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 saturati... 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
In this paper,we study the semilinear subelliptic equation{−△Xu=f(x,u)+g(x,u)inΩ,u=0 on∂Ω,where△X=−∑m i=1 Xi∗Xi is the self-adjoint Hormander operator associated with the vector fields X=(X_(1),X_(2),...,X_(m))sati...In this paper,we study the semilinear subelliptic equation{−△Xu=f(x,u)+g(x,u)inΩ,u=0 on∂Ω,where△X=−∑m i=1 Xi∗Xi is the self-adjoint Hormander operator associated with the vector fields X=(X_(1),X_(2),...,X_(m))satisfying the Hormander's condition,f(x,u)∈C(Ω×R),g(x,u)is a Carathéodory function onΩ×R,andΩis an open bounded domain in R~n with smooth boundary.Combining the perturbation from the symmetry method with the approaches involving the eigenvalue estimate and the Morse index in estimating the minimax values,we obtain two kinds of existence results for multiple weak solutions to the problem above.Furthermore,we discuss the difference between the eigenvalue estimate approach and the Morse index approach in degenerate situations.Compared with the classical elliptic cases,both approaches here have their own strengths in the degenerate cases.This new phenomenon implies that the results in general degenerate cases would be quite different from the situations in classical elliptic cases.展开更多
In this paper,the study of gradient regularity for solutions of a class of elliptic problems of p-Laplace type is offered.In particular,we prove a global result concerning Lorentz-Morrey regularity of the non-homogene...In this paper,the study of gradient regularity for solutions of a class of elliptic problems of p-Laplace type is offered.In particular,we prove a global result concerning Lorentz-Morrey regularity of the non-homogeneous boundary data problem:-div((s^(2)+|▽u|^(2)p-2/2)▽u)=-div(|f|^(p-2)f)+g inΩ,u=h in■Ω,with the(sub-elliptic)degeneracy condition s∈[0,1]and with mixed data f∈L^(p)(Q;R^(n)),g∈Lp/(p-1)(Ω;R^(n))for p∈(1,n).This problem naturally arises in various applications such as dynamics of non-Newtonian fluid theory,electro-rheology,radiation of heat,plastic moulding and many others.Building on the idea of level-set inequality on fractional maximal distribution functions,it enables us to carry out a global regularity result of the solution via fractional maximal operators.Due to the significance of M_(α)and its relation with Riesz potential,estimates via fractional maximal functions allow us to bound oscillations not only for solution but also its fractional derivatives of orderα.Our approach therefore has its own interest.展开更多
Let F be a compact d-set in R^n with 0 〈 d ≤ n, which includes various kinds of fractals. The author establishes an embedding theorem for the Besov spaces Bpq^s（F） of Triebel and the Sobolev spaces W^1,P（F,d,μ）...Let F be a compact d-set in R^n with 0 〈 d ≤ n, which includes various kinds of fractals. The author establishes an embedding theorem for the Besov spaces Bpq^s（F） of Triebel and the Sobolev spaces W^1,P（F,d,μ） of Hajtasz when s 〉 1, 1 〈 p 〈∞ and 0 〈 q ≤ ∞. The author also gives some applications of the estimates of the entropy numbers in the estimates of the eigenvalues of some fractal pseudodifferential operators in the spaces Bpq^0（F） and Fpq^0（F）.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
Let L = -div（AV） be a second order divergence form elliptic operator, and A be an accretive, n × n matrix with bounded measurable complex coefficients in R^n. We obtain the Lp bounds for the commutator generate...Let L = -div（AV） be a second order divergence form elliptic operator, and A be an accretive, n × n matrix with bounded measurable complex coefficients in R^n. We obtain the Lp bounds for the commutator generated by the Kato square root v/L and a Lipschitz function, which recovers a previous result of Calderon, by a different method. In this work, we develop a new theory for the commutators associated to elliptic operators with Lipschitz function. The theory of the commutator with Lipschitz function is distinguished from the analogous elliptic operator theory.展开更多
A version of the ＂Fredholm index = spectral flow＂ theorem is proved for general families of elliptic operators （A（t）}t∈R on closed （compact and without boundary） manifolds. Here we do not require that A（t）,...A version of the ＂Fredholm index = spectral flow＂ theorem is proved for general families of elliptic operators （A（t）}t∈R on closed （compact and without boundary） manifolds. Here we do not require that A（t）, t ∈R or its leading part is self-adjoint.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 divergent-form elliptic operator,where A is an accretive n×n matrix with bounded and measurable complex coefficients on Herein,we prove that the commutator [b,L1/2]of the Kato ...Let L=-div（A▽） be a second-order divergent-form elliptic operator,where A is an accretive n×n matrix with bounded and measurable complex coefficients on Herein,we prove that the commutator [b,L1/2]of the Kato square root L1/2 and b with ▽b∈Ln（Rn）（n> 2）,is bounded from the homogenous Sobolev space L1p（Rn） to Lp（Rn）（p-（L） <p<p+（L））.展开更多
