%0 Journal Article %T A new compact class of open sets under Hausdorff distance and shape optimization %A Donghui Yang %J Mathematics %D 2010 %I arXiv %X In this paper we obtain a new class of open sets, and we prove the class is compact under the Hausdorff distance, then we prove the existence of solutions of some shape optimization for elliptic equations. %U http://arxiv.org/abs/1004.2874v2