Skip to main content

Variational Theory of Weak Geometric Structures: The Measure Method and Its Applications

  • Conference paper

Part of the book series: Progress in Nonlinear Differential Equations and Their Applications ((PNLDE,volume 51))

Abstract

We present a survey of results for different kinds of variational problems where weak geometric structures intervene as a common feature. A unified method is adopted, which has been developed in several papers during recent years; it is based on the use of measures, fitted out with suitable tangential properties and functional spaces. Some of the results are new, and their proof will be given in a forthcoming paper.

This is a preview of subscription content, log in via an institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. W. K. Allard, First variation of a varifold,Annals of Math., 95 (1972), 417–491.

    Article  MathSciNet  MATH  Google Scholar 

  2. H. Attouch and G. Buttazzo, Homogenization of reinforced periodic one-codimensional structures, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 14 (4) (1987), 465–484.

    MathSciNet  MATH  Google Scholar 

  3. L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford University Press (2000).

    MATH  Google Scholar 

  4. A. Bensoussan, J. L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, North-Holland, Amsterdam (1978).

    MATH  Google Scholar 

  5. G. Bouchitté, G. Bellettini and I. Fragalà, BV functions with respect to a measure and relaxation of metric integral functionals, J. Convex Anal., 6 (2) (1999), 349–366.

    Article  MATH  Google Scholar 

  6. G. Bouchitté and G. Buttazzo, Characterization of optimal shapes and masses through Monge-Kantorovich equation, J. Eur. Math. Soc. 3 (2001), 139–168.

    Article  MATH  Google Scholar 

  7. G. Bouchitté, G. Buttazzo and I. Fragalà, Mean curvature of a measure and related variational problems, Ann. Scuola Norm. Sup. Pisa Cl. Sci., XXV (4) (1997), 179–196.

    Google Scholar 

  8. G. Bouchitté, G. Buttazzo and I. Fragalà, Convergence of Sobolev spaces on varying manifolds, to appear on J. Geom. Anal.

    Google Scholar 

  9. G. Bouchitté, G. Buttazzo and I. Fragalà, Bounds on the effective coefficients of homogenized low dimensional structures, Preprint, (2001).

    Google Scholar 

  10. G. Bouchitté, G. Buttazzo and P. Seppecher, Energies with respect to a measure and applications to low dimensional structures, Calc. Var. Partial Differential Equations 5 (1997), 37–54.

    Article  MathSciNet  MATH  Google Scholar 

  11. G. Bouchitté, G. Buttazzo and P. Seppecher, Optimization solutions via MongeKantorovich equation, C. R. Acad. Sci. Paris, Série I 324 (1997), 1185–1191.

    Article  MATH  Google Scholar 

  12. G. Bouchitté and I. Fragalà, Homogenization of thin structures by two-scale method with respect to measures, SIAM J. Math. Anal., 32 (6) (2001), 1198–1226.

    Article  MathSciNet  MATH  Google Scholar 

  13. G. Bouchitté and I. Fragalà, Homogenization of elastic thin structures: a measure-fattening approach, Preprint, (2000).

    Google Scholar 

  14. G. Bouchitté and I. Fragalà, Second order energies on thin structures: variational theory and non-local effects, Paper in preparation.

    Google Scholar 

  15. G. Bouchitté and M. Valadier, Integral representation of convex functionals on a space of measures, J. Funct. Anal. 80 (1988), 398–420.

    Article  MathSciNet  MATH  Google Scholar 

  16. G. Buttazzo, Semicontinuity, Relaxation, and Integral Representation in the Calculus of Variations, Pitman Research Notes in Mathematics Series, Longman Scientific & Technical, Harlow (1989).

    Google Scholar 

  17. P. G. Ciarlet and H. Le Dret, Justification de la condition aux limites d’encastrement d’une plaque par une méthode asymptotique,C. R. Acad. Sci. Paris. Sér. I Math., 307 (20) (1988), 1015–1018.

    MATH  Google Scholar 

  18. D. Cioranescu and J. Saint Jean Paulin, Homogenization of Reticulated Structures. Applied Mathematical Sciences 136, Springer Verlag, Berlin-Heidelberg-New York (1999).

    Book  MATH  Google Scholar 

  19. B. Dacorogna, Direct Methods in the Calculus of Variations, Applied Mathematical Sciences 78, Springer-Verlag, Berlin-Heidelberg-New York (1988).

    Google Scholar 

  20. G. Dal Maso, An Introduction to Γ-convergence, Birkhäuser, Boston (1993).

    Book  Google Scholar 

  21. H. Federer, Geometric Measure Theory, Springer-Verlag, Berlin-Heidelberg-New York (1969).

    MATH  Google Scholar 

  22. H. Federer and W. Fleming, Normal and integral currents, Annals of Math., 72 (1960), 458–520.

    Article  MathSciNet  MATH  Google Scholar 

  23. I. Fragalà, Lower semi continuity of μ-quasiconvex integrals, Preprint, (2000).

    Google Scholar 

  24. I. Fragalà and C. Mantegazza, On some notions of tangent space to a measure, Proc. Roy. Soc Edinburgh, 129A (1999), 331–342.

    Article  Google Scholar 

  25. M. Giaquinta, G. Modica and J. Soucek, Cartesian Currents in the Calculus of Variations, Springer-Verlag, Berlin-Heidelberg-New-York (1998).

    Google Scholar 

  26. V. V. Jikov, S. M. Kozlov and O. A. Oleinik, Homogenization of Differential Operators and Integral Functionals, Springer-Verlag, Berlin-Heidelberg-New York (1994).

    Book  Google Scholar 

  27. H. Le Dret and A. Raoult, The nonlinear membrane model as variational limit of nonlinear three dimensional elasticity, J. Math. Pures Appl., 74 (1995), 549–578.

    MathSciNet  MATH  Google Scholar 

  28. D. Percivale, The variational methods for tensile structures, Preprint, (1991).

    Google Scholar 

  29. D. Preiss, Geometry of measures on \( {\mathbb{R}^n} \) : distribution, rectifiability and densities, Annals of Math., 125 (1987), 573–643.

    Article  MathSciNet  Google Scholar 

  30. L. Simon, Lectures on Geometric Measure Theory, Proc. Centre for Math. Anal., Australian Nat. Univ., 3 (1983).

    MATH  Google Scholar 

  31. M. Valadier, Multiapplications mesurables à valeurs convexes compactes, J. Math. Pures Appl., 50 (1971), 265–297.

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2002 Springer Basel AG

About this paper

Cite this paper

Bouchitté, G., Fragalà, I. (2002). Variational Theory of Weak Geometric Structures: The Measure Method and Its Applications. In: dal Maso, G., Tomarelli, F. (eds) Variational Methods for Discontinuous Structures. Progress in Nonlinear Differential Equations and Their Applications, vol 51. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8193-7_3

Download citation

  • DOI: https://doi.org/10.1007/978-3-0348-8193-7_3

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-9470-8

  • Online ISBN: 978-3-0348-8193-7

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics