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Conformal mappings that satisfy the boundary condition of equality of metrics. (English. Russian original) Zbl 0890.30025
Dokl. Math. 53, No. 2, 194-196 (1996); translation from Dokl. Akad. Nauk 347, No. 3, 295-297 (1996).
The author considers the problem of reconstructing a conformal mapping \(f: D\to G\) by the boundary condition \[ {|dw|\over \varphi (w)}= {|dz |\over \psi (z)},\;w= f(z),\;z\in \partial D, \] where \(D\) and \(G\) are plane domains, \(0\in D\), \(f(0) =0\), \(f'(0)>0\); \(\varphi, \psi\) are given functions. The existence of a solution and its properties in two classes of functions are studied. The proofs are sketched.

MSC:
30E25 Boundary value problems in the complex plane
30C35 General theory of conformal mappings
31A25 Boundary value and inverse problems for harmonic functions in two dimensions
35J55 Systems of elliptic equations, boundary value problems (MSC2000)
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