We consider the Laplace operator in a thin tube of ${\mathbb R}^3$ with a Dirichlet condition on its boundary. We study asymptotically the spectrum ofsuch an operator as the thickness of the tube's cross section goes to zero. In particular weanalyse how the energy levels depend simultaneously on the curvature of the tube's central axisand on the rotation of the cross section with respect to the Frenet frame. The main argument is aΓ-convergence theorem for a suitable sequence of quadratic energies.

We study the stability of a sequence of integralfunctionals on divergence-free matrix valued fields following the directmethods of Γ-convergence. We prove that the Γ-limitis an integral functional on divergence-free matrix valued fields.Moreover, we show that the Γ-limit is also stable undervolume constraint and various type of boundary conditions.