Casimir Energy Densities for Parallel Plate on Background of Conformally Flat Brane-World Geometries and Cosmological Constant Problem

We study the the Casimir effect of a massless conformally coupled scalar field in the background of a conformally flat brane-world geometries, where the scalar field satisfying Robin boundary conditions on two parallel plates. In the general case of Robin boundary conditions formula are derived for the vacuum expectation values of the energy-momentum tensor. Further the surface energy per unit area which located on the branes which was lacking in the previous paper \cite{setsah}, are obtained . As an application of the general formula we have considered the important special case of the dS$_{4+1}$ bulk. In this specific example we demonstrate that there is a solution for the cosmological constant problem