On the degenerate Beltrami equation

Gutlyanskiĭ, V.; Martio, O.; Sugawa, T.; Vuorinen, M.

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

On the degenerate Beltrami equation
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by V. Gutlyanskiĭ, O. Martio, T. Sugawa and M. Vuorinen PDF

Trans. Amer. Math. Soc. 357 (2005), 875-900 Request permission

Abstract:

We study the well-known Beltrami equation under the assumption that its measurable complex-valued coefficient $\mu (z)$ has the norm $\|\mu \|_\infty =1.$ 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 $\mu .$ A uniqueness theorem is also proved when the singular set $\operatorname {Sing} (\mu )$ of $\mu$ is contained in a totally disconnected compact set with an additional thinness condition on $\operatorname {Sing}(\mu ).$

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