**Roland Gachechiladze and Otar Maisaia**

## Some Qualitative Problems of the Linear Elasticity Theory

**abstract:**

The first and the second boundary value problems of
statics are considered. The dependence of the solutions and of the
corresponding
eigenfrequencies of these problems on the elastic constants and density
is investigated. The same dependence is studied for the total deformation
energy and for Green's operators. The following theorem is proved: among
anisotropic elastic convex bodies of a given volume there exists one for which
the first eigenfrequency of the first boundary value problem is minimal.

**Mathematics Subject Classification:**
73C02

**Key words and phrases:**
Elasticity, $n$-dimensional,
boundary value problem, Green's operator, fundamental frequency,
isoperimetric problem.