On the existence and uniqueness theorem for the degenerate Beltrami equation

: Vuorinen M., Gutlyanskij V., Martio O., Sugawa T.

: 2004

: Доклады Академии Наук / Doklady Biological Sciences

: Doklady Akademii Nauk

: 393

: 1

: 7

: 9

: 3

: 0869-5652

: http://api.elsevier.com/content/abstract/scopus_id:4644232974

The analytical theory of quasi-conformal mappings on a complex plane C implies investigating homeomorphic generalized solutions of Beltrami equation (BE) with a measurable complex-valued coefficient μ. In degenerate case, if |μ(z)|<1 for nearly all z∈C and ∥μ∥=ess sup |μ(z)|=1, BE may have no homeomorphic solutions, and in the case of such solution existence it can be non-unique. For degenerate BE the new existence and uniqueness theorems are established, in which, along with |μ(z)|, behavior of arg μ(z) also plays significant role.