Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-05T21:40:17.016Z Has data issue: false hasContentIssue false

Regularity results for equilibria in a variational model for fracture

Published online by Cambridge University Press:  14 November 2011

Emilio Acerbi
Affiliation:
Dipartimento di Matematica, Via Massimo D'Azeglio 85/A, 43100 Parma, Italy
Irene Fonseca
Affiliation:
Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, U.S.A.
Nicola Fusco
Affiliation:
Dipartimento di Matematica ‘U. Dini’, Università di Firenze, Viale Morgagni 67∕a, 50131 Firenze, Italy

Synopsis

In recent years models describing interactions between fracture and damage have been proposed in which the relaxed energy of the material is given by a functional involving bulk and interfacial terms, of the form

where Ω is an open, bounded subset of ℝN, q ≧1, gL∞ (Ω ℝN), λ, β > 0, the bulk energy density F is quasiconvex, K⊂ℝN is closed, and the admissible deformation u:Ω→ ℝN is C1 in Ω\K One of the main issues has to do with regularity properties of the ‘crack site’ K for a minimising pair (K, u). In the scalar case, i.e. when uΩ→ ℝ, similar models were adopted to image segmentation problems, and the regularity of the ‘edge’ set K has been successfully resolved for a quite broad class of convex functions F with growth p > 1 at infinity. In turn, this regularity entails the existence of classical solutions. The methods thus used cannot be carried out to the vectorial case, except for a very restrictive class of integrands. In this paper we deal with a vector-valued case on the plane, obtaining regularity for minimisers of corresponding to polyconvex bulk energy densities of the form

where the convex function h grows linearly at infinity.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 1997

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Ambrosio, L.. A compactness theorem for a new class of functions of bounded variation. Boll. Un. Mat. Ital. B 3 (1989), 857–81.Google Scholar
2Ambrosio, L.. A new proof of the SBV compactness theorem. Calc. Var. Partial Differential Equations 3 (1995), 127–37.CrossRefGoogle Scholar
3Ambrosio, L.. On the lower semicontinuity of quasiconvex integrals in SBV(Ω.ℝk).Nonlinear Anal. (to appear).Google Scholar
4Ambrosio, L., Fusco, N. and Pallara, D.. Partial regularity of free discontinuity sets II (Preprint, Dip. Mat. e. Appl. Napoli, 1995).Google Scholar
5Ambrosio, L. and Pallara, D.. Partial regularity of free discontinuity sets I (Preprint, Scuola Normale Superiore di Pisa, 1994).Google Scholar
6Bauman, P., Owen, N. C. and Phillips, D.. Maximum principles and apriori estimates for a class of problems from nonlinear elasticity. Ann. Inst. H. Poincaré Anal. Non Linéaire 8 (1991), 119–57.Google Scholar
7Blake, A. and Zisserman, A.. Visual Reconstruction (Cambridge, MA: MIT Press, 1985).Google Scholar
8Bonnet, A.. On the regularity of edges in the Mumford—Shah model for image segmentation (to appear).Google Scholar
9Carriero, M. and Leaci, A.. S k-valued maps minimizing the L P norm of the gradient with free discontinuities. Ann. Scuolo Norm. Sup. Pisa 18 (1991), 321–52.Google Scholar
10David, G. and Semmes, S.. On the singular set of minimizers of the Mumford—Shah functional. J. Math. Pures Appl. (to appear).Google Scholar
11Giorgi, E. De. Free Discontinuity Problems in the Calculus of Variations, a collection of papers dedicated to J. L. Lions on the occasion of his 60th birthday (Amsterdam: North Holland, 1991).Google Scholar
12Giorgi, E. De and Ambrosio, L.. Un nuovo tipo di funzionale del calcolo delle variazioni. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei 82 (1988), 199210.Google Scholar
13Giorgi, E. De, Carriero, M. and Leaci, A.. Existence theorem for a minimum problem with free discontinuity set. Arch. Rational Mech. Anal. 108 (1989), 195218.Google Scholar
14Dougherty, M.. Higher integrability of the gradient for minimizers of certain polyconvex functionals in the calculus of variations (Preprint).Google Scholar
15Fonseca, I. and Francfort, G.. Relaxation in BV versus quasiconvexification in W l, P; a model for the interaction between fracture and damage. Calc. Var. Partial Differential Equations 3 (1995), 407–46.Google Scholar
16Fonseca, I. and Fusco, N.. Regularity results for anisotropic image segmentation models. Ann. Scuolo Norm. Sup. Pisa (to appear).Google Scholar
17Giaquinta, M.. Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Annals of Mathematics Studies 105 (Princeton, NJ: Princeton University Press, 1983).Google Scholar
18Mumford, D. and Shah, J.. Optimal approximations by piecewise smooth functions and associated variational problems. Comm. on Pure and App. Math. 42 (1989), 577685.Google Scholar