Landau’s theorem for log-p-harmonic mappings

The main aim of this paper is to prove the existence of Landau-Bloch constant for log-p-harmonic mappings.     

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.     

Weighted integral inequalities for conjugate A-harmonic tensors

We first prove a local weighted integral inequality for conjugate A-harmonic tensors. Then, as an application of our local result, we prove a global weighted integral inequality for conjugate A-harmonic tensors in L^s(μ)-averaging domains, which can be considered as a generalization of the classical result. Finally, we give applications of the above results to quasiregular mappings.     

Gap Phenomena for p-Harmonic Maps

In this paper, we generalize to p-harmonic mapssome gap results known for harmonic maps. In particular, we prove that,under a certain level of energy depending on the curvature of the domainand target manifolds, the only p-harmonic maps are theconstant ones. The main tools are Bochner–Weitzenböck andReilly-type formulas involving the p-Laplace operator.     

Distortion theorems for Bloch mappings on the unit polydisc D

In this article, we establish distortion theorems for some various subfamilies of Bloch mappings defined in the unit polydisc Dⁿ with critical points, which extend the results of Liu and Minda to higher dimensions. We obtain lower bounds of |det (f'(z))| and ? det (f'(z)) for Bloch mapping f. As an application, some lower and upper bounds of Bloch constants for the subfamilies of holomorphic mappings are given.     

Integrability of rotationally symmetric n-harmonic maps

A rotationally symmetric n-harmonic map is a rotationally symmetric p-harmonic map between two n-dimensional model spaces such that p=n. We show that rotationally symmetric n-harmonic maps can be integrated and are n-harmonic diffeomorphism, and apply such results to investigate the asymptotic behaviors of these maps. We also derive this integrability using Lie theory.     

On the minimization of an energy functional

The author studies the minimization of an energy functional which is introduced in the study of ferromagnets and antiferromagnets. It is proved that the map u₀=(u₀₁,u₀₂,0) is a minimizer by setting up the uniform gradient estimate where (u₀₁,u₀₂) is a p-harmonic map.     

Rough Isometry and <Emphasis Type="Italic">p</Emphasis>-Harmonic Boundaries of Complete Riemannian Manifolds

In this paper, we describe the behavior of bounded energy finite solutions for certain nonlinear elliptic operators on a complete Riemannian manifold in terms of its p-harmonic boundary. We also prove that if two complete Riemannian manifolds are roughly isometric to each other, then their p-harmonic boundaries are homeomorphic to each other. In the case, there is a one to one correspondence between the sets of bounded energy finite solutions on such manifolds. In particular, in the case of the Laplacian, it becomes a linear isomorphism between the spaces of bounded harmonic functions with finite Dirichlet integral on the manifolds. This work was supported by grant No. R06-2002-012-01001-0(2002) from the Basic Research Program of the Korea Science & Engineering Foundation.     

A Theorem of Liouville Type for p-Harmonic Morphisms

In this article we prove a Liouville type theorem for p-harmonic morphisms. We show that if : MNis a p-harmonic morphism (p2) from a complete noncompact Riemannian manifold Mof nonnegative Ricci curvature into a Riemannian manifold Nof nonpositive scalar curvature such that the p-energy E _p (), or (2p–2)-energy E _2p–2() is finite, then is constant.     

The method of p -harmonic approximation and optimal interior partial regularity for energy minimizing p -harmonic maps under the controllable growth condition

In this paper, we are concerned with the partial regularity for the weak solutions of energy minimizing p-harmonic maps under the controllable growth condition. We get the interior partial regularity by the p-harmonic approximation method together with the technique used to get the decay estimation on some Degenerate elliptic equations and the obstacle problem by Tan and Yan. In particular, we directly get the optimal regularity. This work was partially supported by the National Natural Science Foundation of China (Grant No. 10531020) and the Program of 985 Innovation Engineering on Information in Xiamen University (2004–2007).     

