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On optimal shape design. (English) Zbl 0849.49021
In the paper the following shape optimization problems are considered: take a large ball \(B\) in \(\mathbb{R}^n\) and consider, for every open subset \(\Omega\) of \(B\), the solution \(u_\Omega\) of the elliptic problem \[ \begin{cases} -\Delta u = f & \text{in \(\Omega\)}\\ u \in H^1_0(\Omega)\end{cases} \] extended by zero on \(B \setminus \Omega\), where \(f \in L^2(B)\) is given. Try to minimize then a given functional \(J(u_\Omega)\) when \(\Omega\) varies among all open bounded subsets of \(B\).
In order to apply the direct methods of the calculus of variations, a topology on the class of admissible domains is needed. The author defines \(\Omega_j \to \Omega\) if and only if the orthogonal projections \(P_{\Omega_j} : H^1_0(B) \to H^1_0 (\Omega_j)\) converge strongly to \(P_\Omega\) as operators. This convergence is not compact; in other words, it may happen that the limit of a sequence of projections \(P_{\Omega_j}\) is not a projection of the form \(P_\Omega\) for some \(\Omega\). The author shows that in the case \(n = 2\) the compactness can be obtained by restricting the class of admissible domains; more precisely, for each positive integer \(k\) the class \({\mathcal A}_k = \{\Omega\) open subset of \(B\), the number of connected components of \(\overline{B} \setminus \Omega\) is \(\leq k\}\) is proved to be compact for the convergence above.
In section 5 the previous result is applied to show the existence of an optimal two-dimensional submarine.
Reviewer: G.Buttazzo (Pisa)

49Q10 Optimization of shapes other than minimal surfaces
49J45 Methods involving semicontinuity and convergence; relaxation