Abstract
We prove the existence of weak solutions to the homogeneous wave equation on a suitable class of time-dependent domains. Using the approach suggested by De Giorgi and developed by Serra and Tilli, such solutions are approximated by minimizers of suitable functionals in space-time.
Funding statement: This material is based on work supported by the Italian Ministry of Education, University, and Research through the Project “Calculus of Variations” (PRIN 2015). The authors are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).
References
[1] L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford University Press, Oxford, 2000. Search in Google Scholar
[2] G. Dal Maso and C. J. Larsen, Existence for wave equations on domains with arbitrary growing cracks, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 22 (2011), 387–408. 10.4171/RLM/606Search in Google Scholar
[3] G. Dal Maso, C. J. Larsen and R. Toader, Existence for constrained dynamic Griffith fracture with a weak maximal dissipation condition, J. Mech. Phys. Solids 95 (2016), 697–707. 10.1016/j.jmps.2016.04.033Search in Google Scholar
[4] G. Dal Maso and R. Toader, On the Cauchy problem for the wave equation on time-dependent domains, preprint SISSA (2018). 10.1016/j.jde.2018.08.056Search in Google Scholar
[5] E. De Giorgi, Conjectures concerning some evolution problems, Duke Math. J. 81 (1996), 255–268. 10.1215/S0012-7094-96-08114-4Search in Google Scholar
[6] L. B. Freund, Dynamic Fracture Mechanics, Cambridge University Press, New York, 1990. 10.1017/CBO9780511546761Search in Google Scholar
[7] S. Nicaise and A.-M. Sändig, Dynamic crack propagation in a 2D elastic body: The out-of-plane case, J. Math. Anal. Appl. 329 (2007), 1–30. 10.1016/j.jmaa.2006.06.043Search in Google Scholar
[8] E. Serra and P. Tilli, Nonlinear wave equations as limits of convex minimization problems: Proof of a conjecture by De Giorgi, Ann. of Math. (2) 175 (2012), 1551–1574. 10.4007/annals.2012.175.3.11Search in Google Scholar
[9] E. Serra and P. Tilli, A minimization approach to hyperbolic Cauchy problems, J. Eur. Math. Soc. (JEMS) 18 (2016), 2019–2044. 10.4171/JEMS/637Search in Google Scholar
© 2020 Walter de Gruyter GmbH, Berlin/Boston
