Shape optimization problems on metric measure spaces

We consider shape optimization problems of the form $\min\big\{J(Ω)\ :\ Ω\subset X,\ m(Ω)\le c\big\},$ 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.

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