Calculus of Variations and Geometric Measure Theory

M. Friedrich - F. Solombrino

Functionals defined on piecewise rigid functions: Integral representation and $\Gamma$-convergence

created by solombrin on 12 Apr 2019
modified on 25 Mar 2020


Published Paper

Inserted: 12 apr 2019
Last Updated: 25 mar 2020

Journal: Archive for Rational Mechanics and Analysis
Volume: 236
Pages: 1325--1387
Year: 2020
Doi: 10.1007/s00205-020-01493-8


We analyse integral representation and $\Gamma$-convergence properties of functionals defined on \emph{piecewise rigid functions}, i.e., functions which are piecewise affine on a Caccioppoli partition where the derivative in each component is constant and lies in a set without rank-one connections. Such functionals account for interfacial energies in the variational modeling of materials which locally show a rigid behavior. Our resulst are based on techniques for $\Gamma$-convergence and a careful adaption of the global method for relaxation (Bouchitt\'eet al. 1998, 2001) to this new setting, under rather general assumptions. They constitute a first step towards the investigation of lower semicontinuity, relaxation, and homogenization for free-discontinuity problems in spaces of (generalized) functions of bounded deformation.