In communication between V. Zorich and the author, the
question was asked to find examples of harmonic gradient mapping from the unit ball onto itself in 3-space. For example, if $u=x^2 +y^2 - 2 z^2$, then $f=
\nabla u=(2x,2y,-4z)$ is injective harmonic gradient mapping from
$\mathbb{B}^3$ onto the ellipsoid.
If $u$ is real-valued function such that $f= \nabla
u=(x,y,z)$, then $u= x^2/2 +y^2/2 +z^2/2 +c$.
In particular, $Id$ is not harmonic gradient mapping.
In complex plane, if $u$ is real-valued harmonic function, then $u_z=\frac{1}{2}(u'_x -u'_y)$ is analytic function and therefore
$ \nabla u=\overline{F}$, where $F= 2 u_z$ is analytic function.
We consider a particular case.
In the paper Harmonic quasiconformal self-mappings and Möbius transformations of the unit ball (David Kalaj Miodrag Mateljević), Pacific Journal of Mathematics .01/2010; 247(2). DOI: 10.2140/pjm.2010.247.389, see
4.2.Perturbation of the identity.
We show that small smooth perturbations of
the boundary value of a holomorphic automorphism
of the unit ball
onto itself induce harmonic quasiconformal mappings.
ABSTRACT Plane harmonic mappings and their connections to quasiconformal mappings have recently been extensively studied, see [P. Duren, Harmonic mappings in the plane. Cambridge Tracts in Mathematics 156. Cambridge: Cambridge University Press (2004; Zbl 1055.31001)] and [D. Kalaj, Math. Z. 260, No. 2, 237–252 (2008; Zbl 1151.30014)]. Little is known of the situation in higher dimensions. The authors prove a regularity result for small K: If f is a K-quasiconformal harmonic mapping of the unit ball B n ,n>2, onto itself, then f is bi-Lipschitz provided that K
- Badges
- Science topic
- Similar topics
- Mathematics