Abstract

We study a variational problem involving material shapes, modeled as Radon measures on a given ambient space, in a force vector field. We prove some properties of absolutely continuity of the optimal shapes in dependence on the distributions of the zeros of the force field. More regularity properties of the optimal shapes, as well as simple geometrical characterizations, are established in the case of central force fields, in dependence on the regularity of the field and on the choice of the ambient space

Author information

Contact details are reproduced from the original publication and may be historical.

Luca Granieri

Dip. di Matematica, Politecnico di Bari, via Orabona 4, 70125 Bari, Italy

l.granieri@poliba.it

Francesco Maddalena

Dip. di Matematica, Politecnico di Bari, via Orabona 4, 70125 Bari, Italy

f.maddalen@poliba.it

References

  1. L. Ambrosio: Introduction to Geometric Measure Theory and Minimal Surfaces. Corso Introduttivo alla Teoria Geometrica della Misura ed alle Superfici Minime, Scuola Normale Superiore, Pisa (1997).Google Scholar
  2. L. Ambrosio, N. Fusco, D. Pallara: Functions of Bounded Variation and Free Discontinuity Problems, Clarendon Press, Oxford (2000).https://doi.org/10.1093/oso/9780198502456.001.0001Google Scholar
  3. L. Ambrosio, M. Novaga, E. Paolini: Some regularity results for minimal crystals, ESAIM, Control Optim. Calc. Var. 8 (2002) 69–103.https://doi.org/10.1051/cocv:2002018Google Scholar
  4. L. Ambrosio, E. Paolini: Partial regularity for quasi minimizers of perimeter, Ric. Mat. 48, suppl. (1999) 167–186.Google Scholar
  5. F. Almgren, J. E. Taylor: Optimal geometry in equilibrium and growth, Fractals 3(4) (1995) 713–723.https://doi.org/10.1142/S0218348X95000631Google Scholar
  6. G. Bouchitté, G. Buttazzo: Characterization of optimal shapes and masses through Monge–Kantorovich equation, J. Eur. Math. Soc. 3 (2001) 139–168.https://doi.org/10.1007/s100970000027Google Scholar
  7. D. Bucur, G. Buttazzo: Variational Methods in Some Shape Optimization Problems, Scuola Normale Superiore, Pisa (2002).Google Scholar
  8. G. Bouchitté, G. Buttazzo, P. Seppecher: Shape optimization solutions via Monge–Kantorovich equation, C. R. Acad. Sci., Paris, Sér. I 324 (1997) 1185–1191.https://doi.org/10.1016/S0764-4442(97)87909-8Google Scholar
  9. G. Bouchitté, G. Buttazzo, P. Seppecher: Energies with respect to a measure and applications to lower dimensional structures, Calc. Var. Partial Differ. Equ. 5 (1997) 37–54.https://doi.org/10.1007/s005260050058Google Scholar
  10. J. E. Brothers, F. Morgan: The isoperimetric theorem for general integrands, Mich. Math. J. 41(1994) 419–431.https://doi.org/10.1307/mmj/1029005070Google Scholar
  11. A. V. Cherkaev: Variational Methods for Structural Optimization, Springer, New York (2000).https://doi.org/10.1007/978-1-4612-1188-4Google Scholar
  12. L. C. Evans, R. F. Gariepy: Measure Theory and Fine Properties of Functions, CRC Press, Boca Raton (1992).Google Scholar
  13. I. Fonseca: The Wulff theorem revisited, Proc. R. Soc. Lond., Ser. A 432 (1991) 125–145.https://doi.org/10.1098/rspa.1991.0009Google Scholar
  14. I. Fonseca, S. Müller: A uniqueness proof for the Wulff theorem, Proc. R. Soc. Edinb., Sect. A 119 (1991) 125–136.https://doi.org/10.1017/S0308210500028365Google Scholar
  15. I. Fragalá, C. Mantegazza: On some notions of tangent space to a measure, Proc. R. Soc. Edinb., Sect. A, Math. 129 (1999) 331–342.https://doi.org/10.1017/S0308210500021387Google Scholar
  16. M. Giaquinta, G. Modica, J. Souček: Cartesian Currents in the Calculus of Variations I. Cartesian Currents, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, A Series of Modern Surveys in Mathematics 37, Springer, Berlin (1998).https://doi.org/10.1007/978-3-662-06218-0Google Scholar
  17. M. A. Grinfeld: The stress driven instability in elastic crystals: Mathematical models and physical manifestations, J. Nonlinear Sci. 3 (1993) 35–83.https://doi.org/10.1007/BF02429859Google Scholar
  18. O. D. Kellog: Foundations of Potential Theory, Springer, Berlin (1967).https://doi.org/10.1007/978-3-642-86748-4Google Scholar
  19. R. J. McCann: Equilibrium shapes for planar crystals in an external field, Commun. Math. Phys. 195 (1998) 699–723.https://doi.org/10.1007/s002200050409Google Scholar
  20. J. Taylor: Existence and structure of solutions to a class of nonelliptic variational problems, Symp. Math. 14 (1974) 499–508.Google Scholar

Suggested citation

L. Granieri, F. Maddalena. “Optimal Shapes in Force Fields.” Journal of Convex Analysis 15 (2008), No. 1, 17–37. https://doi.org/10.68381/jca15002

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2008. Rights now held by Banach Press LLC.