An existence result for a class of shape optimization problems. (English) Zbl 0811.49028

Buttazzo, Giuseppe; Dal Maso, Gianni

Arch. Ration. Mech. Anal. 122, No. 2, 183-195 (1993).

The basic type of problem the authors consider is the minimization of a functional defined on the family of quasi-open subsets of a given bounded, open \(\Omega\subset \mathbb{R}^ n\). A subset of \(\Omega\) is termed quasi-open if the infimum of the capacity of its symmetric difference with open subsets of \(\Omega\) is zero, and the family of quasi-open sets is denoted by \({\mathcal A}(\Omega)\). To obtain an existence result, attention is focused on functionals that are lower semicontinuous with respect to the appropriate notion of convergence, called \(\gamma\)- convergence, which is defined in terms of the resolvent operators for the solution of the Dirichlet problem for the Laplacian. The authors’ main result is that if the functional \(F: {\mathcal A}(\Omega)\to \overline{\mathbb{R}}\) is lower semicontinuous with respect to \(\gamma\)- convergence and satisfies \(F(A)\geq F(B)\) whenever \(A\subset B\), then, for every choice of \(c\) with \(0\leq c\leq |\Omega|\), the minimum of \(F(A)\) over the set of \(A\in {\mathcal A}(\Omega)\) with \(| A|= c\) is attained. Several examples are presented. The proof involves constructing an auxiliary functional on the closed, convex set of functions \({\mathcal K}= \{w\in H^ 1_ 0(\Omega): w\geq 0, 1+\Delta w\geq 0\}\). The minimum of the original problem is the set of points at which a constrained minimizer of the auxiliary functional is positive.

Reviewer: H.Parks (Corvallis)

MSC:

49Q10	Optimization of shapes other than minimal surfaces
35P99	Spectral theory and eigenvalue problems for partial differential equations

Keywords:

shape optimization; quasi-open set; eigenvalues of an elliptic operator; \(\gamma\)-convergence

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