THE GRADIENT ESTIMATE OF SUBELLIPTIC HARMONIC MAPS WITH A POTENTIAL

In this paper,we investigate subelliptic harmonic maps with a potential from noncompact complete sub-Riemannian manifolds corresponding to totally geodesic Riemannian foliations.Under some suitable conditions,we give the gradient estimates of these maps and establish a Liouville type result.

Sem ilinear Equationswith CriticalExponentsonthe Heisenberg Group

We consider the semilinear subelliptic Laplace equation -Δ Hn u=u p+f(x,u) in Ω,u>0 in Ω,u=0 on Ω . Under suitable assumptions on f and p ,some existence and nonexistence results are obtained.

SOLVABILITY OF THE FOURTH ORDER NONLINEAR SUBELLIPTIC EQUATIONS ON THE HEISENBERG GROUP

In this paper,some existence results for the fourth order nonlinear subelliptic equations on the Heisenberg group are given by means of variational methods.

On Riesz Mean Inequalities for Subelliptic Laplacian

In this paper, we mainly focus on the Riesz means of eigenvalues of the subelliptic Laplacian on the Heisenberg group Hn. We establish a trace formula of associated eigenvalues, then we prove differential inequalities, difference inequalities and monotonicity formulas for the Riesz means of eigenvalues of the subelliptic Laplacian.

HARNACK'S INEQUALITY FOR GENERALIZED SUBELLIPTIC SCHRDINGER OPERATORS

We prove a uniform Harnack μu = 0, where △G is a sublaplacian, μ is scale-invariant Kato condition. inequality for nonnegative solutions of △u - a non-negative Radon measure and satisfying

Comparison Principles for Some Fully Nonlinear Sub-Elliptic Equations on the Heisenberg Group

In this paper,we prove a form of the strong comparison principle for a class of fully nonlinear subelliptic operators of the form■H^2,sψ+L(·,ψ,■Hψ)on the Heisenberg group,which include the CR invariant operators.

Quaternion H-type group and differential operatorΔ_λ

We study the relations between the quaternion H-type group and the boundary of the unit ball on the two-dimensional quaternionic space.The orthogonal projection of the space of square integrable functions defined on quaternion H-type group into its subspace of boundary values of q- holomorphic functions is considered.The precise form of Cauchy-Szeg(?)kernel and the orthogonal projection operator is obtained.The fundamental solution for the operatorΔ_λis found.

The Gevrey smoothing effect for the spatially inhomogeneous Boltzmann equations without cut-off

In this article we study the Gevrey regularization effect for the spatially inhomogeneous Boltzmann equation without angular cut-off.This equation is partially elliptic in the velocity direction and degenerates in the spatial variable.We consider the nonlinear Cauchy problem for the fluctuation around the Maxwellian distribution and prove that any solution with mild regularity will become smooth in the Gevrey class at positive time with the Gevrey index depending on the angular singularity.Our proof relies on the symbolic calculus for the collision operator and the global subelliptic estimate for the Cauchy problem of the linearized Boltzmann operator.

Poisson formulas for circular functions on some groups of type H Dedicated to Professor Sheng GONG on the occasion of his 75th birthday

Explicit Poisson kernels are found for the subelliptic Dirichlet problem with boundary data satisfying certain symmetry conditions on balls and halfspaces in some Heisenberg type groups.

MAXIMUM PRINCIPLES OF NONHOMOGENEOUS SUBELLIPTIC P-LAPLACE EQUATIONS AND APPLICATIONS

Maximum principles for weak solutions of nonhomogeneous subelliptic p-Laplace equations related to smooth vector fields {Xj} satisfying the Hoermander condition are proved by the choice of suitable test functions and the adaption of the classical Moser iteration method. Some applications are given in this paper.

Subelliptic Harmonic Maps with Values in Metric Spaces of Nonpositive Curvature

We prove the Holder continuity of a harmonic map from a domain of a sub-Riemannian manifold into a locally compact manifold with nonpositive curvature,and more generally into a non-positively curved metric space in the Alexandrov sense.

Local Uniqueness of Weak Solutions for a Class of Quasilinear Subelliptic Equations

In this note,we obtain some a-priori estimates for gradient of weak solutions to a class of subelliptic quasilinear equations constructed by Ho¨rmander’s vector fields,and then prove local uniqueness of weak solutions.A key ingredient is the estimated about kernel on metirc ＂annulus＂.