Lower semicontinuity in GSBD for nonautonomous surface integrals

We provide a sufficient condition for lower semicontinuity of nonautonomous noncoercive surface energies defined on the space of GSBD(p) functions, whose dependence on the x-variable is W-1,W-1 or even BV: the notion of nonautonomous symmetric joint convexity, which extends the analogous definition devised for autonomous integrands in Friedrich et al. [J. Funct. Anal. 280 (2021) 108929] where the conservativeness of the approximating vector fields is assumed. This condition allows to extend to our setting a nonautonomous chain formula in SBV obtained in Ambrosio et al. [Manuscr. Math. 140 (2013) 461-480], and this is a key tool in the proof of the lower semicontinuity result. This new joint convexity can be checked explicitly for some classes of surface energies arising from variational models of fractures in inhomogeneous materials.

Automated summary

We establish a sufficient condition for the lower semicontinuity of nonautonomous noncoercive surface energies on GSBD(p) functions. The condition extends the definition of nonautonomous symmetric joint convexity, previously introduced for autonomous integrands. We also extend a nonautonomous chain formula in SBV and utilize it to prove the lower semicontinuity result. This work has practical significance in evaluating surface energies arising from variational models of fractures in inhomogeneous materials.