back to the two-soliton solution. The method can also be extended to other types of nonlinear evolution equations in mathematical physics.展开更多
In this paper,we mainly discuss a priori bounds of the following degenerate elliptic equation,a^ ij（x）δij u＋b^ i（x）δiu＋f（x,u）=0,in Ω belong to belong toR^n,（＊）where a^ijδiφδjφ=0 on δΩ,andφis the d...In this paper,we mainly discuss a priori bounds of the following degenerate elliptic equation,a^ ij（x）δij u＋b^ i（x）δiu＋f（x,u）=0,in Ω belong to belong toR^n,（＊）where a^ijδiφδjφ=0 on δΩ,andφis the defining function of δΩ.Imposing suitable conditions on the coefficients and f（x,u）,one can get the L^∞-estimates of（＊）via blow up method.展开更多
The authors study homogenization of some nonlinear partial differential equations of the form -div (?(hx, h2x,Duh)) =f, where a is periodic in the first two arguments and monotone in the third. In particular the case ...The authors study homogenization of some nonlinear partial differential equations of the form -div (?(hx, h2x,Duh)) =f, where a is periodic in the first two arguments and monotone in the third. In particular the case where a satisfies degenerated structure conditions is studied. It is proved that uh converges weakly in W01,1 (?) to the unique solution of a limit problem as h ? '. Moreover, explicit expressions for the limit problem are obtained.展开更多
In this paper, the authors prove the existence of solutions for degenerate elliptic equations of the form-div(a(x)▽_p u(x)) = g(λ, x, |u|^(p-2)u) in R^N, where ▽_pu =|▽u|^(p-2)▽u and a(x) is a degenerate nonnegat...In this paper, the authors prove the existence of solutions for degenerate elliptic equations of the form-div(a(x)▽_p u(x)) = g(λ, x, |u|^(p-2)u) in R^N, where ▽_pu =|▽u|^(p-2)▽u and a(x) is a degenerate nonnegative weight. The authors also investigate a related nonlinear eigenvalue problem obtaining an existence result which contains information about the location and multiplicity of eigensolutions. The proofs of the main results are obtained by using the critical point theory in Sobolev weighted spaces combined with a Caffarelli-Kohn-Nirenberg-type inequality and by using a specific minimax method, but without making use of the Palais-Smale condition.展开更多
基金supported by the Open Project of Key Laboratory of Mathematics Mechanization,CAS under Grant No.KLMM0602
文摘This paper is based on the relations between projection Riccati equations and Weierstrass elliptic equation, combined with the Groebner bases in the symbolic computation.Then the novel method for constructing the Weierstrass elliptic solutions to the nonlinear evolution equations is given by using the above relations.
基金supported by National Natural Science Foundation of China (Grant Nos.11171051 and 91230103)
文摘We sharpen and prove a conjecture suggested by Chen and Xie, which states that in Galerkineigenfunction discretization for -Δu = u3 , when the finite-dimensional subspace is taken as the eigensubspace corresponding to an N-fold eigenvalue of -Δ, the discretized problem has at least 3N-1 distinct nonzero solutions. We also present a related result on the multiplicities of eigenvalues of -Δ.
基金supported by National Natural Science Foundation of China (Grant No.90916027)Special Funds for National Basic Research Program of China (973 Program) (Grant No. 2010CB832702)the State Key Laboratory of Scientific and Engineering Computing
文摘H61der and gradient estimates for the correctors in the homogenization are presented based on the translation invariance and Li-Vogelius＇s gradient estimate. If the coefficients are piecewise smooth and the homogenized solution is smooth enough, the interior error of the first-order expansion is O（e） in the HSlder norm; it is O（e） in W1,∞ based on the Avellaneda-Lin＇s gradient estimate when the coefficients are Lipschitz continuous. These estimates can be partly extended to the nonlinear parabolic equations.
基金Supported by National Natural Science Foundation of China under Grant Nos.11271211,11275072,11435005K.C.Wong Magna Fund in Ningbo University
文摘In this paper, the truncated Painleve analysis and the consistent tanh expansion （CTE） method are developed for the （2＋1）-dimensional breaking soliton equation. As a result, the soliton-cnoidal wave interaction solution of the equation is explicitly given, which is dimcult to be found by other traditional methods. When the value of the Jacobi elliptic function modulus rn = 1, the soliton-cnoidal wave interaction solution reduces back to the two-soliton solution. The method can also be extended to other types of nonlinear evolution equations in mathematical physics.
基金supported by National Natural Science Foundation of China(Grant No.11131005)
文摘In this paper,we mainly discuss a priori bounds of the following degenerate elliptic equation,a^ ij（x）δij u＋b^ i（x）δiu＋f（x,u）=0,in Ω belong to belong toR^n,（＊）where a^ijδiφδjφ=0 on δΩ,andφis the defining function of δΩ.Imposing suitable conditions on the coefficients and f（x,u）,one can get the L^∞-estimates of（＊）via blow up method.
文摘The authors study homogenization of some nonlinear partial differential equations of the form -div (?(hx, h2x,Duh)) =f, where a is periodic in the first two arguments and monotone in the third. In particular the case where a satisfies degenerated structure conditions is studied. It is proved that uh converges weakly in W01,1 (?) to the unique solution of a limit problem as h ? '. Moreover, explicit expressions for the limit problem are obtained.
文摘In this paper, the authors prove the existence of solutions for degenerate elliptic equations of the form-div(a(x)▽_p u(x)) = g(λ, x, |u|^(p-2)u) in R^N, where ▽_pu =|▽u|^(p-2)▽u and a(x) is a degenerate nonnegative weight. The authors also investigate a related nonlinear eigenvalue problem obtaining an existence result which contains information about the location and multiplicity of eigensolutions. The proofs of the main results are obtained by using the critical point theory in Sobolev weighted spaces combined with a Caffarelli-Kohn-Nirenberg-type inequality and by using a specific minimax method, but without making use of the Palais-Smale condition.
