Tengiz Buchukuri, Roland Duduchava
abstract:
In [58], we have revised an asymptotic model of a shell (Koiter,
Sanchez-Palencia, Ciarlet, etc.), based on the the calculus of tangent Günter's
derivatives, developed in the papers of R. Duduchava, D. Mitrea and M. Mitrea
[55,58,64]. As a result, the 2-dimensional shell equation on a mid-surface
$\mathcal{S}$ was written in terms of Günter's
derivatives, unit normal vector field and the Lamé
constants. The principal part of the obtained equation coincides with the Lamé
equation on the Hypersurface $\mathcal{S}$ investigated in [55,58,64].
The present investigation is inspired by the paper of G. Friesecke, R. D. James
and S. Müller [77], where a hierarchy
of Plate Models are derived from nonlinear elasticity by $\Gamma$-Convergence.
The final goal of the present investigation is to derive and investigate 2D
shell equations in terms of Günter's
derivatives by $\Gamma$-Convergence.
As a first step to the final goal, by T. Buchukuri, R. Duduchava and G.
Tephnadze was studied a mixed boundary value problem for the stationary heat
transfer equation in a thin layer around a surface $\mathcal{C}$ with the
boundary (see [16]). It was established what happens to the solution of the
boundary value problem when the thickness of the layer converges to zero. In
particular, there was shown that the $\Gamma$-limit of a mixed type
Dirichlet-Neumann boundary value problem (BVP) for the Laplace equation in the
initial thin layer is a Dirichlet BVP for the Laplace-Beltrami equation on the
surface. The result was derived based on the variational reformulation of the
problem using the Günter's tangent
differential operators on a hypersurface and layers. The similar results were
obtained for the Lamé operator. This approach allows global representation of
basic differential operators and of corresponding boundary value problems in
terms of the standard cartesian coordinates of the ambient Euclidean space
$\mathbb{R}^n$.
Mathematics Subject Classification: 35J05, 35J20, 53A05, 80A20
Key words and phrases: Hypersurface, Günter's derivatives, Lamé equation, $\Gamma$-Convergence, Shell equation