Let n≥2 and let L be a second-order elliptic operator of divergence form with coefficients consisting of both an elliptic symmetric part and a BMO anti-symmetric part in ℝ^(n).In this article,we consider the weighted...Let n≥2 and let L be a second-order elliptic operator of divergence form with coefficients consisting of both an elliptic symmetric part and a BMO anti-symmetric part in ℝ^(n).In this article,we consider the weighted Kato square root problem for L.More precisely,we prove that the square root L^(1/2)satisfies the weighted L^(p)estimates||L^(1/2)(f)||L_(ω)^p(R^(n))≤C||■f||L_(ω)^p(R^(n);R^(n))for any p∈(1,∞)andω∈Ap(ℝ^(n))(the class of Muckenhoupt weights),and that||■f||L_(ω)^p(R^(n);R^(n))≤C||L^(1/2)(f)||L_(ω)^p(R^(n))for any p∈(1,2+ε)andω∈Ap(ℝ^(n))∩RH_(2+ε/p),(R^(n))(the class of reverse Hölder weights),whereε∈(0,∞)is a constant depending only on n and the operator L,and where(2+ε/p)'denotes the Hölder conjugate exponent of 2+ε/p.Moreover,for any given q∈(2,∞),we give a sufficient condition to obtain that||■f||L_(ω)^p(R^(n);R^(n))≤C||L^(1/2)(f)||L_(ω)^p(R^(n))for any p∈(1,q)andω∈A_(p)(R^(n))∩pRH_(q/p),(R^(n)).As an application,we prove that when the coefficient matrix A that appears in L satisfies the small BMO condition,the Riesz transform∇L^(−1/2)is bounded on L_(ω)^(p)(ℝ^(n))for any given p∈(1,∞)andω∈Ap(ℝ^(n)).Furthermore,applications to the weighted L^(2)-regularity problem with the Dirichlet or the Neumann boundary condition are also given.展开更多
It is proved that, for the nondivergence elliptic equations Σi,jn=1aijuxixj=f, if f belongs to the generalized Morrey spaces Lp, (w), then uxixj ∈ Lp, (w), where u is the W2,p-solution of the equations. In order to ...It is proved that, for the nondivergence elliptic equations Σi,jn=1aijuxixj=f, if f belongs to the generalized Morrey spaces Lp, (w), then uxixj ∈ Lp, (w), where u is the W2,p-solution of the equations. In order to obtain this, the author first establish the weighted boundedness for the commutators of some singular integral operators on Lp, (w).展开更多
Let Ω be a bounded open domain in Rn with smooth boundary Ω, X =(X1,X2,... ,Xm) be a system of real smooth vector fields defined on Ω and the bound-ary Ω is non-characteristic for X. If X satisfies the HSrmande...Let Ω be a bounded open domain in Rn with smooth boundary Ω, X =(X1,X2,... ,Xm) be a system of real smooth vector fields defined on Ω and the bound-ary Ω is non-characteristic for X. If X satisfies the HSrmander's condition, then the vectorfield is finitely degenerate and the sum of square operator △X =m∑j=1 X2 j is a finitely de-generate elliptic operator. In this paper, we shall study the sharp estimate of the Dirichlet eigenvalue for a class of general Grushin type degenerate elliptic operators △x on Ω.展开更多
The Jackson-type estimates by using some elliptic operators will be achieved. These results will be used to characterize the regularity of some elliptic operators by means of the approximation degree and the saturatio...The Jackson-type estimates by using some elliptic operators will be achieved. These results will be used to characterize the regularity of some elliptic operators by means of the approximation degree and the saturation class of the multivariate Bernstein operators as well.展开更多
Let L=-div(A▽) be a second order divergence form elliptic operator, where A is an accretive, n×n matrix with bounded measurable complex coefficients on R^n. Let L^α/2 (0 <α< 1) denotes the fractional dif...Let L=-div(A▽) be a second order divergence form elliptic operator, where A is an accretive, n×n matrix with bounded measurable complex coefficients on R^n. Let L^α/2 (0 <α< 1) denotes the fractional differential operator associated with L and (-△)^α/2b ∈ L^n/α(R^n). In this article, we prove that the commutator[b, L^α/2] is bounded from the homogenous Sobolev space Lα%2 (R^n) to L^2(R^n).展开更多
