On the degenerate Beltrami equation

V. Gutlyanskiǐ; O. Martio; T. Sugawa; M. Vuorinen

doi:10.1090/S0002-9947-04-03708-0

On the degenerate Beltrami equation

V. Gutlyanskiǐ, O. Martio, T. Sugawa, M. Vuorinen

INF - Computer and Mathematical Sciences

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48 Citations (Scopus)

Abstract

We study the well-known Beltrami equation under the assumption that its measurable complex-valued coefficient μ(z) has the norm ∥μ∥ _∞ = 1. Sufficient conditions for the existence of a homeomorphic solution to the Beltrami equation on the Riemann sphere are given in terms of the directional dilatation coefficients of μ. A uniqueness theorem is also proved when the singular set Sing(μ) of μ is contained in a totally disconnected compact set with an additional thinness condition on Sing(μ).

Original language	English
Pages (from-to)	875-900
Number of pages	26
Journal	Transactions of the American Mathematical Society
Volume	357
Issue number	3
DOIs	https://doi.org/10.1090/S0002-9947-04-03708-0
Publication status	Published - 2005 Mar

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AU - Martio, O.

AU - Sugawa, T.

AU - Vuorinen, M.

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AB - We study the well-known Beltrami equation under the assumption that its measurable complex-valued coefficient μ(z) has the norm ∥μ∥ ∞ = 1. Sufficient conditions for the existence of a homeomorphic solution to the Beltrami equation on the Riemann sphere are given in terms of the directional dilatation coefficients of μ. A uniqueness theorem is also proved when the singular set Sing(μ) of μ is contained in a totally disconnected compact set with an additional thinness condition on Sing(μ).

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On the degenerate Beltrami equation

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