Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 40, 2015, 3-15

ABSOLUTE CONTINUITY OF MAPPINGS WITH FINITE GEOMETRIC DISTORTION

Ville Tengvall

University of Jyväskylä, Department of Mathematics and Statistics
P.O. Box 35 (MaD), FI-40014 University of Jyväskylä, Finland; ville.tengvall 'at' jyu.fi

Abstract. Suppose that Ω ⊂ Rⁿ is a domain with n ≥ 2. We show that a continuous, sense-preserving, open and discrete mapping of finite geometric outer distortion with K_O(⋅,f) ∈ L_loc^1/(n-1)(Ω) is absolutely continuous on almost every line parallel to the coordinate axes. Moreover, if U ⊂ Ω is an open set with N(f,U) < ∞, then f satisfies the distortion inequality

|Df(x)|ⁿ ≤ C|J(x,f)|K_O(x,f)

for almost every x ∈ U, where the constant C > 0 depends only on n and on the maximum multiplicity N(f,U).

2010 Mathematics Subject Classification: Primary 30C62, 30C65.

Key words: Mappings of finite distortion, moduli inequalities, Q-mappings.

Reference to this article: V. Tengvall: Absolute continuity of mappings with finite geometric distortion. Ann. Acad. Sci. Fenn. Math. 40 (2015), 3-15.

Full document as PDF file

doi:10.5186/aasfm.2015.4018