Let L=-div(A▽) be a second order divergence form elliptic operator with bounded measurable coefficients in R^(n).We establish weighted L^(p) norm inequalities for commutators generated by √L and Lipschitz functions,...Let L=-div(A▽) be a second order divergence form elliptic operator with bounded measurable coefficients in R^(n).We establish weighted L^(p) norm inequalities for commutators generated by √L and Lipschitz functions,where the range of p is different from(1,∞),and we isolate the right class of weights introduced by Auscher and Martell.In this work,we use good-λ inequality with two parameters through the weighted boundedness of Riesz transforms ▽L^(-1/2).Our result recovers,in some sense,a previous result of Hofmann.展开更多
The authors study the existence of solutions for the nonlinear elliptic system {-Mλ+,(D2u)=f(u,v)in Ω,-Mλ+,(D2u)=g(u,v)inΩ,u≥0,v≥0 inΩ,u=v=0 on Ω, where Ω is a bounded convex domain in RN, N ≥ 2. I...The authors study the existence of solutions for the nonlinear elliptic system {-Mλ+,(D2u)=f(u,v)in Ω,-Mλ+,(D2u)=g(u,v)inΩ,u≥0,v≥0 inΩ,u=v=0 on Ω, where Ω is a bounded convex domain in RN, N ≥ 2. It is shown that under some assumptions on f and g, the problem has at least one positive solution (u, v).展开更多
An efficient multigrid finite-differences scheme for solving elliptic Fredholm partial integro-differential equations (PIDE) is discussed. This scheme combines a second-order accurate finite difference discretization ...An efficient multigrid finite-differences scheme for solving elliptic Fredholm partial integro-differential equations (PIDE) is discussed. This scheme combines a second-order accurate finite difference discretization of the PIDE problem with a multigrid scheme that includes a fast multilevel integration of the Fredholm operator allowing the fast solution of the PIDE problem. Theoretical estimates of second-order accuracy and results of local Fourier analysis of convergence of the proposed multigrid scheme are presented. Results of numerical experiments validate these estimates and demonstrate optimal computational complexity of the proposed framework.展开更多
In this paper, we reveal the multiple boundary layer phenomena of the solution of nonlinear higher order elliptic equations with perturbation both in boundary and in operator, and provide a method to find uniformly va...In this paper, we reveal the multiple boundary layer phenomena of the solution of nonlinear higher order elliptic equations with perturbation both in boundary and in operator, and provide a method to find uniformly valid asymptotic solution of arbitrary order for these types of problems.展开更多
In this paper an existence and uniqueness theorem of positive solutions to a class of semilinear elliptic systems is proved. Also, a necessary condition for the existence of the positive solution is obtained. As the a...In this paper an existence and uniqueness theorem of positive solutions to a class of semilinear elliptic systems is proved. Also, a necessary condition for the existence of the positive solution is obtained. As the application of the main theorem, two examples are given.展开更多
Relations of the 3D multi-directional derivatives are studied in this paper. These relations are applied to a geeral second-order linear elliptical operator and the corresponding expression are obtained. These relatio...Relations of the 3D multi-directional derivatives are studied in this paper. These relations are applied to a geeral second-order linear elliptical operator and the corresponding expression are obtained. These relations and expressions play important roles in the meshless finite point method.展开更多
Oscillation criteria for semilinear elliptic differential equations are obtained. The results are extensions of integral averaging technique of Kamenev. General means are employed to establish our results.
The existence and uniqueness of solutions to backward stochastic differential equations with jumps and with unbounded stopping time as terminal under the non_Lipschitz condition are obtained. The convergence of soluti...The existence and uniqueness of solutions to backward stochastic differential equations with jumps and with unbounded stopping time as terminal under the non_Lipschitz condition are obtained. The convergence of solutions and the continuous dependence of solutions on parameters are also derived. Then the probabilistic interpretation of solutions to some kinds of quasi_linear elliptic type integro_differential equations is obtained.展开更多
The regularity for a class of X-elliptic equations with lower order termLu+vu=-∑i,j=1 mXj^*(aij(x)Xiu)+vu=μis studied, where X = {X1,..., Xm} is a family of locally Lipschitz continuous vector fields, v is in...The regularity for a class of X-elliptic equations with lower order termLu+vu=-∑i,j=1 mXj^*(aij(x)Xiu)+vu=μis studied, where X = {X1,..., Xm} is a family of locally Lipschitz continuous vector fields, v is in certain Morrey type space and μ a nonnegative Radon measure. The HSlder continuity of the solution is proved when μ satisfies suitable growth condition, and a converse result on the estimate of μ is obtained when u is in certain HSlder class.展开更多
A class of nonlinear boundary value problems(BVP) for the second_order E 2 class elliptic systems in general form is discussed. By introducing a kind of transformation,this kind of BVP is reduced to a class of genera...A class of nonlinear boundary value problems(BVP) for the second_order E 2 class elliptic systems in general form is discussed. By introducing a kind of transformation,this kind of BVP is reduced to a class of generalized nonlinear Riemann_Hilbert BVP. And then some singular integral operators are introduced to establish the equivalent nonlinear singular integral equations. The solvability is proved under some suitable hypotheses by means of the properties of singular integral operators and the function theoretic methods.展开更多
In this paper, We study a general class of nonlinear degenerated elliptic problems associated with the differential inclusion β(u)-div(α(x, Du)+F(u)) ∈ f in fΩ, where f ∈ L1 (Ω). A vector field a(.,....In this paper, We study a general class of nonlinear degenerated elliptic problems associated with the differential inclusion β(u)-div(α(x, Du)+F(u)) ∈ f in fΩ, where f ∈ L1 (Ω). A vector field a(.,.) is a Carath6odory function. Using truncation techniques and the generalized monotonicity method in the functional spaces we prove the existence of renormalized solutions for general L1-data. Under an additional strict monotonicity assumption uniqueness of the renormalized solution is established.展开更多
Finite difference type preconditioners for spectral element discretizations based on Legendre-Gauss-Lobatto points are analyzed. The latter is employed for the approximation of uniformly elliptic partial differential ...Finite difference type preconditioners for spectral element discretizations based on Legendre-Gauss-Lobatto points are analyzed. The latter is employed for the approximation of uniformly elliptic partial differential problems. In this work, it is shown that the condition number of the resulting preconditioned system is bounded independently of both of the polynomial degrees used in the spectral element method and the element sizes. Several numerical tests verify the h-p independence of the proposed preconditioning.展开更多
The paper is devoted to the proof of the uniqueness theorem for solution of the equation for the non-local ionization source in a glow discharge and a hollow cathode in general 3D geometry. The theorem is applied to w...The paper is devoted to the proof of the uniqueness theorem for solution of the equation for the non-local ionization source in a glow discharge and a hollow cathode in general 3D geometry. The theorem is applied to wide class of electric field configurations, and to the walls of discharge volume, which have a property of incomplete absorption of the electrons. Cathode is regarded as interior singular source, which is placed arbitrarily close to the wall. The existence of solution is considered also. During the proof of the theorem many of useful structure formulae are obtained. Elements of the proof structure, which have arisen, are found to have physical sense. It makes clear physical construction of non-local electron avalanche, which builds a source of ionization in glow discharge at low pressures. Last has decisive significance to understand the hollow cathode discharge configuration and the hollow cathode effect.展开更多
Consider(M,g)as an n-dimensional compact Riemannian manifold.In this paper we are going to study a class of elliptic differential operators which appears naturally in the study of hypersurfaces with constant mean curv...Consider(M,g)as an n-dimensional compact Riemannian manifold.In this paper we are going to study a class of elliptic differential operators which appears naturally in the study of hypersurfaces with constant mean curvature and also the study of variation theory for 1-area functional.展开更多
基金supported by the Key Project of Gansu Provincial National Science Foundation(23JRRA1022)the National Natural Science Foundation of China(12071431)+1 种基金the Fundamental Research Funds for the Central Universities(lzujbky-2021-ey18)the Innovative Groups of Basic Research in Gansu Province(22JR5RA391).
