Asymptoptic behaviour of Sobolev constants for thin curved rods or pipes

Abstract

We study the Sobolev imbedding inequality in a curved rod or pipe with a smooth central curve $\gamma$. Using the variational approach and the two-scale convergence for thin domains we find the limit of the Sobolev imbedding constant W$^{1,r}\hookrightarrow L^{q}$ as $\epsilon$, the ratio between cross section diameter and the lenght of the rod, tends to 0.