On Dirichlet, Poncelet and Abel problems

TL;DR

This paper explores deep interconnections between boundary value problems, algebraic geometry, and physics, revealing conditions for solution uniqueness and linking classical problems like Dirichlet and Pell-Abel to modern mathematical physics.

Contribution

It establishes new equivalences between boundary value problems, algebraic Pell-Abel equations, and moment problems, providing solvability criteria based on rational parameters.

Findings

Solution non-uniqueness linked to Poncelet problem periodicity

Solvability criteria expressed as rational numbers from problem data

Connections to elliptic solutions of integrable systems

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 $\theta\in\Bbb Q$ 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 Pad\'e interpolation.