Existence and relaxation problems in optimal shape design (Q5929992)

The paper deals with optimal shape design of the form NEWLINE\[NEWLINE\min\{J(\Omega, u):\Omega\in{\mathcal B},\;u\in S(\Omega)\},NEWLINE\]NEWLINE where \(J\) is the cost functional, \(\Omega\) varies in a given class \({\mathcal B}\) of admissible domains of \(\mathbb{R}^N\), and \(S(\Omega)\) denotes the set of solutions of a state relation, which can be a PDE, a variational inequality, or a hemivariational inequality.NEWLINENEWLINENEWLINEUnder suitable geometrical constraints on the admissible domains of the class \({\mathcal B}\) it is known that the minimization problem above has a compactness property which makes possible, via direct methods of the calculus of variations, the proof of the existence of an optimal solution.NEWLINENEWLINENEWLINEWithout the geometrical constraints mentioned above, in general no optimal solution exists and a relaxation procedure is needed to study the behaviour of minimizing sequences. In the paper, starting from the results obtained in [\textit{G. Buttazzo} and \textit{G. Dal Maso}, Appl. Math. Optimization 23, No. 1, 17-49 (1991; Zbl 0762.49017)], the author studies the cases when the state equation is of partial differential type (elliptic, parabolic, hyperbolic) with Dirichlet conditions on the free boundary \(\partial\Omega\), presenting a relaxed formulation which involves measures of capacitary type, i.e., nonnegative Borel measures on \(\mathbb{R}^N\) which vanish on all sets of capacity zero.