Skip to main content
SpringerLink
Log in
Book cover

Solvability, Regularity, and Optimal Control of Boundary Value Problems for PDEs pp 431–468Cite as

Smooth and Broken Minimizers of Some Free Discontinuity Problems

  • Chapter
  • First Online:

Part of the book series: Springer INdAM Series ((SINDAMS,volume 22))

Abstract

We show that minimizers of free discontinuity problems with energy dependent on jump integrals and Dirichlet boundary conditions are smooth provided a smallness condition is imposed on data. We examine in detail two examples: the elastic-plastic beam and the elastic-plastic plate with free yield lines. In both examples there is a gap between the condition for solvability (safe load condition) and this smallness condition (load regularity condition) which imply regularity and uniqueness of minimizers. Such gap allows the existence of damaged/creased minimizers. Eventually we produce explicit examples of irregular solutions when the load is in the gap.

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

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   109.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   139.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   139.99
Price excludes VAT (USA)
  • Durable hardcover 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

References

  1. Agmon, S.: The L p approach to the Dirichlet problem. Part I: regularity theorems. Ann. Scuola Normale Sup. Pisa, Cl. Scienze 3a S. 13(4), 405–448 (1959)

    Google Scholar 

  2. Alberti, G., Bouchitté, G., dal Maso, G.: The calibration method for the Mumford-Shah functional and free-discontinuity problems. Calc. Var. Partial Diff. Equ. 16(3), 299–333 (2003)

    Google Scholar 

  3. Amar, M., De Cicco, V.: The uniqueness as a generic property for some one dimensional segmentation problems. Rend. Sem. Univ. Padova 88, 151–173 (1992)

    MATH  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  5. Babadjian, J.F., Chambolle, A., Lemenant, A.: Energy release rate for non smooth cracks in planar elasticity. J. l’Ecole Polytechnique - Math. 2, 117–152 (2015)

    Article  MATH  MathSciNet  Google Scholar 

  6. Barenblatt, G.I.: The formation of equilibrium cracks during brittle fracture, general ideas and hypotheses. Axially symmetric cracks. Appl. Math. Mech. (PMM) 23, 622–636 (1959)

    Google Scholar 

  7. Braides, A., Dal Maso, G., Garroni, A.: Variational formulation of softening phenomena in fracture mechanics: the one-dimensional case. Arch. Rational Mech. Anal. 146, 23–58 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  8. Boccellari, T., Tomarelli, F.: Generic uniqueness of minimizer for Blake & Zisserman functional. Rev. Mat. Complut. 26, 361–408 (2013). doi:10.1007/s13163-012-0103-1

    Article  MATH  MathSciNet  Google Scholar 

  9. Braides, A., Fonseca, I.: Brittle thin films. Appl. Math. Opt. 44(3), 299–323 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  10. Carriero, M., Leaci, A., Tomarelli, F.: Plastic free discontinuities and special bounded hessian. C. R. Acad. Sci. Paris Sér. I Math. 314(8), 595–600 (1992)

    MATH  MathSciNet  Google Scholar 

  11. Carriero, M., Leaci, A., Tomarelli, F.: Special Bounded Hessian and elastic-plastic plate. Rend. Accad. Naz. Sci. XL, Mem. Mat. 5(16), 223–258 (1992)

    Google Scholar 

  12. Carriero, M., Leaci, A., Tomarelli, F.: Strong solution for an elastic plastic plate. Calc. Var. Partial Differ. Equ. 2(2), 219–240 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  13. Carriero, M., Leaci, A., Tomarelli, F.: A second order model in image segmentation: Blake & Zisserman functional. In: Variational Methods for Discontinuous Structures (Como, 1994), Progress in Nonlinear Differential Equations and Their Applications, vol. 25, pp. 55–72. Birkhäuser, Basel (1996)

    Google Scholar 

  14. Carriero, M., Leaci, A., Tomarelli, F.: Necessary conditions for extremals of Blake & Zisserman functional. C.R. Math. Acad. Sci. Paris 334(4), 343–348 (2002). doi:10.1016/S1631-073X(02)02231-8

    Google Scholar 

  15. Carriero, M., Leaci, A., Tomarelli, F.: Calculus of variations and image segmentation. J. Physiol. Paris 97(2–3), 343–353 (2003). doi:10.1016/j.jphysparis.2003.09.008

