Lower semicontinuity for functionals defined on piecewise rigid functions and on $GSBD$

Abstract

In this work, we provide a characterization result for lower semicontinuity of surface energies defined on piecewise rigid functions, i.e., functions which are piecewise affine on a Caccioppoli partition where the derivative in each component is a skew symmetric matrix. This characterization is achieved by means of an integral condition, called BD-ellipticity, which is in the spirit of BV -ellipticity defined by Ambrosio and Braides [5]. By specific examples we show that this novel concept is in fact stronger compared to its BV analog. We further provide a sufficient condition implying BD-ellipticity which we call symmetric joint convexity. This notion can be checked explicitly for certain classes of surface energies which are relevant for applications, e.g., for variational fracture models. Finally, we give a direct proof that surface energies with symmetric jointly convex integrands are lower semicontinuous also on the larger space of GSBD p functions.

Integral representation for energies in linear elasticity with surface discontinuities

“…(ii) As F is lower semicontinuous on W 1,p , the integrand f is quasiconvex [46]. Since F is lower semicontinuous on piecewise rigid functions, the integrand g is BD-elliptic [42] (at least if one can ensure, for instance, that g has a continuous dependence in x). A fortiori, g is BV -elliptic [3].…”