THE WEIGHTED KATO SQUARE ROOT PROBLEMOF ELLIPTIC OPERATORS HAVING A BMOANTI-SYMMETRICPART

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.

DEGREE OF APPROXIMATION ASSOCIATED WITH SOME ELLIPTIC OPERATORS AND ITS APPLICATIONS

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.

LOWER BOUNDS OF DIRICHLET EIGENVALUES FOR A CLASS OF FINITELY DEGENERATE GRUSHIN TYPE ELLIPTIC OPERATORS

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.

ON MIN'S ZETA-FUNCTION AND HERMITE ELLIPTIC OPERATOR

In this note, the authors study some fundamental properties on a Min＇s zeta- function and explore its connection with Hermite elliptic operator.

WEIGHTED NORM INEQUALITIES FOR COMMUTATORS OF THE KATO SQUARE ROOT OF SECOND ORDER ELLIPTIC OPERATORS ON R^(n)

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.

The Commutator of the Kato Square Root for Second Order Elliptic Operators on R^n

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.

Inequalities for Eigenvalues of a System of Equations of Elliptic Operator in Weighted Divergence Form on Metric Measure Space

Let A be a symmetric and positive definite(1,1)tensor on a bounded domain Ω in an ndimensional metric measure space(R^n,<,>,e^-φdv).In this paper,we investigate the Dirichlet eigenvalue problem of a system of equations of elliptic operators in weighted divergence form{LA,φu+α▽(divu)-▽φdivu]=-su,inΩ,u∣aΩ=0,where LA,φ=div(A▽(·))-(A▽φ,▽(·)),α is a nonnegative constant and u is a vector-valued function.Some universal inequalities for eigenvalues of this problem are established.Moreover,as applications of these results,we give some estimates for the upper bound of sk+1 and the gap of sk+1-sk in terms of the first k eigenvalues.Our results contain some results for the Lam′e system and a system of equations of the drifting Laplacian.

L^p-gradient estimates for the commutators of the Kato square roots of second-order elliptic operators on R^n

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)

Dirichlet Eigenvalue Problem of Degenerate Elliptic Operators with Non-Smooth Coefficients

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.

Existence and Uniqueness of Renormalized Solution of Nonlinear Degenerated Elliptic Problems

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.

Relations of 3D directional derivatives and expressions of typical differential operators

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.

The Eigenvalues of a Class of Elliptic Differential Operators

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.

LOW-RANK TENSOR STRUCTURE OF SOLUTIONS TO ELLIPTIC PROBLEMS WITH JUMPING COEFFICIENTS

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.

Positive Solutions for Asymptotically Linear Cone-Degenerate Elliptic Equations

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.

Efficient Reconstruction Methods for Nonlinear Elliptic Cauchy Problems with Piecewise Constant Solutions

In this article,a level-set approach for solving nonlinear elliptic Cauchy problems with piecewise constant solutions is proposed,which allows the definition of a Tikhonov functional on a space of level-set functions.We provide convergence analysis for the Tikhonov approach,including stability and convergence results.Moreover,a numerical investigation of the proposed Tikhonov regularization method is presented.Newton-type methods are used for the solution of the optimality systems,which can be interpreted as stabilized versions of algorithms in a previous work and yield a substantial improvement in performance.The whole approach is focused on three dimensional models,better suited for real life applications.

Some Applications of Besov Spaces on Fractals

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）.

Some mathematical aspects of Anderson localization:boundary effect,multimodality,and bifurcation

Anderson localization is a famous wave phenomenon that describes the absence of diffusion of waves in a disordered medium.Here we generalize the landscape theory of Anderson localization to general elliptic operators and complex boundary conditions using a probabilistic approach,and further investigate some mathematical aspects of Anderson localization that are rarely discussed before.First,we observe that under the Neumann boundary condition,the low energy quantum states are localized on the boundary of the domain with high probability.We provide a detailed explanation of this phenomenon using the concept of extended subregions and obtain an analytical expression of this probability in the one-dimensional case.Second,we find that the quantum states may be localized in multiple different subregions with high probability in the one-dimensional case and we derive an explicit expression of this probability for various boundary conditions.Finally,we examine a bifurcation phenomenon of the localization subregion as the strength of disorder varies.The critical threshold of bifurcation is analytically computed based on a toy model and the dependence of the critical threshold on model parameters is analyzed.

Fredholm Index and Spectral Flow in Non-self-adjoint Case

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.

On the Existence of Feller Semigroups with Discontinuous Coefficients Ⅱ

This paper is devoted to the functional analytic approach to the problem of existence of Markov processes with Dirichlet boundary condition, oblique derivative boundary condition and first-order Wentzell boundary condition for second-order, uniformly elliptic differential operators with discontinuous coefficients. More precisely, we construct Feller semigroups associated with absorption, reflection, drift and sticking phenomena at the boundary. The approach here is distinguished by the extensive use of the ideas and techniques characteristic of the recent developments in the Calderon- Zygmund theory of singular integral operators with non-smooth kernels.

Matched Asymptotic Expansions of the Eigenvalues of a 3-D Boundary-Value Problem Relative to Two Cavities Linked by a Hole of Small Size

In this article,we consider a domain consisting of two cavities linked by a hole of small size.We derive a numerical method to compute an approximation of the eigenvalues of an elliptic operator without refining in the neighborhood of the hole.Several convergence rates are obtained and illustrated by numerical simulations.