On Dirichlet, Poncelet and Abel problems

Vladimir P. Burskii; Alexei S. Zhedanov

doi:10.3934/cpaa.2013.12.1587

Article Contents

2013, Volume 12, Issue 4: 1587-1633. Doi: 10.3934/cpaa.2013.12.1587

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On Dirichlet, Poncelet and Abel problems

Vladimir P. Burskii^1, and
Alexei S. Zhedanov^2,

1.
Institute of Applied Mathematics, Donetsk, 83114
2.
Donetsk Institute for Physics and Technology, Donetsk, 83114

Received: May 31, 2011

Revised: May 31, 2011

Published: June 2013

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Abstract

We propose interconnections between some problems of PDE, geometry, algebra, calculus and physics. Uniqueness of a solution of the Dirichlet problem and of some other boundary value problems for the string equation inside an arbitrary biquadratic algebraic curve is considered. It is shown that a solution is non-unique if and only if a corresponding Poncelet problem for two conics has a periodic trajectory. A set of problems is proven to be equivalent to the above problem. Among them are the solvability problem of the algebraic Pell-Abel equation and the indeterminacy problem of a new moment problem that generalizes the well-known trigonometrical moment problem. Solvability criteria of the above-mentioned problems can be represented in form $\theta\in Q$ where number $\theta=m/n$ is built by means of data for a problem to solve. We also demonstrate close relations of the above-mentioned problems to such problems of modern mathematical physics as elliptic solutions of the Toda chain, static solutions of the classical Heisenberg $XY$-chain and biorthogonal rational functions on elliptic grids in the theory of the Padé interpolation.

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Mathematics Subject Classification: Primary: 14H52, 35L20, 14N15; Secondary: 35L35, 13P05, 34K13, 44A60, 37K10.

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