74 Luxemburg Str.,

83114 Donetsk,

UKRAINE

and

83114 Donetsk,

UKRAINE

E-mail: v30@dn.farlep.net, zhedanov@yahoo.com

**Boundary value problems for string equation, Poncelet
problem and Pell-Abel equation: unique answer**

**Abstract:**

This work is devoted to a connection between ill-posed boundary value
problems in a bounded semialgebraic domain for partial differential equations
and the Poncelet problem, recently revealed by authors. The Poncelet problem
is one of famous problems of projective geometry and it by itself has numerous
links with a set of different problems of analysis and physics. Investigations
of ill-posed boundary value problems in bounded domains for partial differential
equations go back to J. Hadamard. The solution uniqueness of the Dirichlet
problem for the string equation $u_{xy} =0$ in $\Omega$, $u|_C =\phi$ on
$C=\partial \Omega $ in a bounded domain $\Omega$ is connected with properties
of John automorphism $T:\partial \Omega \to \partial\Omega$. In particular,
there is the following *sufficient condition of uniqueness*: The homogeneous
Dirichlet problem has only trivial solution in the space $C^2(\overline\Omega)$
if the set of periodic points of $T$ on $C$ is finite or denumerable. We
consider this problem in a bounded semialgebraic domain, the boundary of
which is given by some bi-quadratic algebraic curve $F(x,y):=\sum_{i,k=0}^2
a_{ik}x^iy^k=0. $ We show the John mapping in this case is the same as
Ponselet mapping in some rational parametrizations of conics. From it we
obtain

**Theorem.** *For generic bi-quadratic curve the Dirichlet problem
has non-unique solution if and only if corresponding Poncelet problem has
periodic trajectory.*

From Poncelet theorem we obtain if there exists some periodical point
then each point of $C$ is periodical with the same period. On the other
hand a Baxter parametrization allows write the Poncelet mapping by means
of elliptic Jacoby functions and obtain a criterion of existence of periodical
points and a criterion of

uniqueness breakdown for above the Dirichlet problem.

In turn the solution uniqueness of the Dirichlet problem is equivalent
to solution uniqueness of some class of boundary value problems for the
same equation on $C$ and is equivalent to an indeterminacy of some moment
problem on $C$: $\exists \alpha(s) \ne 0$, $\forall k=0,1, \ldots \
\int_C [x(s)]^k\alpha(s) ds= \int_C

[y(s)]^k\alpha(s)ds=0$, where $(x,y)$ are Cartesian coordinates of
point on $C$ parametrized by $s$.

Except for that a Cayley determinant criterion of periodicity of Poncelet problem for case of even period can be understood as a criterion of solvable for algebraic Pell-Abel equation $P^2-RQ^2=1$, where for given polinomial $R$ of the order 4 it is required to find polinomials $P$, $Q$. The last problem has connections with a lot of different problems of analysis also.

This work is supported in part by the Russian Foundation for Basic Research (RFBR) grant no. 03-01-00780.