Abstract

Lower semicontinuity of surface energies in integral form is known to be equivalent to BV-ellipticity of the surface density. In this paper, we prove that BV-ellipticity coincides with the simpler notion of biconvexity for a class of densities that depend only on the jump height and jump normal, and are positively 1-homogeneous in the first argument. The second main result is the analogous statement in the setting of bounded deformations, where we show that BD-ellipticity reduces to symmetric biconvexity. Our techniques are primarily inspired by constructions from the analysis of structured deformations and the general theory of free discontinuity problems.

Author information

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Dominik Engl

Mathematisch-Geographische Fakultät, Katholische Universität, Eichstätt-Ingolstadt, Germany

dominik.engl@ku.de

Carolin Kreisbeck

Mathematisch-Geographische Fakultät, Katholische Universität, Eichstätt-Ingolstadt, Germany

carolin.kreisbeck@ku.de

Keywords

  • Interfacial energy
  • lower semicontinuity
  • BV- and BD-ellipticity
  • biconvexity

Mathematics Subject Classification

49J45; 26B25, 9Q20, 70G75

References

  1. F. J. Almgren, Jr.: Existence and Regularity Almost Everywhere of Solutions to Elliptic Variational Problems with Constraints, Memoirs of the AMS 4/165, American Mathematical Society, Providence (1976).https://doi.org/10.1090/memo/0165Google Scholar
  2. L. Ambrosio, A. Braides: Functionals defined on partitions in sets of finite perimeter. I: Integral representation and Γ-convergence, J. Math. Pures Appl., Ser. 9, 69/3 (1990) 285–305.Google Scholar
  3. L. Ambrosio, A. Braides: Functionals defined on partitions in sets of finite perimeter. II: Semicontinuity, relaxation and homogenization, J. Math. Pures Appl., Ser. 9, 69/3 (1990) 307–333.Google Scholar
  4. L. Ambrosio, A. Braides: Energies in SBV and variational models in fracture mechanics, in: Homogenization and Applications to Material Sciences (Nice 1995), GAKUTO Internat. Ser. Math. Sci. Appl. Vol. 9, Gakkōtosho, Tokyo (1995) 1–22.Google Scholar
  5. L. Ambrosio, N. Fusco, D. Pallara: Functions of Bounded Variation and Free Discontinuity Problems, Oxford Mathematical Monographs, Oxford University Press, New York (2000).https://doi.org/10.1093/oso/9780198502456.001.0001Google Scholar
  6. G. Bouchitté, I. Fonseca, L. Mascarenhas: A global method for relaxation, Arch. Rational Mech. Analysis 145/1 (1998) 51–98.https://doi.org/10.1007/s002050050124Google Scholar
  7. A. Braides, A. Defranceschi, E. Vitali: Homogenization of free discontinuity problems, Arch. Rational Mech. Analysis 135/4 (1996) 297–356.https://doi.org/10.1007/BF02198476Google Scholar
  8. D. G. Caraballo: Crystals and polycrystals in Rn : lower semicontinuity and existence, J. Geom. Analysis 18/1 (2008) 68–88.https://doi.org/10.1007/s12220-007-9006-7Google Scholar
  9. D. G. Caraballo: BV-ellipticity and lower semicontinuity of surface energy of Caccioppoli partitions of Rn, J. Geom. Analysis 23/1 (2013) 202–220.https://doi.org/10.1007/s12220-011-9242-8Google Scholar
  10. D. G. Caraballo: Existence of surface energy minimizing partitions of Rn satisfying volume constraints, Trans. Amer. Math. Soc. 369/3 (2017) 1517–1546.https://doi.org/10.1090/tran/6630Google Scholar
  11. R. Choksi, I. Fonseca: Bulk and interfacial energy densities for structured deformations of continua, Arch. Rational Mech. Analysis 138/1 (1997) 37–103.https://doi.org/10.1007/s002050050036Google Scholar
  12. B. Dacorogna: Direct Methods in the Calculus of Variations, 2nd ed., Applied Mathematical Sciences Vol. 78, Springer, New York (2008).Google Scholar
  13. G. Del Piero, D. R. Owen: Structured deformations of continua, Arch. Rational Mech. Analysis 124/2 (1993) 99–155.https://doi.org/10.1007/BF00375133Google Scholar
  14. L. C. Evans, R. F. Gariepy: Measure Theory and Fine Properties of Functions, Studies in Advanced Mathematics, CRC Press, Boca Raton (1992).Google Scholar
  15. M. Friedrich, M. Perugini, F. Solombrino: Lower semicontinuity for functionals defined on piecewise rigid functions and on GSBD, J. Funct. Analysis 280/7 (2021), art.no. 108929, 45p.https://doi.org/10.1016/j.jfa.2021.108929Google Scholar
  16. M. Friedrich, F. Solombrino: Functionals defined on piecewise rigid functions: integral representation and Γ-convergence, Arch. Ration. Mech. Analysis 236/3 (2020) 1325–1387.https://doi.org/10.1007/s00205-020-01493-8Google Scholar
  17. G. Gargiulo, E. Zappale: A lower semicontinuity result in SBD, J. Convex Analysis 15/1 (2008) 191–200.https://doi.org/10.68381/jca15013Google Scholar
  18. G. Gargiulo, E. Zappale: A lower semicontinuity result in SBD for surface integral functionals of fracture mechanics, Asymptotic Analysis 72/3-4 (2011) 231–249.https://doi.org/10.3233/ASY-2011-1032Google Scholar
  19. G. Gargiulo, E. Zappale: Some sufficient conditions for lower semicontinuity in SBD and applications to minimum problems, Math. Methods Appl. Sci. 34/12 (2011) 1541–1552.https://doi.org/10.1002/mma.1464Google Scholar
  20. J. Kristensen, F. Rindler: Piecewise affine approximations for functions of bounded variation, Numer. Math. 132/2 (2016) 329–346.https://doi.org/10.1007/s00211-015-0721-xGoogle Scholar
  21. J. Matias, M. Morandotti, D. R. Owen: Energetic Relaxation to Structured Deformations. A Multiscale Geometrical Basis for Variational Problems in Continuum Mechanics, SpringerBriefs on PDEs and Data Science, Springer, Singapore (2023).https://doi.org/10.1007/978-981-19-8800-4Google Scholar
  22. M. Silhavy: The general form of the relaxation of a purely interfacial energy for structured deformations, Math. Mech. Complex Systems 5 (2017) 191–215.https://doi.org/10.2140/memocs.2017.5.191Google Scholar

Suggested citation

D. Engl, C. Kreisbeck, M. Morandotti. “Characterizing BV- and BD-Ellipticity for a Class of Positively 1-Homogeneous Surface Energy Densities.” Journal of Convex Analysis 33 (2026), No. 1&2, 29–55. https://doi.org/10.68381/jca33003

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Published by Heldermann Verlag, 2026. Rights now held by Banach Press.