Weighted Hardy-Littlewood inequality for -harmonic tensors

In this paper we prove a local weighted integral inequality for conjugate -harmonic tensors similar to the Hardy and Littlewood integral inequality for conjugate harmonic functions. Then by using the local weighted integral inequality, we prove a global weighted integral inequality for conjugate -harmonic tensors in John domains.

     

Weighted integral inequalities for solutions of the A-harmonic equation

We prove the parametric versions of -weighted integral inequalities for differential forms satisfying the A-harmonic equation. These results can be considered as extensions of the classical inequalities for Sobolev functions.     

共轭 A -调和张量的双权积分不等式

该文引进一类新的权函数 $-A^{\lambda_{3}}_{r}(\lambda_{1},\lambda-{2},\Omega)$ -权, 证明了共轭$\cal A$ -调和张量的局部加权积分不等式.作为局部结果的应用, 证明了在有界区域$\Omega$中共轭$\cal A$ -调和张量的整体加权积分不等式.这些结果可看成是经典结果的推广.最后, 给出了上述结果在拟正则映射理论中的应用.     

Harnack inequality and Liouville properties for L-harmonic operator

For any complete manifold with nonnegative Bakry-Emery's Ricci curvature, we prove the gradient estimate of L-harmonic function. As application, we use this gradient estimate to deduce the localized version of the Harnack inequality for L-harmonic operator and some Liouville properties of positive or bounded L-harmonic function.     

Regularity for nonlinear elliptic systems with dini coefficients under natural growth condition for the case: 1 < m < 2

In this article, we consider interior regularity for weak solutions to nonlinear elliptic systems of divergence type with Dini continuous coefficients under natural growth condition for the case 1 < m < 2. All estimates in the case of m ≥ 2 is no longer suitable, and we can't obtain the Caccioppoli's second inequality by using these techniques developed in the case of m ≥ 2. But the Caccioppoli's second inequality is the key to use A-harmonic approximation method. Thus, we adopt another technique introduced by Acerbi and Fcsco to overcome the difficulty and we also overcome those difficulties due to Dini condition. And then we apply the A-harmonic approximation method to prove partial regularity of weak solutions.     

L(μ)-averaging domains

In this paper we first introduce L^s(μ)-averaging domains which are generalizations of L^s-averaging domains introduced by S.G. Staples. We characterize L^s(μ)-averaging domains using the quasihyperbolic metric. As applications, we prove norm inequalities for conjugate A-harmonic tensors in L^s(μ)-averaging domains which can be considered as generalizations of the Hardy and Littlewood theorem for conjugate harmonic functions. Finally, we give applications to quasiconformal and quasiregular mappings.     

共轭A-调和张量的双权Hardy-Littlewood不等式

本文证明了共轭A-调和张量的局部双权积分不等式,此结果类似于共轭调和函数的古典Hardy-Littlewood不等式.作为局部结果的应用,还证明了John域上的共轭A-调和张量的全局双权积分不等式.     

A r -Weighted Poincaré-Type Inequalities for Differential Forms in Some Domains

We prove local weighted integral inequalities for differential forms. Then byusing the local results, we prove global weighted integral inequalities for differential forms in L _s (μ)-averaging domains and in John domains, respectively, which can be considered as generalizations of the classical Poincaré-type inequality.     

Weak <Emphasis Type="Italic">WT</Emphasis><Subscript>2</Subscript>-class of differential forms and weakly <Emphasis Type="Italic">A</Emphasis>-harmonic tensors

In this paper, we give the definition of weak WT2-class of differential forms, and then obtain its weak reverse Holder inequality. As an application, we give an alternative proof of the higher integrability result of weakly A-harmonic tensors due to B. Stroffolini.     

Weighted Caccioppoli-type estimates and weak reverse Hölder inequalities for -harmonic tensors

We obtain a local weighted Caccioppoli-type estimate and prove the weighted version of the weak reverse Hölder inequality for -harmonic tensors.