We study the separability properties of solutions to elliptic equations with piecewise constant coefficients in Rd, d ≥ 2. The separation rank of the solution to diffusion equation with variable coefficients is prese...We study the separability properties of solutions to elliptic equations with piecewise constant coefficients in Rd, d ≥ 2. The separation rank of the solution to diffusion equation with variable coefficients is presented.展开更多
In the 1970’s, Folland and Stein studied a family of subelliptic scalar operators $ \mathcal{L}_\lambda $ which arise naturally in the $ \bar \partial _b $ -complex. They introduced weighted Sobolev spaces as the nat...In the 1970’s, Folland and Stein studied a family of subelliptic scalar operators $ \mathcal{L}_\lambda $ which arise naturally in the $ \bar \partial _b $ -complex. They introduced weighted Sobolev spaces as the natural spaces for this complex, and then obtained sharp estimates for $ \bar \partial _b $ in these spaces using integral kernels and approximate inverses. In the 1990’s, Rumin introduced a differential complex for compact contact manifolds, showed that the Folland-Stein operators are central to the analysis for the corresponding Laplace operator, and derived the necessary estimates for the Laplacian from the Folland Stein analysis. In this paper, we give a self-contained derivation of sharp estimates in the anisotropic Folland-Stein spaces for the operators studied by Rumin using integration by parts and a modified approach to bootstrapping.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
In this paper,the authors study the asymptotically linear elliptic equation on manifold with conical singularities-ΔBu+λu=a(z)f(u),u≥0 in R+N,where N=n+1≥3,λ>0,z=(t,x_(1),…,x_(n)),and ΔB=(t■t)^(2)+■^(2)x_(...In this paper,the authors study the asymptotically linear elliptic equation on manifold with conical singularities-ΔBu+λu=a(z)f(u),u≥0 in R+N,where N=n+1≥3,λ>0,z=(t,x_(1),…,x_(n)),and ΔB=(t■t)^(2)+■^(2)x_(1)+…+■^(2)x_(n).Combining properties of cone-degenerate operator,the Pohozaev manifold and qualitative properties of the ground state solution for the limit equation,we obtain a positive solution under some suitable conditions on a and f.展开更多
The aim of this review is to introduce some recent results in eigenvalues problems for a class of degenerate elliptic operators with non-smooth coefficients,we present the explicit estimates of the lower bound and upp...The aim of this review is to introduce some recent results in eigenvalues problems for a class of degenerate elliptic operators with non-smooth coefficients,we present the explicit estimates of the lower bound and upper bound for its Dirichlet eigenvalues.展开更多
In References [1] and [2], B. V. Boyarski and I have studied separately the Dirichlet problem and the Neumann problem for the following system of second order linear elliptic
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 National Natural Science Foundation of China(Grant No.12131017)supported by National Natural Science Foundation of China(Grant No.12201607)+3 种基金National Key R&D Program of China(Grant No.2022YFA1005602)Knowledge Innovation Program of Wuhan-Shuguang Project(Grant No.2023010201020286)China Postdoctoral Science Foundation(Grant No.2023T160655)supported by China National Postdoctoral Program for Innovative Talents(Grant No.BX20230270)。
文摘In this paper,we study the semilinear subelliptic equation{−△Xu=f(x,u)+g(x,u)inΩ,u=0 on∂Ω,where△X=−∑m i=1 Xi∗Xi is the self-adjoint Hormander operator associated with the vector fields X=(X_(1),X_(2),...,X_(m))satisfying the Hormander's condition,f(x,u)∈C(Ω×R),g(x,u)is a Carathéodory function onΩ×R,andΩis an open bounded domain in R~n with smooth boundary.Combining the perturbation from the symmetry method with the approaches involving the eigenvalue estimate and the Morse index in estimating the minimax values,we obtain two kinds of existence results for multiple weak solutions to the problem above.Furthermore,we discuss the difference between the eigenvalue estimate approach and the Morse index approach in degenerate situations.Compared with the classical elliptic cases,both approaches here have their own strengths in the degenerate cases.This new phenomenon implies that the results in general degenerate cases would be quite different from the situations in classical elliptic cases.
基金supported by Ministry of Education and Training(Vietnam),under grant number B2023-SPS-01。
文摘In this paper,the study of gradient regularity for solutions of a class of elliptic problems of p-Laplace type is offered.In particular,we prove a global result concerning Lorentz-Morrey regularity of the non-homogeneous boundary data problem:-div((s^(2)+|▽u|^(2)p-2/2)▽u)=-div(|f|^(p-2)f)+g inΩ,u=h in■Ω,with the(sub-elliptic)degeneracy condition s∈[0,1]and with mixed data f∈L^(p)(Q;R^(n)),g∈Lp/(p-1)(Ω;R^(n))for p∈(1,n).This problem naturally arises in various applications such as dynamics of non-Newtonian fluid theory,electro-rheology,radiation of heat,plastic moulding and many others.Building on the idea of level-set inequality on fractional maximal distribution functions,it enables us to carry out a global regularity result of the solution via fractional maximal operators.Due to the significance of M_(α)and its relation with Riesz potential,estimates via fractional maximal functions allow us to bound oscillations not only for solution but also its fractional derivatives of orderα.Our approach therefore has its own interest.
