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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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Abstract

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

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

Dates
First available in Project Euclid: 17 December 2008

Permanent link to this document
https://projecteuclid.org/euclid.pm/1229531050

Mathematical Reviews number (MathSciNet)
MR2474121

Zentralblatt MATH identifier
1189.30053

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

Keywords
Quasiconformal Hausdorff measure removability

Citation

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.https://projecteuclid.org/euclid.pm/1229531050


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