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Nonlinear quasiconformal glue theorems

  • Wen Guo-chun
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 1039)

Abstract

Let Γ0, Γ1, …, ΓN, L1, …, LM be the boundary contours of an (N + M + 1)-connected plane domain D, where Γ1, …, ΓN, L1, …, LM are situated inside Γ0. In D there are some mutually exclusive contours γ1,…,γn,11,…,1m. We assume that and denote where Open image in new window and Open image in new window are the domains surrounded by γj and 1j, respectively. We deal with the nonlinear uniformly elliptic complex equation of the first order Open image in new window , F = Q (z, w, wz)wz, zεD. We suppose that it satisfies the condition C: 1) Q(z,w,s) is continuous in wεIE (the whole plane) for almost every point zεD and sεIE, and is measurable in zεD for all continuous functions w(z) and all measurable functions s(z) in D+\{zo}, zo εD+; 2) the equation satisfies the uniformly elliptic condition. We prove then that the equation has a homeomorphic solution w(z) which maps quasiconformally D+ and D onto G+ and G, respectively, with w(zo) = ∞, zoεD+, and satisfies the gluing conditions where α(t) maps each of γj, lj, Lj, and Γj topologically onto itself; they give positive shifts on γ ∪ Γ and reverse shifts on l ∪ L, etc.

Keywords

Positive Shift Complex Equation Boundary Contour Reverse Shift Glue Condition 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

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    HUANG Hai-yang: On compound problems of analytic functions with shift, to appear.Google Scholar
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    Litvenčuk, G. S.: Boundary value problems and integral equations with shift [in Russian], Izdat. "Nauka" Fiz.-Mat. Lit., Moscow 1977.Google Scholar
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    LU Chien-ke: On compound boundary problems, Scientia Sinica 14 (1965), 1545–1555.MathSciNetGoogle Scholar
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    WEN Guo-chun: The singular case of Riemann — Hilbert boundary value problems, Acta Sci. Natur. Univ. Pekin. 1981, No. 4, 1–14.Google Scholar
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    —: On linear and nonlinear compound boundary value problems with shift, ibid. 1982, No. 2, 1–12.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1983

Authors and Affiliations

  • Wen Guo-chun
    • 1
  1. 1.Institute of MathematicsPeking UniversityBeijingPeople's Republic of China

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