No Swiss-cheese universe on the brane

We study the possibility of brane-world generalization of the Einstein-Straus Swiss-cheese cosmological model. We find the modifications induced by the brane-world scenario. At a first glance only the motion of the boundary is modified and the fluid in the exterior region is allowed to have pressure. The general relativistic Einstein-Straus model emerges in the low density limit. By imposing that the brane is static, a combination of the junction conditions and modified cosmological evolution leads to the conclusion that the brane is flat. Thus no static Swiss-cheese universe can exist on the brane. The conclusion is not altered by the introduction of a cosmological constant in the FLRW regions. This result mimics a similar general relativistic result: static Einstein-Straus universes do not exist.