Shape optimization problems on metric measure spaces

We consider shape optimization problems of the form min J(Ω): Ω ⊂ X, m(Ω) c , where X is a metric measure space and J is a suitable shape functional. We adapt the notions of γ - convergence and weak-γ convergence to this new general abstract setting to prove the existence of an optimal domain. Several examples are pointed out and discussed. © 2012 Elsevier Inc. All rights reserved