On subelliptic harmonic maps with potential

The Harmonic Gbc Function Map is a Bijection If the Target Domain is Convex

In this article we consider classes of harmonic and subharmonic functions introduced with using integral operators Riman-Liouville by Professor M. Djrbashyan when α > 0. These classes are significant generalizations of already well known classes of harmonic and subharmonic functions match up with them only in a particular case. In our article we consider angular and chordal limits of harmonic and subharmonic functions got by using Riman-Liouville integral operators. A set of the points at which, probably, these limits don’t exist are characterized by using a linear measure of zero.

The aim of this paper is twofold. First, we investigate the properties of the composition of harmonic mappings with harmonic mappings, and the composition of biharmonic mappings with harmonic mappings. Second, we consider the Goodman-Saff conjecture for biharmonic mappings in the unit disk. In fact, we show that the answer to the Goodman-Saff conjecture is positive for a special class of univalently biharmonic mappings which contains the set of all harmonic univalent mappings.

This survey paper deals with the unitary treatment of some classes of linear partial differential equations on Klein surfaces, which are the most general two-manifolds that support harmonic functions. We are mainly concerned with the study of harmonic functions with Dirichlet or Neumann boundary conditions. In such a way, the present paper extends several classical results to the abstract setting of dianalytic manifolds of complex dimension 1. We bring together systematically and concisely the Green and Neumann functions, the harmonic kernel function, and the harmonic measure on Klein surfaces. The technique to extend these concepts is to apply the classical methods to the complex double of a Klein surface, which is a Riemann surface endowed with an antianalytic involution. The existence of a harmonic function on a Riemann surface is given by the property that a harmonic function remains invariant under biholomorphic mappings. Similar problems on Riemann surfaces were developed by L.V. Ahlfors and L. Sario. The analysis developed in this paper offers perspectives to the qualitative analysis of other classes of linear or nonlinear elliptic equations on Klein surfaces.

ABSTRACTIn this work, first, we study the existence of ‐harmonic maps between Riemannian manifolds whose codomain admits a conformal vector field or a strictly convex function. Then, Liouville theorems for Sacks–Uhlenbeck ‐harmonic maps from a noncompact complete Riemannian manifold are given. Lastly, the instability of Sacks–Uhlenbeck ‐harmonic maps from (or to) a sphere and a product of spheres is explored.

The Oshima–Sekiguchi and Liouville theorems on Heintze groups

In this paper, we investigate maximum principle, convergence of sequences and angular limits for harmonic Bloch mappings. First, we give the maximum principle of harmonic Bloch mappings, which is a generalization of the classical maximum principle for harmonic mappings. Second, by using the maximum principle of harmonic Bloch mappings, we investigate the convergence of sequences for harmonic Bloch mappings. Finally, we discuss the angular limits of harmonic Bloch mappings. We show that the asymptotic values and angular limits are identical for harmonic Bloch mappings, and we further prove a result that applies also if there is no asymptotic value. A sufficient condition for a harmonic Bloch mapping has a finite angular limit is also given.

Harmonic maps between pseudo-Riemannian surfaces

In this note we prove that the maximum principle of Jager–Kaul for harmonic maps holds for a more general class of maps, $$V$$ -harmonic maps. This includes Hermitian harmonic maps (Jost and Yau, Acta Math 170:221–254, 1993), Weyl harmonic maps (Kokarev, Proc Lond Math Soc 99:168–194, 2009), affine harmonic maps Jost and Simsir (Analysis (Munich) 29:185–197, 2009), and Finsler maps from a Finsler manifold into a Riemannian manifold. With this maximum principle we establish the existence of $$V$$ -harmonic maps into regular balls.

We prove existence and uniqueness of a solution of the Dirichlet problem for separately $(\alpha, \beta)$-harmonic functions on $\mathbb D^n$ with boundary data in $C(\mathbb T^n)$ using $(\alpha, \beta)$-Poisson kernel $P_{\alpha, \beta} (z, \zeta)$. A characterization by hypergeometric functions of separately $(\alpha, \beta)$-harmonic functions which are also $m$-homogeneous is given, it is used to obtain series expansion of separately $(\alpha, \beta)$-harmonic functions. Basic $H^p$ theory of such functions is developed: integral representations by measures and $L^p$ functions on $\mathbb T^n$, norm and weak$^\ast$ convergence at the distinguished boundary $\mathbb T^n$. Weak $(1,1)$-type estimate for a restricted non-tangential maximal function $M_{A, B}^{\mathrm{NT}}$ is derived. We show that slice functions $u(z_1, …, z_k, \zeta_{k+1}, …, \zeta_n)$, where some of the variables are fixed, belong in the appropriate space of separately $(\alpha', \beta')$-harmonic functions of $k$ variables. We prove a Fatou type theorem on a. e. existence of restricted non-tangential limits for these functions and a corresponding result for unrestricted limit at a point in $\mathbb T^n$. Our results extend earlier results for $(\alpha, \beta)$-harmonic functions in the disc and for $n$-harmonic functions in $\mathbb D^n$.

