An approximation result for special functions with bounded deformation

✍ Scribed by Antonin Chambolle

• Publisher: Elsevier Science
• Year: 2004
• Tongue: English
• Weight: 270 KB
• Volume: 83
• Category: Article
• ISSN: 0021-7824
• DOI: 10.1016/j.matpur.2004.02.004

✦ Synopsis

A "special displacement with bounded deformation" is a function u : Ω ⊂ R N → R N whose symmetrized gradient is a bounded measure which coincides, outside a (N -1)-dimensional rectifiable "jump set" J u , with a summable function e(u). We show that in dimension N = 2, when u and e(u) are square integrable, and the total length H 1 (J u ) is finite, then such a displacement is approximated with a sequence (u n ) n 1 of piecewise continuous displacements whose jump sets J u n are (relatively) closed, with u n and e(u n ) converging strongly in L 2 , respectively to u and e(u), and the lengths H 1 (J u n ) converging to H 1 (J u ). As an application, we approximate with a sequence of elliptic functionals a functional which appears in the theory of brittle fracture in linearized elasticity.