# Partial regularity for symmetric quasiconvex functionals on BD

## TL;DR

This paper proves the first partial regularity results for symmetric quasiconvex functionals of linear growth on the space of functions of bounded deformation (BD), addressing unique challenges posed by Ornstein's Non-Inequality.

## Contribution

It extends partial regularity results from BV to BD functionals and establishes regularity for superlinear growth functionals via reduction techniques.

## Key findings

- First partial regularity results for symmetric quasiconvex BD functionals.
- Extension of BV regularity results to the BD setting.
- Partial regularity for superlinear growth functionals achieved through reduction.

## Abstract

We establish the first partial regularity results for (strongly) symmetric quasiconvex functionals of linear growth on BD, the space of functions of bounded deformation. By Rindler's foundational work (Lower semicontinuity for integral functionals in the space of functions of bounded deformation via rigidity and Young measures, Arch. Ration. Mech. Anal. 202 (2011), no. 1, 63-113), symmetric quasiconvexity is the foremost notion as to sequential weak*-lower semicontinuity of functionals on BD. The overarching main difficulty here is Ornstein's Non-Inequality, implying that the BD-case is genuinely different from the study of variational integrals on BV. In particular, this paper extends the recent work of Kristensen and the author (Partial regularity for BV-Minimizers, Arch. Ration. Mech. Anal. 232 (2019), Issue 3, 1429-1473) from the BV- to the BD-situation. Alongside, we establish partial regularity results for strongly quasiconvex functionals of superlinear growth by reduction to the full gradient case, which might be of independent interest.