     

N-Capacity, N-harmonic radius and N-harmonic transplantation

For the N-Laplacian ΔN on a regular domain the N-Green's function exists. This allows us to define the N-Robin's function and the N-harmonic radius. We show their basic properties and extend the method of the harmonic transplantation to that of the N-harmonic transplantation. The method is used to estimate the first eigenvalue of the N-Laplacian ΔN. We also give another proof of the isoperimetric inequality for the N-capacity and give the isoperimetric inequality for the N-harmonic radius.     

A regularity classification of boundary points for p-harmonic functions and quasiminimizers

In this paper it is shown that irregular boundary points for p-harmonic functions as well as for quasiminimizers can be divided into semiregular and strongly irregular points with vastly different boundary behaviour. This division is emphasized by a large number of characterizations of semiregular points. The results hold in complete metric spaces equipped with a doubling measure supporting a Poincaré inequality. They also apply to Cheeger p-harmonic functions and in the Euclidean setting to A-harmonic functions, with the usual assumptions on A.     

Constancy of p-harmonic maps of finite q-energy into non-positively curved manifolds

We investigate p-harmonic maps, p ≥ 2, from a complete non-compact manifold into a non-positively curved target. First, we establish a uniqueness result for the p-harmonic representative in the homotopy class of a constant map. Next, we derive a Caccioppoli inequality for the energy density of a p-harmonic map and we prove a companion Liouville type theorem, provided the domain manifold supports a Sobolev–Poincaré inequality. Finally, we obtain energy estimates for a p-harmonic map converging, with a certain speed, to a given point.      

Smooth metric measure spaces with weighted Poincaré inequality

In this paper, we study smooth metric measure space (M, g, e ^?f dv) satisfying a weighted Poincaré inequality and establish a rigidity theorem for such a space under a suitable Bakry–Émery curvature lower bound. We also consider the space of f-harmonic functions with finite energy and prove a structure theorem.     

The homogeneous dirichlet problem for quasilinear anisotropic degenerate parabolic-hyperbolic equation with L initial value

The aim of this paper is to prove the well-posedness (existence and uniqueness) of the L^p entropy solution to the homogeneous Dirichlet problems for the anisotropic degenerate parabolic-hyperbolic equations with L^p initial value. We use the device of doubling variables and some technical analysis to prove the uniqueness result. Moreover we can prove that the L^p entropy solution can be obtained as the limit of solutions of the corresponding regularized equations of nondegenerate parabolic type.     

The L resolvents of elliptic operators with uniformly continuous coefficients

We consider the strongly elliptic operator A of order 2m in the divergence form with bounded measurable coefficients and assume that the coefficients of top order are uniformly continuous. For 1<p<∞, A is a bounded linear operator from the L^p Sobolev space H^m,p into H^−m,p. We will prove that (A−λ)⁻¹ exists in H^−m,p for some λ and estimate its operator norm.     

On C classifications of hyperbolic vector fields

In this paper we study smooth classification of hyperbolic vector fields based on their linear approximations only and obtain the following. On Rⁿ, n?5, with only two kinds of exceptions, any two hyperbolic vector fields with generic nonlinear parts and where A_i are n×n matrices, are C¹ conjugate to each other if and only if A₁ and A₂ are strictly similar, and they are C¹ orbitally equivalent if and only if A₁ and A₂ are similar.     

Linear relations for p-harmonic functions

We discuss the p-harmonicity of the linear combination of p-harmonic functions in the Euclidean space and on a tree. If p≠2, the p-harmonicity is non-linear, i.e., the linear combination of p-harmonic functions need not be p-harmonic. In spite of this non-linear nature, we find some p-harmonic functions whose linear combinations become p-harmonic.     

Multiple solutions for semi-linear corner degenerate elliptic equations

The present paper is concerned with the existence of multiple solutions for semi-linear corner-degenerate elliptic equations with subcritical conditions. First, we introduce the corner type weighted p-Sobolev spaces and discuss the properties of continuous embedding, compactness and spectrum. Then, we prove the corner type Sobolev inequality and Poincaré inequality, which are important in the proof of the main result.