Lower semicontinuity inGSBDfor nonautonomous surface integrals

Abstract

We provide a sufficient condition for lower semicontinuity of nonautonomous noncoercive surface energies defined on the space ofGSBDpfunctions, whose dependence on thex-variable isW1,1or evenBV: the notion ofnonautonomous symmetric joint convexity, which extends the analogous definition devised for autonomous integrands in Friedrichet 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 inSBVobtained in Ambrosioet 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.