Appendix: Shape optimization problems for graphs

Abstract

In the previous chapters we discussed a wide variety of spectral optimization problems. In particular, we have a theory, which can be successfully applied to study the existence of optimal sets in the very general context of metric measure spaces. The variables in this case were always subsets of a given ambient space, since most of the geometric and analytical objects can be viewed as subspaces of some bigger space, this is quite a reasonable assumption. The more restrictive assumption, and the one that provided enough structure to develop the theory, concerns the cost functionals. More precisely, to each subset Ω of the ambient space X we associate in a specific way a subspace H(Ω) of some prescribed functional space H on X. The cost functionals with respect to which we optimize are in fact of the form F(Ω) = F(H(Ω)), where ℱ is a functional on the subspaces of H.