Lengths, areas and Lipschitz-type spaces of planar harmonic mappings

Suppose $$p\ge 1$$ , $$w=P[F]$$ is a harmonic mapping of the unit disk $${\mathbb D}$$ satisfying F is absolutely continuous and $${\dot{F}}\in L^p(0, 2\pi )$$ , where $${\dot{F}}(e^{it})=\frac{\mathrm {d}}{\mathrm {d}t}F(e^{it})$$ . In this paper, we obtain Bergman norm estimates of the partial derivatives for w, i.e., $$\Vert w_z\Vert _{L^p}$$ and $$\Vert \overline{w_{{\bar{z}}}}\Vert _{L^p}$$ , where $$1\le p<2$$ . Furthermore, if w is a harmonic quasiregular mapping of $${\mathbb {D}}$$ , then we show that $$w_z$$ and $$\overline{w_{{\bar{z}}}}$$ are in the Hardy space $$H^p$$ , where $$1\le p\le \infty $$ . The corresponding Hardy norm estimates, $$\Vert w_z\Vert _{p}$$ and $$\Vert \overline{w_{{\bar{z}}}}\Vert _{p}$$ , are also obtained.

We establish existence and uniqueness theorems for V-harmonic maps from complete noncompact manifolds. This class of maps includes Hermitian harmonic maps, Weyl harmonic maps, affine harmonic maps, and Finsler harmonic maps from a Finsler manifold into a Riemannian manifold. We also obtain a Liouville type theorem for V-harmonic maps. In addition, we prove a V-Laplacian comparison theorem under the Bakry-Emery Ricci condition.

Haptic devices allow a user to feel either reaction forces from virtual interactions or reaction forces reflected from a remote site during a bilateral teleoperation task. Also, guiding forces can be exerted to train the user in the performance of a virtual task or to assist him/her to safely teleoperate a robot. The generation of guiding forces relies on the existence of a motion plan that provides the direction to be followed to reach the goal from any free configuration of the configuration space ( ${\mathcal C}$ -space). This paper proposes a method to obtain such a plan that interleaves a sampling-based exploration of ${\mathcal C}$ -space with an efficient computation of harmonic functions. A deterministic sampling sequence (with a bias based on harmonic function values) is used to obtain a hierarchical cell decomposition model of ${\mathcal C}$ -space. A harmonic function is iteratively computed over the partially known model using a novel approach. The harmonic function is the navigation function used as motion plan. The approach has been implemented in a planner (called the Kautham planner) that, given an initial and a goal configuration, provides: (a) a channel of cells connecting the cell that contains the initial configuration with the cell that contains the goal configuration; (b) two harmonic functions over the whole ${\mathcal C}$ -space, one that guides motions towards the channel and another that guides motions within the channel towards the goal; and (c) a path computed over a roadmap built with the free samples of the channel. The harmonic functions and the solution path are then used to generate the guiding forces for the haptic device. The planning approach is illustrated with examples on 2D and 3D workspaces.

We lay the foundations of Fatou theory in one and several complex variables. We describe the main contributions contained in E. M. Stein’s book Boundary Behavior of Holomorphic Functions, published in 1972 and still a source of inspiration. We also give an account of his contributions to the study of the boundary behavior of harmonic functions. The point of this paper is not simply to exposit well-known ideas. Rather, we completely reorganize the subject in order to bring out the profound contributions of E. M. Stein to the study of the boundary behavior both of holomorphic and harmonic functions in one and several variables. In an appendix, we provide a self-contained proof of a new result which is relevant to the differentiation of integrals, a topic which, as witnessed in Stein’s work, and especially by the aforementioned book, has deep connections with the boundary behavior of harmonic and holomorphic functions.