On Dirichlet problem for string equation, Poncelet problem, Pell-Abel equation, and some other related problems

In a plane domain bounded by a biquadratic curve, we consider the problem of the uniqueness of a solution of the Dirichlet problem for the string equation. We show that this problem is equivalent to the classical Poncelet problem in projective geometry for two appropriate ellipses and also to the problem of the solvability of the Pell-Abel algebraic equation; some other related problems are also considered. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 4, pp. 435–450, April, 2006.