    Article  MATH  Google Scholar 

  16. Carriero, M., Leaci, A., Tomarelli, F.: Second order variational problems with free discontinuity and free gradient discontinuity. In: Calculus of Variations: Topics from the Mathematical heritage of Ennio De Giorgi. Quaderni di Matematica, vol. 14, pp. 135–186. Department of Mathematics, Seconda University of Napoli, Caserta (2004)

    Google Scholar 

  17. Carriero, M., Leaci, A., Tomarelli, F.: Euler equations for Blake and Zisserman functional. Calc. Var. Partial Differ. Equ. 32(1), 81–110 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  18. Carriero, M., Leaci, A., Tomarelli, F.: A Dirichlet problem with free gradient discontinuity. Adv. Math. Sci. Appl. 20(1), 107–141 (2010)

    MATH  MathSciNet  Google Scholar 

  19. Carriero, M., Leaci, A., Tomarelli, F.: A candidate local minimizer of Blake & Zisserman functional. J. Math. Pures Appl. 96, 58–87 (2011). doi:10.1016/j.matpur.2011.01.005

    Article  MATH  MathSciNet  Google Scholar 

  20. Carriero, M., Leaci, A., Tomarelli, F.: Image inpainting via variational approximation of a Dirichlet problem with free discontinuity. Adv. Calc.Var. 7(3), 267–295 (2014)

    Google Scholar 

  21. Carriero, M., Leaci, A., Tomarelli, F.: A survey on the Blake–Zisserman functional. Milan J. Math. 83, 397–420 (2015). doi:10.1007/s00032-015-0246-x

    Article  MATH  MathSciNet  Google Scholar 

  22. Ciarlet, P.G.: Mathematical Elasticity, vol II: Theory of Plates. Studies in Mathematics and Its Applications. North-Holland, Amsterdam (1997).

    MATH  Google Scholar 

  23. Dal Maso, G.: Generalised functions of bounded deformation. J. Eur. Math. Soc. 15(5), 1943–1997 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  24. Dal Maso, G., Lazzaroni, G.: Crack growth with non-interpenetration: a simplified proof for the pure Neumann problem. Discr. Cont. Continuous Dyn. Syst. 31(4), 1219–1231 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  25. Dal Maso, G., Francfort, G., Toader, R.: Quasistatic crack growth in nonlinear elasticity. Arch. Rat. Mech Anal. 176(2), 165–225 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  26. De Giorgi, E.: Free discontinuity problems in calculus of variations. In: Dautray, R. (ed.) Frontiers in Pure & Applied Mathematics, pp. 55–61. North-Holland, Amsterdam (1991)

    Google Scholar 

  27. De Giorgi, E., Ambrosio, L.: Un nuovo tipo di funzionale del Calcolo delle Variazioni. Atti Accad. Naz. Lincei, Rend. Cl. Sci. Fis. Mat. Natur. 82, 199–210 (1988)

    Google Scholar 

  28. De Giorgi, E., Carriero, M., Leaci, A.: Existence theorem for a minimum problem with free discontinuity set. Arch. Rational Mech. Anal. 108, 195–218 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  29. Del Piero, G., Truskinovsky, L.: A one-dimensional model for localized and distributed failure. J. Phys. IV France 8, 95–102 (1998)

    Google Scholar 

  30. Del Piero, G.: Interface energies and structured deformations in plasticity. In: Variational Methods for Discontinuous Structures. PNLDE, vol. 51, pp. 103–116. Birkhäuser, Basel (2002)

    Google Scholar 

  31. Fonseca, I., Leoni, G., Paroni, R.: On hessian matrices in the space BH. Commun. Contemp. Math. 7, 401–420 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  32. Francfort, G., Marigo, J.J.: Revisiting brittle fracture as an energy minimization problem. J. Mech. Phys. Solids 46, 1319–1342 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  33. Griffith, A.A.: The phenomenon of rupture and flow in solids. Phyl. Trans. Roy. Soc. A 221, 163–198 (1920)

    Article  Google Scholar 

  34. Lü, Z.-X., Yang, X.-P.: Existence of free discontinuity problems in SBD(Ω). Nonlinear Anal. 71, 332–340 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  35. Maddalena, F., Percivale, D., Puglisi, G., Truskinovsky, L.: Mechanics of reversible unzipping. Continuum Mech. Thermodyn. 21, 251–268 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  36. Maddalena, F., Percivale, D., Tomarelli, F.: Elastic structures in adhesion interaction. In: Frediani, A., Buttazzo, G. (eds.) Variational Analysis and Aerospace Engineering. Springer Optimization and Its Applications, vol. 66, pp. 289–304. Springer, Berlin (2012). ISBN 978-1-4614-2434-5

