A domain optimization problem with boundary penalty cost functional ✩

In this paper, we deal with a domain optimization problem governed by a boundary-value problem of Laplace equation. We prove the existence of the optimal domain for the given cost functional with a boundary penalty. At the same time, we also point out that there is no optimal domain in the absence of the boundary penalty. At last, necessary condition of the optimal domain is given.  2004 Published by Elsevier Inc.