In this paper, we are concerned with the Riesz means of Dirichlet eigenvalues for the sub-Laplace operator on the Engel group and deriver different inequalities for Riesz means. The Weyl-type estimates for means of ei...In this paper, we are concerned with the Riesz means of Dirichlet eigenvalues for the sub-Laplace operator on the Engel group and deriver different inequalities for Riesz means. The Weyl-type estimates for means of eigenvalues are given.展开更多
Let Γ?R;be a regular anisotropic fractal. We discuss the problem of the negative spectrum for the Schr?dinger operators associated with the formal expression H;=id-?+βtr;,β∈R,acting in the anisotropic Sobolev spac...Let Γ?R;be a regular anisotropic fractal. We discuss the problem of the negative spectrum for the Schr?dinger operators associated with the formal expression H;=id-?+βtr;,β∈R,acting in the anisotropic Sobolev space W;（R;）, where ? is the Dirichlet Laplanian in R;and tr;is a fractal potential（distribution） supported by Γ.展开更多
There are both loss and dispersion characteristics for most dielectric media. In quantum theory the loss in medium is generally described by Langevin force in the Langevin noise （LN） scheme by which the quantization...There are both loss and dispersion characteristics for most dielectric media. In quantum theory the loss in medium is generally described by Langevin force in the Langevin noise （LN） scheme by which the quantization of the radiation field in various homogeneous absorbing dielectrics can be successfully actualized. However, it is invalid for the anisotropic dispersion medium. This paper extends the LN theory to an anisotropic dispersion medium and presented the quantization of the radiation field as well as the transformation relation between the homogeneous and anisotropic dispersion media.展开更多
We consider a nonlinear Robin problem driven by the anisotropic(p,q)-Laplacian and with a reaction exhibiting the competing effects of a parametric sublinear(concave) term and of a superlinear(convex) term.We prove a ...We consider a nonlinear Robin problem driven by the anisotropic(p,q)-Laplacian and with a reaction exhibiting the competing effects of a parametric sublinear(concave) term and of a superlinear(convex) term.We prove a bifurcation-type theorem describing the changes in the set of positive solutions as the parameter varies.We also prove the existence of a minimal positive solution and determine the monotonicity and continuity properties of the minimal solution map.展开更多
Two concepts of phenomenological optics of homogeneous, anisotropic and dispersive media are compared, the younger and more general concept of media with spatial dispersion and the older concept of (bi)-anisotropic me...Two concepts of phenomenological optics of homogeneous, anisotropic and dispersive media are compared, the younger and more general concept of media with spatial dispersion and the older concept of (bi)-anisotropic media with material tensors for electric and magnetic induction which only depend on the frequency. The general algebraic form of the polarization vectors for the electric field and their one-dimensional projection operators is discussed without the degenerate cases of optic axis for which they become two-dimensional projection operators. Group velocity and diffraction coefficients in an approximate equation for the slowly varying amplitudes of beam solutions are calculated. As special case a polariton permittivity for isotropic media with frequency dispersion but without losses is discussed for the usual passive case and for the active case (occupation inversion of two energy levels that goes in direction of laser theory) and the group velocity is calculated. For this active case, regions of frequency and wave vector with group velocities greater than that of light in vacuum were found. This is not fully understood and due to large diffraction is likely only to realize in guided resonator form. The notion of “negative refraction” is shortly discussed but we did not find agreement with its assessment in the original paper.展开更多
In this paper we give the conditions on the pair （~1, W2） which ensures the boundedness of the anisotropic maximal operator and anisotropic singular integral operators from one generalized Morrey space Mp,w1 to anot...In this paper we give the conditions on the pair （~1, W2） which ensures the boundedness of the anisotropic maximal operator and anisotropic singular integral operators from one generalized Morrey space Mp,w1 to another Mp,w2, 1 〈 p 〈 ∞, and from the space M1,w1 to the weak space M1,w2.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
The fundamental mechanical equations were studied under the mechanical space. The differential stress operator and strain operator were obtained. There were strain energy operator and Hamilton operator for elastic bod...The fundamental mechanical equations were studied under the mechanical space. The differential stress operator and strain operator were obtained. There were strain energy operator and Hamilton operator for elastic body in same way, and the following results were testified. 1) The equilibrium equation of force is equivalent to the harmony equation of deformation under the mechanical space. They are all the basic mode of eigen equation of stress or strain operator. 2) The eigen value of stress or strain operator is corresponding to the order of kinetic energy of elastic body, and the elastic wave represents the non basic mode. 3) The eigen functions of stress operator or strain operator corresponding to some kinetic energy order are fields of modal stress or modal strain in same order. 4) The eigen equations of strain energy operator are the fundamental equations of elastic mechanics which are expressed with the potential functions. [展开更多
Let A be an expansive dilation. We define weakly strongly singular integral kernels and study the action of the operators induced by these kernels on anisotropic Hardy spaces associated with A.
