Calculus of Variations and Geometric Measure Theory

M. Friedrich - M. Perugini - F. Solombrino

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

created by solombrin on 19 Feb 2020
modified on 22 Jan 2021


Published Paper

Inserted: 19 feb 2020
Last Updated: 22 jan 2021

Journal: Journal of Functional Analysis
Volume: 280
Number: 7
Pages: 108929
Year: 2021
Doi: 10.1016/j.jfa.2021.108929

ArXiv: 2002.08133 PDF


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. 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.