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Beltrami equations with coefficient in the Sobolev space $W^{1,p}$

A. Clop, D. Faraco, J. Mateu, J. Orobitg, and X. Zhong

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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.

Article information

Publ. Mat. Volume 53, Number 1 (2009), 197-230.

First available in Project Euclid: 17 December 2008

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 30C62: Quasiconformal mappings in the plane 35J15: Second-order elliptic equations 35J70: Degenerate elliptic equations

Quasiconformal Hausdorff measure removability


Clop, A.; Faraco, D.; Mateu, J.; Orobitg, J.; Zhong, X. Beltrami equations with coefficient in the Sobolev space $W^{1,p}$. Publ. Mat. 53 (2009), no. 1, 197--230.

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