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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)

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