**David Natroshvili, Guram Sadunishvili, Irine Sigua**

## Some Remarks
Concerning Jones Eigenfrequencies and Jones Modes

**Abstract:**

Three-dimensional fluid-solid interaction problems with regard for thermal
stresses are considered. An elastic structure is assumed to be a bounded
homogeneous isotropic body occupying a domain $\Omega^+\subset\mathbb{R}^3$,
where the thermoelastic four dimensional field is defined, while in the
unbounded exterior domain $\Omega^-=\mathbb{R}^3\setminus\ov{\Omega^+}$ there is
defined the scalar (acoustic pressure) field. These two fields satisfy the
differential equations of steady state oscillations in the corresponding domains
along with the transmission conditions of special type on the interface
$\partial\Omega^{\pm}$. We show that uniqueness of solutions strongly depends on
the geometry of the boundary $\pa\Omega^{\pm}$. In particular, we prove that for
the corresponding homogeneous transmission problem for a ball there exist
infinitely many exceptional values of the oscillation parameter (Jones
eigenfrequencies). The corresponding eigenvectors (Jones modes) are written
explicitly. On the other hand, we show that if the boundary surface
$\partial\Omega^+$ contains two flat, non-parallel sub-manifolds then there are
no Jones eigenfrequencies for such domains.

**Keywords:**

Elasticity, thermoelasticity, fluid-solid interaction, Jones eigenfrequencies,
Jones modes.

**MSC 2000:** 74F10, 74F05