文摘In this paper, we are concerned with the Riesz means of Dirichlet eigenvalues for the sub-Laplace operator on the Engel group and deriver different inequalities for Riesz means. The Weyl-type estimates for means of eigenvalues are given.
基金supported by the Natural Science Foundation of the Jiangsu Higher Education Institutions of China(Grant No.13KJB110010)the Pre Study Foundation of Nanjing University of Finance&Economics(Grant No.YYJ2013016)the Priority Academic Program Development of Jiangsu Higher Education Institutions(PAPD)
文摘Let Γ?R;be a regular anisotropic fractal. We discuss the problem of the negative spectrum for the Schr?dinger operators associated with the formal expression H;=id-?+βtr;,β∈R,acting in the anisotropic Sobolev space W;（R;）, where ? is the Dirichlet Laplanian in R;and tr;is a fractal potential（distribution） supported by Γ.
基金Project supported by the National Natural Science Foundation of China (Grant No 10574010)
文摘There are both loss and dispersion characteristics for most dielectric media. In quantum theory the loss in medium is generally described by Langevin force in the Langevin noise （LN） scheme by which the quantization of the radiation field in various homogeneous absorbing dielectrics can be successfully actualized. However, it is invalid for the anisotropic dispersion medium. This paper extends the LN theory to an anisotropic dispersion medium and presented the quantization of the radiation field as well as the transformation relation between the homogeneous and anisotropic dispersion media.
基金supported by NNSF of China(12071413)NSF of Guangxi(2018GXNSFDA138002)。
文摘We consider a nonlinear Robin problem driven by the anisotropic(p,q)-Laplacian and with a reaction exhibiting the competing effects of a parametric sublinear(concave) term and of a superlinear(convex) term.We prove a bifurcation-type theorem describing the changes in the set of positive solutions as the parameter varies.We also prove the existence of a minimal positive solution and determine the monotonicity and continuity properties of the minimal solution map.
文摘Two concepts of phenomenological optics of homogeneous, anisotropic and dispersive media are compared, the younger and more general concept of media with spatial dispersion and the older concept of (bi)-anisotropic media with material tensors for electric and magnetic induction which only depend on the frequency. The general algebraic form of the polarization vectors for the electric field and their one-dimensional projection operators is discussed without the degenerate cases of optic axis for which they become two-dimensional projection operators. Group velocity and diffraction coefficients in an approximate equation for the slowly varying amplitudes of beam solutions are calculated. As special case a polariton permittivity for isotropic media with frequency dispersion but without losses is discussed for the usual passive case and for the active case (occupation inversion of two energy levels that goes in direction of laser theory) and the group velocity is calculated. For this active case, regions of frequency and wave vector with group velocities greater than that of light in vacuum were found. This is not fully understood and due to large diffraction is likely only to realize in guided resonator form. The notion of “negative refraction” is shortly discussed but we did not find agreement with its assessment in the original paper.
文摘In this paper we give the conditions on the pair （~1, W2） which ensures the boundedness of the anisotropic maximal operator and anisotropic singular integral operators from one generalized Morrey space Mp,w1 to another Mp,w2, 1 〈 p 〈 ∞, and from the space M1,w1 to the weak space M1,w2.
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.
文摘The fundamental mechanical equations were studied under the mechanical space. The differential stress operator and strain operator were obtained. There were strain energy operator and Hamilton operator for elastic body in same way, and the following results were testified. 1) The equilibrium equation of force is equivalent to the harmony equation of deformation under the mechanical space. They are all the basic mode of eigen equation of stress or strain operator. 2) The eigen value of stress or strain operator is corresponding to the order of kinetic energy of elastic body, and the elastic wave represents the non basic mode. 3) The eigen functions of stress operator or strain operator corresponding to some kinetic energy order are fields of modal stress or modal strain in same order. 4) The eigen equations of strain energy operator are the fundamental equations of elastic mechanics which are expressed with the potential functions. [
基金supported by NSF of China (Grant No.10571015)RFDP of China (Grant No.20050027025)NSF of Zhejiang Province (Grant No.Y7080325)
文摘Let A be an expansive dilation. We define weakly strongly singular integral kernels and study the action of the operators induced by these kernels on anisotropic Hardy spaces associated with A.
