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Lower-semicontinuity in domain optimization problems. (English) Zbl 0629.49006
In the present paper, lower semicontinuity of certain classes of functionals is studied when the domain of integration, which defines the functionals, is not fixed. For this purpose, a certain class of domains introduced by D. Chenais [J. Math. Anal. Appl. 52, 189-219 (1975; Zbl 0317.49005)] is employed. For this class of domains, a basic lemma is proved which plays an essential role in the derivations of the lower- semicontinuity theorems. These theorems are applied to the study of the existence of the optimal domain in domain optimization problems; a boundary-value problem of Neumann type or Dirichlet type is the main constraint in these optimization problems.

49J45 Methods involving semicontinuity and convergence; relaxation
49K20 Optimality conditions for problems involving partial differential equations
49K40 Sensitivity, stability, well-posedness
35J25 Boundary value problems for second-order elliptic equations
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[1] Cea, J.,Problems of Shape Optimal Design, Optimization of Distributed Parameter Structures, Edited by E. J. Haug and J. Cea, Sijthoff and Noordhoff, Alphen aan den Rijn, The Netherlands, Vol. 2, pp. 1005-1048, 1981.
[2] Pironneau, O.,On Optimum Problems in Stokes Flow, Journal of Fluid Mechanics, Vol. 59, pp. 117-128, 1973. · Zbl 0274.76022
[3] Pironneau, O.,On Optimum Design in Fluid Mechanics, Journal of Fluid Mechanics, Vol. 64, pp. 97-110, 1974. · Zbl 0281.76020
[4] Rousselet, B.,Response Dynamique et Optimisation de Domaine, Preprints of 3rd IFAC Symposium on Control of Distributed Parameter Systems, Edited by J. P. Babary and L. LeLetty, Toulouse, France, 1982.
[5] Rousselet, B.,Shape Design Sensitivity of a Membrane, Journal of Optimization Theory and Applications, Vol. 40, pp. 595-623, 1983. · Zbl 0497.73097
[6] Zolesio, J. P.,The Material Derivative (or Speed) Method for Shape Optimization, Optimization of Distributed Parameter Structures, Edited by E. J. Haug and J. Cea, Sijthoff and Noordhoff, Alphen aan den Rijn, The Netherlands, Vol. 2, pp. 1089-1151, 1981.
[7] Fujii, N.,Necessary Conditions for a Domain Optimization Problem in Elliptic Boundary-Value Problems, SIAM Journal on Control and Optimization, Vol. 24, pp. 346-360, 1986. · Zbl 0587.49023
[8] Chenais, D.,On the Existence of a Solution in a Domain Identification Problem, Journal of Mathematical Analysis and Applications, Vol. 52, pp. 189-219, 1975. · Zbl 0317.49005
[9] Chenais, D.,Hom?omorphisme entre Ouverts Lipschitziens, Annali di Matematica Pura e Applicata (IV), Vol. 118, pp. 343-398, 1978.
[10] Serrin, J.,On the Definition and the Properties of Certain Variational Integrals, Transaction of the American Mathematical Society, Vol. 101, pp. 139-167, 1961. · Zbl 0102.04601
[11] Morrey, C. B.,Multiple Integrals in the Calculus of Variations, Springer-Verlag, Berlin, Germany, 1966. · Zbl 0142.38701
[12] Ladyzhenskaya, O. A., andUral’tseva, N. N.,Linear and Quasilinear Elliptic Equations, Academic Press, New York, New York, 1968.
[13] Courant, R.,Methods of Mathematical Phsyics, Vol. 2, Interscience, New York, New York, 1962. · Zbl 0099.29504
[14] Mizohata, S.,The Theory of Partial Differential Equations, Cambridge University Press, London, England, 1973. · Zbl 0263.35001
[15] Polya, G.,Torsional Rigidicity, Principal Frequency, Electrostatic Capacity, and Symmetrization, Quarterly of Applied Mathematics, Vol. 6, pp. 267-277, 1948. · Zbl 0037.25301
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