Shape Optimization Problems with Internal Constraint

Abstract

We consider shape optimization problems with internal inclusion constraints, of the form $$\min\big\{J(Ω)\ :\ \Dr\subsetΩ\subset\R^d,\ |Ω|=m\big\},$$ where the set $\Dr$ is fixed, possibly unbounded, and $J$ depends on $\Omega$ via the spectrum of the Dirichlet Laplacian. We analyze the existence of a solution and its qualitative properties, and rise some open questions.

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