Nonlinear mixed Cherepanov boundary-value problem

ABSTRACT

We consider a nonlinear boundary-value problem consisting in the determination of a function $w\left(z\right)$ that is meromorphic in the upper half-plane, satisfies the homogeneous Hilbert boundary condition on the set L of n intervals of the real axis and has a given module on the set $M=\mathbb{R}\setminus \overline{L}$. This problem was stated and solved in Cherepanov. Cherepanov proved that the required solution with a given number and location of its interior zeros and poles and with integrable singularities at all endpoints of L exists if and only if n−1 solvability conditions are satisfied. We prove that this problem is unconditionally solvable. A particular solution, $w_0\left(z\right)$, is found in the class of meromorphic functions with a properly chosen location of n−1 zeros and poles. Namely, we have shown that these zeros and poles are defined as the solution of some solvable real analogue of the Jacobi inversion problem. A general meromorphic solution of the Cherepanov problem is obtained with the help of the particular solution $w_0\left(z\right)$. The problem of a possible decrease in the number of zeros and poles of the desired solution is investigated.

COMMUNICATED BY:

• Y. Antipov

KEYWORDS:

• Nonlinear mixed boundary-value problem
• analytic functions
• closed form solution

AMS SUBJECT CLASSIFICATION:

• 30E25

Acknowledgements

The work is performed according to the Russian Government Program of Competitive Growth of Kazan Federal University. We consider it our pleasant duty to thank professors V. V. Mityushev, S. V. Rogosin and E. Wegert for sending us copies of their monographs [6,13], respectively. Helpful comments of two anonymous Referees and Associated Editor are highly appreciated.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 The H-continuity of $a\left(x\right)$ in a neighbourhood of infinity means that $a\left(1/x\right)$ is a H-continuous function in a neighbourhood of zero (see [2,3]).

2 Here and below $\lfloor x\rfloor$ denotes the largest integer less than or equal to x