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

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