Previous |  Up |  Next


closure; pre-image; mappings
The closures of the pre-images associated with families of mappings in different topologies of normed spaces are considered. The question of finding a description of these closures by means of families of the same kind as original ones is studied. It is shown that for the case of the weak topology this question may be reduced to finding an appropriate closure of a given family. There are discussed various situations when the description may be obtained for the case of the strong topology. An example of a family is constructed which shows that it is, in general, impossible to find such a description for this case.
[1] Buttazzo G.: Semicontinuity, Relaxation and Integral Representation in the Calculus of Variations. Pitman Res. Notes in Math. 207 Longmann Harlow (1989). MR 1020296 | Zbl 0669.49005
[2] Dal Maso G.: An Introduction to $\Gamma $-convergence. Birkhäuser Boston (1993). MR 1201152 | Zbl 0816.49001
[3] Defranceschi A.: An introduction to homogenization and $G$-convergence. School on Homogenization Lecture notes of the courses held at ICTP, Trieste, September 6-17, 1993 63-122.
[4] Engelking R.: General Topology. Monografie Matematyczne, tom 60 Państwowe Wydawnictwo Naukowe Warszawa (1977). MR 0500779 | Zbl 0373.54002
[5] Fattorini H.O.: Infinite Dimensional Optimization Theory and Optimal Control. Cambridge Univ. Press Cambridge (1997).
[6] Gajewski H., Gröger K., Zacharias K.: Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen. vol. 38 of Mathematische Lehrbücher und Monographien Akademie-Verlag Berlin (1974). MR 0636412
[7] Gamkrelidze R.V.: Principles of Optimal Control (in Russian). Tbilisi University Tbilisi (1975). MR 0686791
[8] Ioffe A.D., Tikhomirov V.M.: Extension of variational problems (in Russian). Trudy Moskov. Mat. Obšč. 18 187-246 (1968). MR 0254702
[9] Lions J.L.: Optimal Control of Systems Governed by Partial Differential Equations. Springer-Verlag (1971), Berlin. MR 0271512 | Zbl 0203.09001
[10] Lurie K.A.: The extension of optimization problems containing controls in coefficients. Proc. Royal Soc. Edinburgh Sect. A 114 81-97 (1990). MR 1051609
[11] Raitums U.: Optimal Control Problems for Elliptic Equations (in Russian). Zinatne Riga (1989). MR 1042986
[12] Raitums U.: On the projections of multivalued maps. J. Optim. Theory Appl. 92 3 633-660 (1997). MR 1432612 | Zbl 0885.49001
[13] Roubíček T.: Relaxation in Optimization Theory and Variational Calculus. W. de Gruyter Berlin (1997). MR 1458067
[14] Tartar L.: Problems de contrôle des coefficientes dans les équations aux dérivées partieles. Lecture Notes Economical and Mathematical Systems 107 (1975), 420-426. MR 0428166
[15] Tartar L.: Remarks on optimal design problems. in Calculus of Variations, Homogenization and Continuum Mechanics G. Buttazzo and G. Bouchitte and P. Suquet Singapore (1994), World Scientific 279-296. MR 1428706 | Zbl 0884.49015
[16] Warga J.: Optimal Control of Differential and Functional Equations. Academic Press New York (1972). MR 0372708 | Zbl 0253.49001
[17] Zhikov V.V., Kozlov S.M., Olejnik O.A.: Homogenization of Differential Operators and Integral Functionals. Springer-Verlag Berlin (1994). MR 1329546 | Zbl 0838.35001
Partner of
EuDML logo