Lower semicontinuity in $GSBD$ for nonautonomous surface integrals

TL;DR

This paper establishes a sufficient condition called nonautonomous symmetric joint convexity for the lower semicontinuity of noncoercive surface energies in $GSBD^p$, extending previous autonomous results and applicable to fracture models in inhomogeneous materials.

Contribution

It introduces a new convexity condition for nonautonomous surface energies in $GSBD^p$, enabling lower semicontinuity results and extending chain formulas to nonautonomous settings.

Findings

The nonautonomous symmetric joint convexity condition is explicitly checkable for certain fracture energy classes.

The condition extends chain formulas from autonomous to nonautonomous surface energies.

The results apply to variational models of fractures in inhomogeneous materials.

Abstract

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 $W^{1,1}$ or even $BV$: the notion of nonautonomous symmetric joint convexity, which extends the analogous definition devised for autonomous integrands in arXiv:2002.08133 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 arXiv:1512.02839, 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.