Second-Order Structured Deformations in the Space of Functions of Bounded Hessian

  • Irene Fonseca
  • Adrian Hagerty
  • Roberto ParoniEmail author


This work addresses second-order structured deformations in the framework of the space of special functions of bounded Hessian, BH. An integral representation result is obtained in BH in the vein of the global method for relaxation of Bouchitté et al. (Arch. Ration. Mech. Anal. 145:51–98, 1998) and is applied to a relaxation problem in the context of structured deformations.


Relaxation Lower semicontinuity Second-order structured deformations Functions of bounded Hessian 

Mathematics Subject Classification

49J45 49S05 74Q99 



This paper is part of the A. Hagerty’s Ph.D. thesis at Carnegie Mellon University. The authors acknowledge the Center for Nonlinear Analysis where part of this work was carried out. The research of I. Fonseca was partially funded by the National Science Foundation under Grant No. DMS-1411646. The research of A. Hagerty was partially funded by National Science Foundation under PIRE Grant Nos. OISE-0967140 and DMS-1411646. The research of R. Paroni was partially funded from the Università di Pisa through the Grant PRA_2018_61.


  1. Ball, J.M., Currie, J.C., Olver, P.J.: Null Lagrangians, weak continuity, and variational problems of arbitrary order. J. Funct. Anal. 41, 135–174 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  2. Barroso, A.C., Matias, J., Morandotti, M., Owen, D.R.: Second-order structured deformations: relaxation, integral representation and applications. Arch. Ration. Mech. Anal. 225, 1025–1072 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  3. Bouchitté, G., Fonseca, I., Mascarenhas, L.: A global method for relaxation. Arch. Ration. Mech. Anal. 145, 51–98 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  4. Bouchitté, G., Fonseca, I., Leoni, G., Mascarenhas, L.: A global method for relaxation in \(W^{1, p}\) and in \({{\rm SBV}}_p\). Arch. Ration. Mech. Anal. 165, 187–242 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  5. Choksi, R., Fonseca, I.: Bulk and interfacial energy densities for structured deformations of continua. Arch. Ration. Mech. Anal. 138, 37–103 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  6. Dal Maso, G., Fonseca, I., Leoni, G.: Nonlocal character of the reduced theory of thin films with higher order perturbations. Adv. Calc. Var. 3, 287–319 (2010)MathSciNetzbMATHGoogle Scholar
  7. Del Piero, G., Owen, D.R.: Structured deformations of continua. Arch. Ration. Mech. Anal. 124, 99–155 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  8. Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions. CRC Press, Boca Raton (2015)CrossRefzbMATHGoogle Scholar
  9. Fonseca, I., Müller, S.: Relaxation of quasiconvex functionals in \({\text{ BV }}({\Omega }, {\text{ R }}^p)\) for integrands \(f(x, u,\nabla u)\). Arch. Ration. Mech. Anal. 123, 1–49 (1993)CrossRefzbMATHGoogle Scholar
  10. Fonseca, I., Leoni, G., Paroni, R.: On Hessian matrices in the space \(BH\). Commun. Contemp. Math. 7, 401–420 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  11. Hagerty, A.: Relaxation of functionals in the space of vector-valued functions of bounded Hessian. Calc. Var. Partial Differ. Equ. 58, 4 (2019)MathSciNetCrossRefzbMATHGoogle Scholar
  12. Kristensen, J., Rindler, F.: Relaxation of signed integral functionals in BV. Calc. Var. Partial Differ. Equ. 37, 29–62 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  13. Owen, D.R., Paroni, R.: Second-order structured deformations. Arch. Ration. Mech. Anal. 155, 215–235 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  14. Paroni, R.: Second-Order Structured Deformations: Approximation Theorems and Energetics, pp. 177–202. Springer, Vienna (2004)Google Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mathematical SciencesCarnegie Mellon UniversityPittsburghUSA
  2. 2.Dipartimento di Ingegneria Civile e IndustrialeUniversità di PisaPisaItaly

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