Milan Journal of Mathematics

, Volume 85, Issue 1, pp 149–185 | Cite as

Linear Hyperbolic Systems in Domains with Growing Cracks

  • Maicol CaponiEmail author


We consider the hyperbolic system ü \({ - {\rm div} (\mathbb{A} \nabla u) = f}\) in the time varying cracked domain \({\Omega \backslash \Gamma_t}\), where the set \({\Omega \subset \mathbb{R}^d}\) is open, bounded, and with Lipschitz boundary, the cracks \({\Gamma_t, t \in [0, T]}\), are closed subsets of \({\bar{\Omega}}\), increasing with respect to inclusion, and \({u(t) : \Omega \backslash \Gamma_t \rightarrow \mathbb{R}^d}\) for every \({t \in [0, T]}\). We assume the existence of suitable regular changes of variables, which reduce our problem to the transformed system \({ - {\rm div} (\mathbb{B}\nabla v) + a\nabla v - 2 \nabla \dot{v}b = g}\) on the fixed domain \({\Omega \backslash \Gamma_0}\). Under these assumptions, we obtain existence and uniqueness of weak solutions for these two problems. Moreover, we show an energy equality for the functions v, which allows us to prove a continuous dependence result for both systems. The same study has already been carried out in [3, 7] in the scalar case.

Mathematics Subject Classification (2010)

35L53 35A01 35A02 35Q74 74R10 


Second order linear hyperbolic systems dynamic fracture mechanics cracking domains 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Adams R.A.: Sobolev spaces. Pure and Applied mathematics, Vol. 65. Academic Press, New York-London, (1975)Google Scholar
  2. 2.
    Dal Maso G., Larsen C.J.: Existence for wave equations on domains with arbitrary growing cracks. Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 22, 387–408 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    G. Dal Maso and I. Lucardesi: The wave equation on domains with cracks growing on a prescribed path: existence, uniqueness, and continuous dependence on the data. Appl. Math. Res. Express (2016), doi: 10.1093/amrx/abw006.
  4. 4.
    R. Dautray and J.L. Lions: Analyse mathématique et calcul numérique pour les sciences et les techniques. Vol. 8. Évolution: semi-groupe, variationnel. Masson, Paris, 1988.Google Scholar
  5. 5.
    Ladyzenskaya O.A.: On integral estimates, convergence, approximate methods, and solutions in functionals for linear elliptic operators. Vestnik Leningrad. Univ. 13, 60–69 (1958)MathSciNetGoogle Scholar
  6. 6.
    Lions J.L., Magenes E.: Non-homogeneous boundary value problems and applications. Vol. I. Die Grundlehren der mathematischen Wissenschaften, Band 181. Springer-Verlag, New York-Heidelberg (1972)CrossRefGoogle Scholar
  7. 7.
    Nicaise S., Sändig A.M.: Dynamic crack propagation in a 2D elastic body: the out-of-plane case. J. Math. Anal. Appl. 329, 1–30 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Oleinik O.A., Shamaev A.S., Yosifian G.A.: Mathematical problems in elasticity and homogenization. Studies in Mathematics and its Applications, 26. North-Holland Publishing Co., Amsterdam (1992)zbMATHGoogle Scholar

Copyright information

© Springer International Publishing 2017

Authors and Affiliations

  1. 1.SISSATriesteItaly

Personalised recommendations