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A growth of harmonic functions in the whole space ℝn is examined. We found the estimate for a uniform norm of spherical harmonics in terms of the best approximation of harmonic function in the ball by harmonic polynomials. An approximation error of harmonic function in the ball is estimated by the maximum modulus of an entire harmonic function in space, as well as the maximum modulus of an entire harmonic function in space in terms of the maximum modulus of some entire function of one complex variable or the maximal term of its power series. These results allowed us to obtain the necessary and sufficient conditions under which a harmonic function in the ball of an n-dimensional space, n≥3, can be continued to the entire harmonic one. This result is formulated in terms of the best approximation of the given function by harmonic polynomials. In order to characterize growth of an entire harmonic function, we used the generalized and the lower generalized orders. Formulae for the generalized and the lower generalized orders of an entire harmonic function in space are expressed in terms of the approximation error by harmonic polynomials of the function that continues. We also investigated the growth of functions of slow increase. The obtained results are analogues to classical results, which are known for the entire functions of one complex variable.The conducted research is important due to the fact that the harmonic functions occupy a special place not only in many mathematical studies, but also when applying mathematical analysis to physics and mechanics, where these functions are often employed to describe various stationary processes

Let $(W,q, \mathcal{D})$ be a 4-dimensional Walker manifold. After providing a characterization and some examples for several special $(1,1)$-tensor fields on $(W,q, \mathcal{D})$, we prove that the proper almost complex structure $J$, introduced by Matsushita, is harmonic in the sense of Garcia-Rio et al. if and only if the almost Hermitian structure $(J,q)$ is almost Kahler. We classify all harmonic functions locally defined on $(W,q, \mathcal{D})$. We deal with the harmonicity of quadratic maps defined on $\mathbb{R}^4$ (endowed with a Walker metric $q$) to the $n$-dimensional semi-Euclidean space of index $r$, and then between local charts of two 4-dimensional Walker manifolds. We obtain here the necessary and sufficient conditions under which these maps are harmonic, horizontally weakly conformal, or harmonic morphisms with respect to $q$.

We define two transforms of non-conformal harmonic maps from a surface into the 3-sphere. With these transforms one can construct, from one such harmonic map, a sequence of harmonic maps. We show that there is a correspondence between harmonic maps into the 3-sphere, H-surfaces in Euclidean 3-space and almost complex surfaces in the nearly Kahler manifold . As a consequence we can construct sequences of H-surfaces and almost complex surfaces.

The present paper establishes connections between various subclasses of harmonic univalent functions by applying certain integral operator involving the generalized Bessel functions of first kind. To be more precise, we investigate such connections with harmonic starlike and harmonic convex mappings in the plane.

We prove that given any continuous data f on the harmonic boundary of a complete Riemannian manifold with image within a ball in the normal range, there exists a harmonic map from the manifold into the ball taking the same boundary value at each harmonic boundary point as that of f.

This study is concerned with univalent harmonic mappings on the unit disc of the complex plane. We examine a family of harmonic polynomials that have some interesting geometric properties, e.g. the boundary curve is concave except for cusps, that is dense in the family of harmonic mappings.

The article introduces substitutes for differential 1-forms on self-adjoint harmonic spaces by means of tensor products and energy norms. A representation using indicators of open sets connects 1-forms and topology. Finally, an orthogonal decomposition into exact and harmonic forms is provided, along with a description of harmonic forms in terms of locally harmonic functions.

We consider boundary value problems in a disk and in a ring for homogeneous equations with the Laplace operator of the first and second orders. Solutions are represented in terms of bases of harmonic wavelets in Hardy spaces of harmonic functions in a disk and in a ring, which were constructed earlier.

In this paper we study the distribution of values of harmonic functions in non-Euclidean disks. We introduce the notion of a P′-sequence, which enables us to characterize the class of normal harmonic functions defined in the unit disk. Moreover, we obtain sufficient conditions for the existence of such sequences and give examples which show that these conditions are essential in the stated theorems.

In this note, we introduce an approximation of harmonic maps via a sequence of exponentially harmonic maps. We then reestablish the existence theorem of harmonic maps due to Eells and Sampson.

We describe the asymptotic behavior of functions of the Royden p-algebra in terms of p-extremal length. We also prove that each bounded -harmonic function with finite energy on a complete Riemannian manifold is uniquely determined by the behavior of the function along p-almost every curve.

This paper investigates positive harmonic functions on domains that are complementary to a subset of a cylindrical surface. It characterizes, both in terms of harmonic measure and of a Wiener-type criterion, those domains that admit minimal harmonic functions with exponential growth. Illustrative examples are provided. Two applications are also given. The first of these concerns minimal harmonic functions associated with an irregular boundary point, and amplifies a recent construction of Gardiner and Hansen. The second concerns the possible non-approximability of positive harmonic functions by integrable positive harmonic functions.

