Quasiconformal harmonic functions between convex domains

Quasiconformal Harmonic Functions between Convex Domains

We generalize Martio’s paper [14]. Indeed the problem studied in this paper is under which conditions on a homeomorphism f between the unit circle S1 := {z : |z| = 1} and a fix convex Jordan curve γ the harmonic extension of f is a quasiconformal mapping. In addition, we give some results for some classes of harmonic diffeomorphisms. Further, we give some results concerning harmonic quasiconfor...

Convex functions and quasiconformal mappings

Continuing our investigation of quasiconformal mappings with convex potentials, we obtain a new characterization of quasiuniformly convex functions and improve our earlier results on the existence of quasiconformal mappings with prescribed sets of singularities.

Harmonic Measure in Convex Domains

Introduction. Let Q be an open, convex subset of R^. At almost every point x of dQ, with respect to surface measure da, there is a unique outer unit normal 6. The map g: dQ -* S given by g(x) = 9 and defined almost everywhere is called the Gauss map. (S, n = N 1, is the unit sphere in R^.) Suppose that the origin 0 belongs to Q. Harmonic measure for fit at 0 is the probability measure co such t...

Certain Convex Harmonic Functions

On Quasiconformal Harmonic Maps between Surfaces

It is proved the following theorem, if w is a quasiconformal harmonic mappings between two Riemann surfaces with smooth boundary and aproximate analytic metric, then w is a quasi-isometry with respect to Euclidean metric.