# American Institute of Mathematical Sciences

September  2010, 9(5): 1391-1397. doi: 10.3934/cpaa.2010.9.1391

## Planar ACL-homeomorphisms : Critical points of their components

 1 Dipartimento di Matematica e Appl. “R. Caccioppoli”, Via Cintia- Monte S.Angelo, 80126 Napoli, Italy, Italy, Italy

Received  September 2009 Revised  October 2009 Published  May 2010

We study planar homeomorphisms $f: \Omega\subset R^2$ onto $\to \Omega' \subset R^2$, $f=(u,v)$, which are absolutely continuous on lines parallel to the axes (ACL) together with their inverse $f^{-1}$. The main result is that $u$ and $v$ have almost everywhere the same critical points. This generalizes a previous result ([6]) concerning bisobolev mappings. Moreover we construct an example of a planar ACL-homeomorphism not belonging to the Sobolev class $W_{l o c}^{1,1}$.
Citation: Gioconda Moscariello, Antonia Passarelli di Napoli, Carlo Sbordone. Planar ACL-homeomorphisms : Critical points of their components. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1391-1397. doi: 10.3934/cpaa.2010.9.1391
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