Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 44, 2019, 797-803

QUASICONFORMAL MAPPINGS WITH CONTROLLED LAPLACIAN AND HÖLDER CONTINUITY

David Kalaj and Arsen Zlatičanin

School of Mathematical Sciences, Huaqiao University, Quanzhou 362021, P.R. China
Permanent position: University of Montenegro, Faculty of Natural Sciences and Mathematics
Dzordza Vasingtona b.b. 81000 Podgorica, Montenegro; davidk 'at' ucg.ac.me

University Luigj Gurakuqi, Department of Mathematics
Shkodra, Albania; arsen_zn 'at' yahoo.fr

Abstract. We prove that every K-quasiconformal mapping w of the unit ball B ⊂ Rⁿ, n ≥ 2 onto a C²-Jordan domain Ω is Hölder continuous with constant α = 2 – n/p, provided its weak Laplacian Δw is in L^p(Bⁿ) for some n/2 < p < n. In particular it is Hölder continuous for every 0 < α < 1 provided that Δw ∈ Lⁿ(Bⁿ). Finally for p > n, we prove that w is Lipschitz continuous, a result, whose proof has been already sketched in [16] by the first author and Saksman. The paper contains the proofs of some results announced in [17].

2010 Mathematics Subject Classification: Primary 30C65.

Key words: Lipschitz continuity, Poisson equation.

Reference to this article: D. Kalaj and A. Zlatičanin: Quasiconformal mappings with controlled Laplacian and Hölder continuity. Ann. Acad. Sci. Fenn. Math. 44 (2019), 797-803.

Full document as PDF file

https://doi.org/10.5186/aasfm.2019.4440