    Chapter  Google Scholar 

  37. Maddalena, F., Percivale, D., Tomarelli, F.: Local and non-local energies in adhesive interaction. IMA J. Appl. Math. 81, 1051–1075 (2016)

    Article  MathSciNet  Google Scholar 

  38. Percivale, D.: Upper and lower bounds for Poincaré-type constants in BH. J.Convex Anal. 17, 1089–1111 (2010)

    Google Scholar 

  39. Percivale, D., Tomarelli, F.: Scaled Korn-Poincaré inequality in BD and a model of elastic plastic cantilever. Asymptot. Anal. 23(3–4), 291–311 (2000)

    MATH  MathSciNet  Google Scholar 

  40. Percivale, D., Tomarelli, F.: From SBD to SBH: the elastic plastic plate. Interfaces Free Bound. 4(2), 137–165 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  41. Percivale, D., Tomarelli, F.: From special bounded deformation to special bounded Hessian: the elastic plastic beam. Math. Models Methods Appl. Sci. 15(7), 1009–1058 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  42. Percivale, D., Tomarelli, F.: Smooth and creased equilibria for elastic-plastic plates and beams. In: Variational Problems in Material Science. Progress in Nonlinear Differential Equations and Their Applications, vol. 68, pp. 127–136. Birkhäuser, Basel (2006)

    Google Scholar 

  43. Percivale, D., Tomarelli, F.: A variational principle for plastic hinges in a beam. Math. Models Methods Appl. Sci. 19(12), 2263–2297 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  44. Percivale, D., Tomarelli, F.: Plastic hinges in a beam. In: Buttazzo, G., Frediani, A. (eds.) Variational Analysis and Aerospace Engineering. Springer Optimization and Its Applications, vol. 33, pp. 343–348. Springer, Berlin (2009). ISBN 978-0-387-95856-9.

    Chapter  Google Scholar 

  45. Savaré, G., Tomarelli, F.: Superposition and chain rule for bounded Hessian functions. Adv. Math. 140(12), 237–281 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  46. Save, M.A., Massonet, C.E.: Plastic Analysis and Design of Plates, Shells and Disks. Applied Mathematics and Mechanics. North-Holland, Amsterdam (1972)

    Google Scholar 

  47. Temam, R.: Problèmes Mathematiques en Plasticité. Gauthier-Vllars, Paris (1983)

    MATH  Google Scholar 

  48. Villaggio, P.: Qualitative Methods in Elasticity. Nordhoff, Leyden (1977)

    MATH  Google Scholar 

Download references

Acknowledgements

This paper is dedicated to Gianni Gilardi on the occasion of his 70th Birthday.

Author information

Authors and Affiliations

  1. Dipartimento di Ingegneria Meccanica, Università di Genova, Piazzale Kennedy, Fiera del Mare, Padiglione D, 16129, Genova, Italy

    Danilo Percivale

  2. Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci, 20133, Milano, Italy

    Franco Tomarelli

Authors
  1. Danilo Percivale
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Franco Tomarelli
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Franco Tomarelli .

Editor information

Editors and Affiliations

  1. Dipartimento di Matematica, Università di Pavia, Pavia, Italy

    Pierluigi Colli

  2. Dipartimento di Matematica, Università di Bologna , Bologna, Italy

    Angelo Favini

  3. Dipartimento di Matematica, Università di Pavia, Pavia, Italy

    Elisabetta Rocca

  4. Dipartimento di Matematica, Università di Pavia, Pavia, Italy

    Giulio Schimperna

  5. für Angewandte Analysis und Stochastik (WIAS), Weierstraß-Institut , Berlin, Germany

    Jürgen Sprekels

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this chapter

Cite this chapter

Percivale, D., Tomarelli, F. (2017). Smooth and Broken Minimizers of Some Free Discontinuity Problems. In: Colli, P., Favini, A., Rocca, E., Schimperna, G., Sprekels, J. (eds) Solvability, Regularity, and Optimal Control of Boundary Value Problems for PDEs. Springer INdAM Series, vol 22. Springer, Cham. https://doi.org/10.1007/978-3-319-64489-9_17

Download citation

Publish with us

Policies and ethics

Search

Navigation