文摘Let n≥2 and let L be a second-order elliptic operator of divergence form with coefficients consisting of both an elliptic symmetric part and a BMO anti-symmetric part in ℝ^(n).In this article,we consider the weighted Kato square root problem for L.More precisely,we prove that the square root L^(1/2)satisfies the weighted L^(p)estimates||L^(1/2)(f)||L_(ω)^p(R^(n))≤C||■f||L_(ω)^p(R^(n);R^(n))for any p∈(1,∞)andω∈Ap(ℝ^(n))(the class of Muckenhoupt weights),and that||■f||L_(ω)^p(R^(n);R^(n))≤C||L^(1/2)(f)||L_(ω)^p(R^(n))for any p∈(1,2+ε)andω∈Ap(ℝ^(n))∩RH_(2+ε/p),(R^(n))(the class of reverse Hölder weights),whereε∈(0,∞)is a constant depending only on n and the operator L,and where(2+ε/p)'denotes the Hölder conjugate exponent of 2+ε/p.Moreover,for any given q∈(2,∞),we give a sufficient condition to obtain that||■f||L_(ω)^p(R^(n);R^(n))≤C||L^(1/2)(f)||L_(ω)^p(R^(n))for any p∈(1,q)andω∈A_(p)(R^(n))∩pRH_(q/p),(R^(n)).As an application,we prove that when the coefficient matrix A that appears in L satisfies the small BMO condition,the Riesz transform∇L^(−1/2)is bounded on L_(ω)^(p)(ℝ^(n))for any given p∈(1,∞)andω∈Ap(ℝ^(n)).Furthermore,applications to the weighted L^(2)-regularity problem with the Dirichlet or the Neumann boundary condition are also given.
文摘It is proved that, for the nondivergence elliptic equations Σi,jn=1aijuxixj=f, if f belongs to the generalized Morrey spaces Lp, (w), then uxixj ∈ Lp, (w), where u is the W2,p-solution of the equations. In order to obtain this, the author first establish the weighted boundedness for the commutators of some singular integral operators on Lp, (w).
基金partially supported by the NSFC(11631011,11626251)
文摘Let Ω be a bounded open domain in Rn with smooth boundary Ω, X =(X1,X2,... ,Xm) be a system of real smooth vector fields defined on Ω and the bound-ary Ω is non-characteristic for X. If X satisfies the HSrmander's condition, then the vectorfield is finitely degenerate and the sum of square operator △X =m∑j=1 X2 j is a finitely de-generate elliptic operator. In this paper, we shall study the sharp estimate of the Dirichlet eigenvalue for a class of general Grushin type degenerate elliptic operators △x on Ω.
文摘The Jackson-type estimates by using some elliptic operators will be achieved. These results will be used to characterize the regularity of some elliptic operators by means of the approximation degree and the saturation class of the multivariate Bernstein operators as well.
基金supported by NSFC(11471033),NCET of China(NCET-11-0574)the Fundamental Research Funds for the Central Universities(FRF-BR-16-011A)
文摘Let L=-div(A▽) be a second order divergence form elliptic operator, where A is an accretive, n×n matrix with bounded measurable complex coefficients on R^n. Let L^α/2 (0 <α< 1) denotes the fractional differential operator associated with L and (-△)^α/2b ∈ L^n/α(R^n). In this article, we prove that the commutator[b, L^α/2] is bounded from the homogenous Sobolev space Lα%2 (R^n) to L^2(R^n).
基金supported by NSFC(1187109611471033)+4 种基金supported by NSFC(113710571147103311571160)SRFDP(20130003110003)the Fundamental Research Funds for the Central Universities(2014KJJCA10)。
文摘Let L=-div(A▽) be a second order divergence form elliptic operator with bounded measurable coefficients in R^(n).We establish weighted L^(p) norm inequalities for commutators generated by √L and Lipschitz functions,where the range of p is different from(1,∞),and we isolate the right class of weights introduced by Auscher and Martell.In this work,we use good-λ inequality with two parameters through the weighted boundedness of Riesz transforms ▽L^(-1/2).Our result recovers,in some sense,a previous result of Hofmann.
基金supported by National Natural Sciences Foundations of China (10571175,10631030)
文摘The authors study the existence of solutions for the nonlinear elliptic system {-Mλ+,(D2u)=f(u,v)in Ω,-Mλ+,(D2u)=g(u,v)inΩ,u≥0,v≥0 inΩ,u=v=0 on Ω, where Ω is a bounded convex domain in RN, N ≥ 2. It is shown that under some assumptions on f and g, the problem has at least one positive solution (u, v).
文摘An efficient multigrid finite-differences scheme for solving elliptic Fredholm partial integro-differential equations (PIDE) is discussed. This scheme combines a second-order accurate finite difference discretization of the PIDE problem with a multigrid scheme that includes a fast multilevel integration of the Fredholm operator allowing the fast solution of the PIDE problem. Theoretical estimates of second-order accuracy and results of local Fourier analysis of convergence of the proposed multigrid scheme are presented. Results of numerical experiments validate these estimates and demonstrate optimal computational complexity of the proposed framework.