This paper describes the solvability of Dirichlet problems for Laplace's equation when the boundary data is not smooth enough for the existence of a weak solution in H1Ω. Scales of spaces of harmonic functions and of boundary traces are defined and the solutions are characterized as limits of classical harmonic functions in special norms. The generalized harmonic functions, and their norms, are defined using series expansions involving harmonic Steklov eigenfunctions on the domain. It is shown that the usual trace operator has a continuous extension to an isometric isomorphism of specific spaces. This provides a characterization of the generalized solutions of harmonic Dirichlet problems. Numerical simulations of a model problem are described. This problem is related to the dewetting of thin films and the associated phenomenology is described.

Let S Ĥ be the family of all planar harmonic, univalent and sense-preserving mappings f=h+g¯ where h and g are analytic funtions in the open unit disk. The purpose of this article is to investigate connections between the theory of harmonic mappings in the plane and hypergeometric functions by applying certain planar harmonic convolution operators on various subclasses of S Ĥ . Dedicated to Professor Eveyln M. Silvia, late Professor of Mathematics, UC Davis February 8, 1948–January 21, 2006

We present two transforms of non-conformal harmonic maps from a surface into the 3-sphere. With these transforms one can construct from one non-conformal harmonic map a sequence of non-conformal harmonic maps. We also discuss the correspondence between non-conformal harmonic maps into the 3-sphere, H-surfaces in Euclidean 3-space and almost complex surfaces in the nearly Kähler manifold S3 × S3. ispartof: International Journal of Geometric Methods in Modern Physics vol:12 issue:8 pages:1560012-1560017 ispartof: location:SPAIN, Univ Granada, Granada status: published

The aim of this paper is to extend the notion of pseudo harmonic morphism introduced by Loubeau in [15] (see also [7, 4]) to the case when the source manifold is an admissible Riemannian polyhedron. We define these maps to be harmonic, as in [9], and pseudo-horizontally weakly conformal, see Sec. 3. We characterize them by means of germs of harmonic functions on the source polyhedron (see [13] for a precise definition) and germs of holomorphic functions on the Kähler target manifold, similarly to [15, 7].

A necessary and sufficient condition is given for a planar harmonic mapping f to be locally decomposable as a univalent harmonic mapping F of an analytic function φ.

In spite of the abundance of publications on harmonic mappings of man- ifolds, at present there exists neither a theory of harmonic diffeomorphisms, nor a definition of infinitesimal harmonic transformation of a Riemannian manifold, to say nothing of the theory of groups of such transformations. In the paper, this gap is partially filled, and a new subject of investigations is announced.

This paper introduces a family of area-type integrals over cones. These are used to investigate non-tangential boundary behaviour of harmonic functions on a half-space, extending results of Stein and Brossard.AMS 2000 Mathematics subject classification: Primary 31B25

This is a continuation of the Cambridge Tract “Harmonic maps between Riemannian polyhedra”, by J. Eells and the present author. The variational solution to the Dirichlet problem for harmonic maps with countinuous boundary data is shown to be continuous up to the boundary, and thereby uniquely determined. The domain space is a compact admissible Riemannian polyhedron with boundary, while the target can be, for example, a simply connected complete geodesic space of nonpositive Alexandrov curvature; alternatively, the target may have upper bounded curvature provided that the maps have a suitably small range. Essentially in the former setting it is further shown that a harmonic map pulls convex functions in the target back to subharmonic functions in the domain.

For a non-compact harmonic manifold M, we establish an integral formula for the derivative of a harmonic function on M. As an application we show that for the harmonic spaces having minimal horospheres, bounded harmonic functions are constant. The main result of this article states that the harmonic spaces having polynomial volume growth are flat. In other words, if the volume density function Θ of M has polynomial growth, then M is flat. This partially answers a question of Szabo namely, which density functions determine the metric of a harmonic manifold. Finally, we give some natural conditions which ensure polynomial growth of the volume function.

In this paper we prove that when the Ricci curvature of a Riemannian manifoldM n is almost nonnegative, and a ballB L (p)⊂M n is close in Gromov-Hausdorff distance to a Euclidean ball, then the gradient of the harmonic functionb defined in [ChCo1] does not vanish. In particular, these functions can serve as harmonic coordinates on balls sufficiently close to an Euclidean ball. The proof, is based on a monotonicity theorem that generalizes monotonicity of the frequency for harmonic functions onR n .

unavailable at this time... Mathematics Subject Classification (2000): Primary 31B05; Secondary 47A16 Quaestiones Mathematicae 25 (2002), 527-530

Let M be a Riemannian manifold with asymptotically non-negative curvature. We study the asymptotic behavior of the energy densities of a harmonic map and an exponentially harmonic function on M. We prove that the energy density of a bounded harmonic map vanishes at infinity when the target is a Cartan- Hadamard manifold. Also we prove that the energy density of a bounded exponentially harmonic function vanishes at infinity.

A family of sense–preserving complex valued harmonic polynomials that are close-to-convex on the open unit disk is introduced. By taking limits as the degree of the polynomials tends to infinity, a famiiy of close-to-convex harmonic mappings is also obtained. Similar results are considered for mappings of the punctured disk.

We give necessary and sufficient conditions for a conformal foliation locally generated by conformal vector fields to produce harmonic morphisms. Natural constructions of harmonic maps and morphisms are thus obtained. Also we obtain reducibility results for harmonic morphisms induced by (infinitesimal) conformal actions on Einstein manifolds.

