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The nontrivial solvability of the homogeneous Dirichlet problem in a unit disk $$ K\subset {{\mathbb{R}}^2} $$ in positive Sobolev spaces is studied for a typeless differential equation of arbitrary even order 2m, m ≥ 2, with constant complex-valued coefficients and homogeneous symbol. The detailed proofs of the criteria of nontrivial solvability of the problem are given in various cases that form a complete picture. The example considered by A. V. Bitsadze is generalized for the equations of arbitrary even order 2m, m ≥ 2.

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 Pade interpolation.

In the present paper we consider a system of equations given for one unknown function that have generalized D'Alambert representation of solutions. We establish necessary and sufficient conditions for nontrivial solvability of the Dirichlet problem for such system in the unit ball. A simple necessary condition is obtained which says: for any i and j the angle between the vectors ai and aj must be a π-rational number.

We consider conditions for uniqueness of the solution of the Dirichlet or the Neumann problem for 2-dimensional wave equation inside of bi-quadratic algebraic curve. We show that the solution is non-trivial if and only if corresponding Poncelet problem for two conics associated with the curve has periodic trajectory and if and only if corresponding Pell-Abel equation has a solution., Published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA/