TY - JOUR
T1 - Beltrami equations with coefficient in the sobolev space W1, p
AU - Clop, A.
AU - Faraco, D.
AU - Mateu, J.
AU - Orobitg, J.
AU - Zhong, X.
PY - 2009/1/1
Y1 - 2009/1/1
N2 - We study the removable singularities for solutions to the Beltrami equation ∂̄f = μ∂f, where μ is a bounded function, ∥μ∥∞ ≤ K-1/K+1 < 1, and such that μ ∈ W1, p for some p ≤ 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 μ ∈ W1, p, 2K 2/K2 + 1 < p ≤ 2, preserve compact sets of σ-finite length and vanishing analytic capacity, even though they need not be bilipschitz.
AB - We study the removable singularities for solutions to the Beltrami equation ∂̄f = μ∂f, where μ is a bounded function, ∥μ∥∞ ≤ K-1/K+1 < 1, and such that μ ∈ W1, p for some p ≤ 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 μ ∈ W1, p, 2K 2/K2 + 1 < p ≤ 2, preserve compact sets of σ-finite length and vanishing analytic capacity, even though they need not be bilipschitz.
KW - Hausdorff measure
KW - Quasiconformal
KW - Removability
U2 - https://doi.org/10.5565/PUBLMAT_53109_09
DO - https://doi.org/10.5565/PUBLMAT_53109_09
M3 - Article
SN - 0214-1493
VL - 53
SP - 197
EP - 230
JO - Publicacions Matematiques
JF - Publicacions Matematiques
ER -