Open Access
2009 Beltrami equations with coefficient in the Sobolev space $W^{1,p}$
A. Clop, D. Faraco, J. Mateu, J. Orobitg, X. Zhong
Publ. Mat. 53(1): 197-230 (2009).


We study the removable singularities for solutions to the Beltrami equation $\overline\partial f=\mu\, \partial f$, where $\mu$ is a bounded function, $\|\mu\|_\infty\leq\frac{K-1}{K+1}<1$, and such that $\mu\in W^{1,p}$ for some $p\leq 2$. Our results are based on an extended version of the well known Weyl's lemma, asserting that distributional solutions are actually true solutions. Our main result is that quasiconformal mappings with compactly supported Beltrami coefficient $\mu\in W^{1,p}$, $\frac{2K^2}{K^2+1}<p\leq 2$, preserve compact sets of $\sigma$-finite length and vanishing analytic capacity, even though they need not be bilipschitz.


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A. Clop. D. Faraco. J. Mateu. J. Orobitg. X. Zhong. "Beltrami equations with coefficient in the Sobolev space $W^{1,p}$." Publ. Mat. 53 (1) 197 - 230, 2009.


Published: 2009
First available in Project Euclid: 17 December 2008

zbMATH: 1189.30053
MathSciNet: MR2474121

Primary: 30C62 , 35J15 , 35J70

Keywords: Hausdorff measure , Quasiconformal , Removability

Rights: Copyright © 2009 Universitat Autònoma de Barcelona, Departament de Matemàtiques

Vol.53 • No. 1 • 2009
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