On ACL and hyper Q-homeomorphisms

D.A. Kovtoniuk (Institute of Applied Mathematics and Mechanics of NAS of Ukraine)

Abstract


We show that, if homeomorphism $$$f$$$ of domain $$$D \subset \mathbb{R}^n$$$, $$$n \geqslant 2$$$, is a hyper Q-homeomorphism with $$$Q \in L_{loc}^1$$$, then $$$f \in ACL$$$. As a consequence, such homeomorphism has partial derivatives almost everywhere and approximative differential. Besides, it is possible to prove that external dilatation $$$K_O(x, f)$$$ of mapping $$$f$$$ is almost everywhere majorated by function $$$Q^{n-1}$$$ and $$$f \in W_{loc}^{1,1}$$$.

References


Astala K., Iwaniec T., Koskela O., Martin J. "Mappings of BMO-bounded distortion",Math. Ann., 2000; 317: pp. 703-726. doi:10.1007/PL00004420

Bekkenbakh E., Bellman R. Inequalities, 1965. (in Russian)

Caraman P. $$$n$$$-Dimensional Quasiconformal Mappings, Abacus Press, Turnbridge Wells, 1974.

David G. "Solutions de l'equation de Beltrami avec $$$\| \mu \|_{\infty} = 1$$$", Ann. Acad. Sci. Fenn. Ser. AI. Math., 1988; 13(1): pp. 25-70.

Gehring F.W., Iwaniec T. "The limits of mappings with finite distortion", Ann. Acad. Sci. Fenn. Ser. AI. Math., 1999; 24: pp. 253-264.

Heinonen J., Koskela P. "Sobolev mappings with integrable dilatations", Arch. Ration. Mech. Anal., 1993; 125: pp. 81-97. doi:10.1007/BF00411478

Iwaniec T., Koskela P., Onninen J. "Mappings of finite distortion: compactness", Ann. Acad. Sci. Fenn. Ser. AI. Math., 2002; 27(2): pp. 391-417.

Iwaniec T., Koskela P., Onninen J. "Mappings of finite distortion: monotonicity and continuity", Invent. Math., 2001; 144(3): pp. 507-531. doi:10.1007/s002220100130

Iwaniec T., Sverak V. "On mappings with integrable dilatation", Proc. Amer. Math. Soc., 1993; 118(1): pp. 181-188. doi:10.2307/2160025

Kauhanen J., Koskela P., Maly J. "Mappings of finite distortion: discreteness and openness", Arch. Rat. Mech. Anal., 2001; 160: pp. 135-151. doi:10.1007/s002050100162

Kauhanen J., Koskela P., Maly J. "Mappings of finite distortion: condition N", Michigan Маth. J., 2001; 49: pp. 169-181. doi:10.1307/mmj/1008719040

Kovtonyuk D., Ryazanov V. "On mappings with finite hyperarea distortion", Trudy IPMM of NAS of Ukraine, 2004; 9: pp. 102-111.

Manfredi J.J., Villamor E. "Mappings with integrable dilatation in higher dimensions", Bull. Amer. Math. Soc., 1995; 32(2): pp. 235-240. doi:10.1090/S0273-0979-1995-00583-5

Manfredi J.J., Villamor E. "An extension of Reshetnyak's theorem", Indiana Univ. Math. J., 1998; 47(3): pp. 1131-1145.

Martio O., Ryazanov V., Srebro U. "To the theory of Q-homeomorphisms", Doklady RAN, 2001; 381(1): pp. 20-22. (in Russian)

Maz'ya V. Sobolev classes, Springer, Berlin-New York, 1985.

Rado T., Reichelderfer P.V. Continuous transformations in analysis, Springer, Berlin, 1955.

Saks S. Theory of the Integral, Dover Publ. Inc. New York, 1964.

Ziemer W.P. "Extremal length and conformal capacity", Trans. Amer. Math. Soc., 1967; 126(3): pp. 460-473. doi:10.2307/1994309




DOI: https://doi.org/10.15421/240713

  

Refbacks

  • There are currently no refbacks.


Copyright (c) 2007 D.A. Kovtoniuk

Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License.


Registered in


ISSN (Online): 2664-5009
ISSN (Print): 2664-4991
DNU