×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] T. Bagby: Quasi topologies and rational approximation. J. Funct. Anal. 10 (1972) 259-268. · Zbl 0266.30024 · doi:10.1016/0022-1236(72)90025-0
[2] G. Buttazzo: Semicontinuity, Relaxation and Integral Representation in the Calculus of Variations. Pitman Res. Notes in Math. 207, Longman, Harlow, 1989. · Zbl 0669.49005
[3] G. Buttazzo: Relaxed formulation for a class of shape optimization problems. Proceedings of ?Boundary Control and Boundary Variations? (Sophia-Antipolis, 1991), Lecture Notes in Control and Information Sci. 178, Springer-Verlag, Berlin, 1992.
[4] G. Buttazzo & G. Dal Maso: Shape optimization for Dirichlet problems: relaxed formulation and optimality conditions. Appl. Math. Optim. 23 (1991) 17-49. · Zbl 0762.49017 · doi:10.1007/BF01442391
[5] M. Chipot & G. Dal Maso: Relaxed shape optimization: the case of nonnegative data for the Dirichlet problem. Adv. Math. Sci. Appl. 1 (1992) 47-81. · Zbl 0769.35013
[6] R. Courant & D. Hilbert: Methods of Mathematical Physics. Wiley, New York, 1953. · Zbl 0051.28802
[7] G. Dal Maso: On the integral representation of certain local functionals. Ricerche Mat. 32 (1983) 85-113. · Zbl 0543.49001
[8] G. Dal Maso: Some necessary and sufficient conditions for the convergence of sequences of unilateral convex sets. J. Funct. Anal. 62 (1985) 119-159. · Zbl 0582.49008 · doi:10.1016/0022-1236(85)90001-1
[9] G. Dal Maso: An Introduction to ?-convergence. Birkhäuser, Boston, 1993. · Zbl 0816.49001
[10] N. Dunford & J. T. Schwartz: Linear Operators. Interscience, New York, 1957.
[11] S. Finzi Vita: Numerical shape optimization for relaxed Dirichlet problems. Math. Mod. Meth. Appl. Sci. 3 (1993), 19-34. · Zbl 0781.49023 · doi:10.1142/S0218202593000035
[12] B. Fuglede: The quasi topology associated with a countably subadditive set function. Ann. Ins. Fourier 21 (1971) 123-169. · Zbl 0197.19401
[13] D. Kinderlehrer & G. Stampacchia: An Introduction to Variational Inequalities and their Applications. Academic Press, New York, 1980. · Zbl 0457.35001
[14] U. Mosco: Convergence of convex sets and of solutions of variational inequalities. Adv. in Math. 3 (1969) 510-585. · Zbl 0192.49101 · doi:10.1016/0001-8708(69)90009-7
[15] O. Pironneau: Optimal Shape Design for Elliptic Systems. Springer-Verlag, Berlin, 1984. · Zbl 0534.49001
[16] V. ?verák: On optimal shape design. Preprint, Heriot-Watt University, Edinburgh, 1992, and C. R. Acad. Sci. Paris Sér. I Math. 315 (1992) 545-549.
[17] W. P. Ziemer: Weakly Differentiable Functions. Springer-Verlag, Berlin, 1989.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.