Abstract
Given a bounded open subset Ω of R n, we prove the existence of a minimum point for a functional F defined on the family A(Ω) of all “quasiopen” subsets of Ω, under the assumption that F is decreasing with respect to set inclusion and that F is lower semicontinuous on A(Ω) with respect to a suitable topology, related to the resolvents of the Laplace operator with Dirichlet boundary condition. Applications are given to the existence of sets of prescribed volume with minimal k th eigenvalue (or with minimal capacity) with respect to a given elliptic operator.
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References
T. Bagby: Quasi topologies and rational approximation. J. Funct. Anal. 10 (1972) 259–268.
G. Buttazzo: Semicontinuity, Relaxation and Integral Representation in the Calculus of Variations. Pitman Res. Notes in Math. 207, Longman, Harlow, 1989.
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.
G. Buttazzo & G. Dal Maso: Shape optimization for Dirichlet problems: relaxed formulation and optimality conditions. Appl. Math. Optim. 23 (1991) 17–49.
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.
R. Courant & D. Hilbert: Methods of Mathematical Physics. Wiley, New York, 1953.
G. Dal Maso: On the integral representation of certain local functionals. Ricerche Mat. 32 (1983) 85–113.
G. Dal Maso: Some necessary and sufficient conditions for the convergence of sequences of unilateral convex sets. J. Funct. Anal. 62 (1985) 119–159.
G. Dal Maso: An Introduction to Γ-convergence. Birkhäuser, Boston, 1993.
N. Dunford & J. T. Schwartz: Linear Operators. Interscience, New York, 1957.
S. Finzi Vita: Numerical shape optimization for relaxed Dirichlet problems. Math. Mod. Meth. Appl. Sci. 3 (1993), 19–34.
B. Fuglede: The quasi topology associated with a countably subadditive set function. Ann. Ins. Fourier 21 (1971) 123–169.
D. Kinderlehrer & G. Stampacchia: An Introduction to Variational Inequalities and their Applications. Academic Press, New York, 1980.
U. Mosco: Convergence of convex sets and of solutions of variational inequalities. Adv. in Math. 3 (1969) 510–585.
O. Pironneau: Optimal Shape Design for Elliptic Systems. Springer-Verlag, Berlin, 1984.
V. Šverák: On optimal shape design. Preprint, Heriot-Watt University, Edinburgh, 1992, and C. R. Acad. Sci. Paris Sér. I Math. 315 (1992) 545–549.
W. P. Ziemer: Weakly Differentiable Functions. Springer-Verlag, Berlin, 1989.
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Communicated by H. Brezis
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Buttazzo, G., Dal Maso, G. An existence result for a class of shape optimization problems. Arch. Rational Mech. Anal. 122, 183–195 (1993). https://doi.org/10.1007/BF00378167
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DOI: https://doi.org/10.1007/BF00378167