基金The author is partially supported by NNSF(10271015) RFDP(20020027004)of China
文摘Let F be a compact d-set in R^n with 0 〈 d ≤ n, which includes various kinds of fractals. The author establishes an embedding theorem for the Besov spaces Bpq^s（F） of Triebel and the Sobolev spaces W^1,P（F,d,μ） of Hajtasz when s 〉 1, 1 〈 p 〈∞ and 0 〈 q ≤ ∞. The author also gives some applications of the estimates of the entropy numbers in the estimates of the eigenvalues of some fractal pseudodifferential operators in the spaces Bpq^0（F） and Fpq^0（F）.
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.
基金supported by NSF of China(Grant No.11471033)supported by NSF of China(Grant Nos.11371057,11571160)+4 种基金NCET of China(Grant No.NCET-11-0574)the Fundamental Research Funds for the Central Universities(FRF-TP-12-006B)the Fundamental Research Funds for the Central Universities(Grant No.2014KJJCA10)SRFDP of China(Grant No.20130003110003)supported by NSF(Grant No.DMS 1101244)
文摘Let L = -div（AV） be a second order divergence form elliptic operator, and A be an accretive, n × n matrix with bounded measurable complex coefficients in R^n. We obtain the Lp bounds for the commutator generated by the Kato square root v/L and a Lipschitz function, which recovers a previous result of Calderon, by a different method. In this work, we develop a new theory for the commutators associated to elliptic operators with Lipschitz function. The theory of the commutator with Lipschitz function is distinguished from the analogous elliptic operator theory.
文摘A version of the ＂Fredholm index = spectral flow＂ theorem is proved for general families of elliptic operators （A（t）}t∈R on closed （compact and without boundary） manifolds. Here we do not require that A（t）, t ∈R or its leading part is self-adjoint.
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 National Natural Science Foundation of China (Grant No. 11471033)Program for New Century Excellent Talents in University of China (Grant No. NCET-11-0574)+1 种基金the Fundamental Research Funds for the Central Universities (Grant No. FRF-BR-17-001B)the Fundamental Research Funds for Doctoral Candidate of University of Science and Technology Beijing (Grant No. FRF-BR-17018)
文摘Let L=-div（A▽） be a second-order divergent-form elliptic operator,where A is an accretive n×n matrix with bounded and measurable complex coefficients on Herein,we prove that the commutator [b,L1/2]of the Kato square root L1/2 and b with ▽b∈Ln（Rn）（n> 2）,is bounded from the homogenous Sobolev space L1p（Rn） to Lp（Rn）（p-（L） <p<p+（L））.
文摘We study the separability properties of solutions to elliptic equations with piecewise constant coefficients in Rd, d ≥ 2. The separation rank of the solution to diffusion equation with variable coefficients is presented.
基金This work was supported by NSERC(Grant No.RGPIN/9319-2005)
文摘In the 1970’s, Folland and Stein studied a family of subelliptic scalar operators $ \mathcal{L}_\lambda $ which arise naturally in the $ \bar \partial _b $ -complex. They introduced weighted Sobolev spaces as the natural spaces for this complex, and then obtained sharp estimates for $ \bar \partial _b $ in these spaces using integral kernels and approximate inverses. In the 1990’s, Rumin introduced a differential complex for compact contact manifolds, showed that the Folland-Stein operators are central to the analysis for the corresponding Laplace operator, and derived the necessary estimates for the Laplacian from the Folland Stein analysis. In this paper, we give a self-contained derivation of sharp estimates in the anisotropic Folland-Stein spaces for the operators studied by Rumin using integration by parts and a modified approach to bootstrapping.
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 the National Natural Science Foundation of China(Nos.11631011,11601402,12171183,11831009,12071364,11871387)。
文摘In this paper,the authors study the asymptotically linear elliptic equation on manifold with conical singularities-ΔBu+λu=a(z)f(u),u≥0 in R+N,where N=n+1≥3,λ>0,z=(t,x_(1),…,x_(n)),and ΔB=(t■t)^(2)+■^(2)x_(1)+…+■^(2)x_(n).Combining properties of cone-degenerate operator,the Pohozaev manifold and qualitative properties of the ground state solution for the limit equation,we obtain a positive solution under some suitable conditions on a and f.
基金supported by the National Natural Science Foundation of China(Grant Nos.11631011,11626251)the China Postdoctoral Science Foundation(Grant No.2020M672398).
文摘The aim of this review is to introduce some recent results in eigenvalues problems for a class of degenerate elliptic operators with non-smooth coefficients,we present the explicit estimates of the lower bound and upper bound for its Dirichlet eigenvalues.
文摘In References [1] and [2], B. V. Boyarski and I have studied separately the Dirichlet problem and the Neumann problem for the following system of second order linear elliptic