Bloch constant of holomorphic mappings on the unit polydisk of ℂ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

In this paper, we give a definition of Bloch mappings defined in the unit polydisk D ⁿ , which generalizes the concept of Bloch functions defined in the unit disk D. It is known that Bloch theorem fails unless we have some restrictive assumption on holomorphic mappings in several complex variables. We shall establish the corresponding distortion theorems for subfamilies β(K) and β _loc(K) of Bloch mappings defined in the polydisk D ⁿ , which extend the distortion theorems of Liu and Minda to higher dimensions. As an application, we obtain lower and upper bounds of Bloch constants for various subfamilies of Bloch mappings defined in D ⁿ . In particular, our results reduce to the classical results of Ahlfors and Landau when n = 1. This work was supported by the National Natural Science Foundation of China (Grant No. 10571164) and Specialized Research Fund for the Doctoral Program of Higher Education (SRFDP) (Grant No. 20050358052)     

Boundary Harnack Principle for <Emphasis Type="Italic">p</Emphasis>-harmonic Functions in Smooth Euclidean Domains

We establish a scale-invariant version of the boundary Harnack principle for p-harmonic functions in Euclidean C ^1,1-domains and obtain estimates for the decay rates of positive p-harmonic functions vanishing on a segment of the boundary in terms of the distance to the boundary. We use these estimates to study the behavior of conformal Martin kernel functions and positive p-superharmonic functions near the boundary of the domain. H. A. was partially supported by Grant-in-Aid for (B) (2) (No. 15340046) Japan Society for the Promotion of Science. N. S. was partially supported by NSF grant DMS-0355027. X. Z. was partially supported by the Taft foundation.     

Gradient estimates for positive smooth f-harmonic functions

For Riemannian manifolds with a measure, we study the gradient estimates for positive smooth f-harmonic functions when the ∞-Bakry-Emery Ricci tensor and Ricci tensor are bounded from below, generalizing the classical ones of Yau (i.e., when f is constant).     

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.     

On rubin's harmonic analysis and its related positive definite functions

In this paper, a new formulation of the Rubin's q-translation is given, which leads to a reliable q-harmonic analysis. Next, related q-positive definite functions are introduced and studied, and a Bochner's theorem is proved.     

On ε quasi-convex mappings in the unit ball of a complex Banach space

In this paper, a new class of biholomorphic mappings named “ε quasi-convex mapping” is introduced in the unit ball of a complex Banach space. Meanwhile, the definition of ε-starlike mapping is generalized from ε∈[0,1] to ε∈[−1,1]. It is proved that the class of ε quasi-convex mappings is a proper subset of the class of starlike mappings and contains the class of ε starlike mappings properly for some ε∈[−1,0)∪(0,1]. We give a geometric explanation for ε-starlike mapping with ε∈[−1,1] and prove that the generalized Roper-Suffridge extension operator preserves the biholomorphic ε starlikeness on some domains in Banach spaces for ε∈[−1,1]. We also give some concrete examples of ε quasi-convex mappings or ε starlike mappings for ε∈[−1,1] in Banach spaces or Cn. Furthermore, some other properties of ε quasi-convex mapping or ε-starlike mapping are obtained. These results generalize the related works of some authors.     

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.      

Classification and Liouville type theorems for <Emphasis Type="Italic">p</Emphasis>-harmonic morphisms

We prove firstly the classification theorem for p-harmonic morphisms between Euclidean domains. Secondly, we show that if is a p-harmonic morphism (p ≥ 2) from a complete Riemannian manifold M of nonnegative Ricci curvature into a Riemannian manifold N of non-positive scalar curvature such that the L ^q -energy is finite, then is constant, which improve the corresponding result due to G. Choi, G. Yun in (Geometriae Dedicata 101 (2003), 53–59).      

Liouville type theorems for p-harmonic maps

Let M be a complete Riemannian manifold and let N be a Riemannian manifold of non-positive sectional curvature. Assume that at all x∈M and at some point x₀∈M, where μ₀>0 is the least eigenvalue of the Laplacian acting on L²-functions on M. Let 2?q?p. Then any q-harmonic map of finite q-energy is constant. Moreover, if N is a Riemannian manifold of non-positive scalar curvature, then any q-harmonic morphism of finite q-energy is constant.     

The generalized Roper-Suffridge extension operator in Banach spaces (II)

In this paper, we generalize the Roper-Suffridge extension operator from Cn to Banach spaces. It is proved that this operator preserves the biholomorphic ? starlikeness on some domains in Banach spaces. From these, we may construct a lots of concrete examples about biholomorphic ? starlike mappings on some domains Ω in Cn, or Hilbert spaces, or Banach spaces from univalent ? starlike functions on the unit disc U in C. Meanwhile, the growth theorems of the corresponding mappings are given. Some results of Gong and Liu, Roper and Suffridge, Graham et al. in Cn are extended to Hilbert spaces or Banach spaces.