文摘In this paper, we reveal the multiple boundary layer phenomena of the solution of nonlinear higher order elliptic equations with perturbation both in boundary and in operator, and provide a method to find uniformly valid asymptotic solution of arbitrary order for these types of problems.
基金The project supported by NNSF of China(10071080)
文摘In this paper an existence and uniqueness theorem of positive solutions to a class of semilinear elliptic systems is proved. Also, a necessary condition for the existence of the positive solution is obtained. As the application of the main theorem, two examples are given.
基金Supported by the National Natural Science Foundation of China 1060100910701014+1 种基金10871029)the Foundation of China Academy of Engineering Physics (2007B09008)
文摘Relations of the 3D multi-directional derivatives are studied in this paper. These relations are applied to a geeral second-order linear elliptical operator and the corresponding expression are obtained. These relations and expressions play important roles in the meshless finite point method.
文摘Oscillation criteria for semilinear elliptic differential equations are obtained. The results are extensions of integral averaging technique of Kamenev. General means are employed to establish our results.
文摘The existence and uniqueness of solutions to backward stochastic differential equations with jumps and with unbounded stopping time as terminal under the non_Lipschitz condition are obtained. The convergence of solutions and the continuous dependence of solutions on parameters are also derived. Then the probabilistic interpretation of solutions to some kinds of quasi_linear elliptic type integro_differential equations is obtained.
基金Supported by the China State Scholarship (2003833095)Department of Education of Zhejiang Province(20051495)
文摘The regularity for a class of X-elliptic equations with lower order termLu+vu=-∑i,j=1 mXj^*(aij(x)Xiu)+vu=μis studied, where X = {X1,..., Xm} is a family of locally Lipschitz continuous vector fields, v is in certain Morrey type space and μ a nonnegative Radon measure. The HSlder continuity of the solution is proved when μ satisfies suitable growth condition, and a converse result on the estimate of μ is obtained when u is in certain HSlder class.
文摘A class of nonlinear boundary value problems(BVP) for the second_order E 2 class elliptic systems in general form is discussed. By introducing a kind of transformation,this kind of BVP is reduced to a class of generalized nonlinear Riemann_Hilbert BVP. And then some singular integral operators are introduced to establish the equivalent nonlinear singular integral equations. The solvability is proved under some suitable hypotheses by means of the properties of singular integral operators and the function theoretic methods.
文摘In this paper, We study a general class of nonlinear degenerated elliptic problems associated with the differential inclusion β(u)-div(α(x, Du)+F(u)) ∈ f in fΩ, where f ∈ L1 (Ω). A vector field a(.,.) is a Carath6odory function. Using truncation techniques and the generalized monotonicity method in the functional spaces we prove the existence of renormalized solutions for general L1-data. Under an additional strict monotonicity assumption uniqueness of the renormalized solution is established.
文摘Finite difference type preconditioners for spectral element discretizations based on Legendre-Gauss-Lobatto points are analyzed. The latter is employed for the approximation of uniformly elliptic partial differential problems. In this work, it is shown that the condition number of the resulting preconditioned system is bounded independently of both of the polynomial degrees used in the spectral element method and the element sizes. Several numerical tests verify the h-p independence of the proposed preconditioning.
文摘The paper is devoted to the proof of the uniqueness theorem for solution of the equation for the non-local ionization source in a glow discharge and a hollow cathode in general 3D geometry. The theorem is applied to wide class of electric field configurations, and to the walls of discharge volume, which have a property of incomplete absorption of the electrons. Cathode is regarded as interior singular source, which is placed arbitrarily close to the wall. The existence of solution is considered also. During the proof of the theorem many of useful structure formulae are obtained. Elements of the proof structure, which have arisen, are found to have physical sense. It makes clear physical construction of non-local electron avalanche, which builds a source of ionization in glow discharge at low pressures. Last has decisive significance to understand the hollow cathode discharge configuration and the hollow cathode effect.
文摘Consider(M,g)as an n-dimensional compact Riemannian manifold.In this paper we are going to study a class of elliptic differential operators which appears naturally in the study of hypersurfaces with constant mean curvature and also the study of variation theory for 1-area functional.