We prove that the dimension of harmonic functions with finite Dirichlet integral is invariant under rough isometries between Riemannian manifolds satisfying the local conditions, expounded below. This result directly generalizes those of Kanai, of Grigor'yan, and of Holopainen. We also prove that the dimension of harmonic functions with finite Dirichlet integral is preserved under rough isometries between a Riemannian manifold satisfying the same local conditions and a graph of bounded degree; and between graphs of bounded degree. These results generalize those of Holopainen and Soardi, and of Soardi, respectively.

A harmonic morphism is a map between Riemannian manifolds which preserves Laplace's equation. We compare the properties of harmonic morphisms with those of the better known harmonic maps, seeing that they behave in some ways “dual” to the latter. In particular, we give representation theorems for harmonic morphisms in low dimensions which suggest that the equations might be soluble in some cases by integrable-system techniques in a similar way to that used in harmonic map theory. Bibliography: 38 titles.

In this paper, we consider the harmonic maps with potential from compact Riemannian manifold with boundary into a convex ball in any Riemannian manifold. We will establish some general properties such as the maximum principles, uniqueness and existence for these maps, and as an application of them, we derive existence and uniqueness result for the Dirichlet problem of the Landau-Lifshitz equations.

(1999). Boundary partial regularity for a class of harmonic maps. Communications in Partial Differential Equations: Vol. 24, No. 1-2, pp. 355-368.

Harmonic maps from B 1 (0, ℝ3) to a smooth compact target manifold N with uniformly small scaled energy (see assumption (2) below) are shown to be unique for their boundary values.

We analyze the boundary behavior of harmonic functions in a domain whose boundary is locally given by a graph of a Holder continuous function. In particular we give a non-probabilistic proof of a Harnack-type principle, due to Banuelos et al. and study some properties of the harmonic measure.

It is shown that a complete simply connected negatively curved manifold supports nontrivial bounded harmonic functions if the singular set of the ideal boundary is disconnected.

In this paper we prove a local weighted integral inequality for conjugate A A -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 A A -harmonic tensors in John domains.

Given a complete manifold with a particular structure at infinity, we give the dimension of the space of harmonic functions with prescribed polynomial growth.

A necessary and sufficient condition is found for a complex-valued harmonic function to be decomposable as an analytic function followed by a univalent harmonic mapping. In the theory of quasiconformal mappings, it is proved that for any measurable function 1a with 11,uII ,, 0 there. According to a theorem of Lewy [3], the Jacobian of a univalent harmonic map in the plane can never vanish, so J(z) > 0 for sense-preserving univalent harmonic maps. It is instructive to begin with two simple examples. Example 1. Let f be the harmonic polynomial f(z) = z2 +3z Then f has dilatation a(z) = z, and f is sense-preserving in the unit disk ID = {z IzI < 1}. We claim that f has no decomposition of the desired form in any neighborhood of the origin. Suppose on the contrary that f = F o p, where (p is analytic near the origin and F is harmonic and univalent on the range of p. Then F is sensepreserving because f is. We may suppose without loss of generality that p(0) = 0. Received by the editors October 10, 1994. 1991 Mathematics Subject Classification. Primary 30C99; Secondary 31A05, 30C65.

A $p$ times continuously differentiable complex-valued function $F=u+iv$ in a domain $D\subseteq ℂ$ is $p$ -harmonic if $F$ satisfies the $p$ -harmonic equation $\mathrm{\Delta }\cdots \mathrm{\Delta }F=0$ , where $p$ is a positive integer. By using the generalized Salagean differential operator, we introduce a class of $p$ -harmonic functions and investigate necessary and sufficient coefficient conditions, distortion bounds, extreme points, and convex combination of the class.

For Riemannian manifoldsMandN, admitting a submersionϕwith compact fibres, we introduce the projection of a function via its decomposition into horizontal and vertical components. By comparing the Laplacians onMandN, we determine conditions under which a harmonic function onU=ϕ−1(V)⊂Mprojects down, via its horizontal component, to a harmonic function onV⊂N.

Publisher Summary This chapter discusses fundamental matters such as the Dirichlet energy and tension tensor of a unit tangent vector field on a Riemannian manifold, first and second variation formulae, and the harmonic vector fields system. The study of the weak solutions to this system (existence and local properties) is missing from the present day mathematical literature. Various instances are investigated where harmonic vector fields occur and to generalizations. Any unit vector field that is a harmonic map is also a harmonic vector field. The study of harmonic map system is more appropriate on a Hermitian manifold and that results in Hermitian harmonic maps to be useful in studying rigidity of complete Hermitian manifolds.

It is shown that the space hp(D,X) has the Kadec-Klee property with respect to pointwise norm convergence in the Banach space X if and only if X has the Radon-Nikodym property and every point of the unit sphere of X is a denting point of the unit ball of X. In addition, it is shown that hp(D,X) is locally uniformly rotund if and only if X is locally uniformly rotund and has the Radon-Nikodym property.