An optimal design problem with perimeter penalization

  • Luigi Ambrosio
  • Giuseppe Buttazzo


We study the optimal design problem of finding the minimal energy configuration for a mixture of two conducting materials when a perimeter penalization of the unknown domain is added. We show that in this situation an optimal domain exists and that, under suitable assumptions on the data, it is an open set.

Mathematics Subject Classification

49J45 49N60 49Q20 


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  1. [1]
    Alt, H.W., Caffarelli, L.A.: Existence and regularity results for a minimum problem with free boundary. J. Reine Angew. Math.325, 107–144 (1981)Google Scholar
  2. [2]
    Buttazzo, G.: Semicontinuity, relaxation and integral representation in the calculus of variations. Pitman Res. Notes Math. Ser.207, Harlow: Longman 1989Google Scholar
  3. [3]
    Campanato, S.: Sistemi Ellittici in Forma Divergenza. Regolarità all'Interno. Quaderno della Scuola Normale Superiore di Pisa, Pisa 1980Google Scholar
  4. [4]
    De Giorgi, E.: Sulla differenziabilità e l'analiticità degli integrali multipli regolari. Mem. Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur.3, 25–43 (1957)Google Scholar
  5. [5]
    De Giorgi, E, Carrierio, M., Leaci, A.: Existence theorem for a minimum problem with free discontinuity set. Arch. Rational Mech. Anal.108, 195–218 (1989)Google Scholar
  6. [6]
    Federer, H.: Geometric measure theory. Berlin Heidelberg New York: Springer 1969Google Scholar
  7. [7]
    Giaquinta, M.: Multiple integrals in the calculus of variations. Princeton: Princeton University Press 1983Google Scholar
  8. [8]
    Giaquinta, M, Giusti, E.: On the regularity of minima of variational integrals. Acta Math.148, 31–46 (1982)Google Scholar
  9. [9]
    Giaquinta, M., Modica, G.: Regularity results for some classes of higher order nonlinear elliptic systems. J. Reine Angew. Math.311/312, 145–169 (1979)Google Scholar
  10. [10]
    Giusti, E.: Minimal surfaces and functions of bounded variation. Boston: Birkhäuser 1984Google Scholar
  11. [11]
    Ioffe, A.D.: On lower semicontinuity of integral functionals I. SIAM J. Control Optim.15, 521–538 (1977)Google Scholar
  12. [12]
    Kinderlehrer, D., Stampacchia, G.: Variational inequalities and applications. New York: Academic Press 1980Google Scholar
  13. [13]
    Kohn, R.V., Strang, G.: Optimal design and relaxation of variational problems, I,II,III. Commun. Pure Appl. Math.39, 113–137, 139–182, 353–377 (1986)Google Scholar
  14. [14]
    Ladyzhenskaya, O.A., Uraltseva, N.N.: Linear and quasilinear elliptic equations. New York: Academic Press 1968Google Scholar
  15. [15]
    Murat, F., Tartar, L.: Optimality conditions and homogenization. Proceedings of “Nonlinear Variational Problems”, Isola d'Elba 1983 (Res. Notes in Math.127, pp. 1–8) London: Pitman 1985Google Scholar

Copyright information

© Springer-Verlag 1993

Authors and Affiliations

  • Luigi Ambrosio
    • 1
  • Giuseppe Buttazzo
    • 2
  1. 1.II Università di Roma-MatematicaRomaItaly
  2. 2.Dipartimento di MatematicaPisaItaly

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