Partial Regularity for $\mathbb{A}$-quasiconvex Functionals

TL;DR

This paper proves partial Hölder regularity for local minimizers of variational problems involving strongly quasi-convex integrands with linear growth, where the full gradient is replaced by a differential operator, under the assumption of $ ext{C}$-ellipticity.

Contribution

It extends partial regularity results to functionals involving a differential operator $ ext{A}$, generalizing previous work on symmetric quasiconvex and BV functionals.

Findings

Established partial Hölder regularity for $ ext{A}$-quasiconvex functionals.

Adapted methods from recent partial regularity research.

Applicable to variational problems with linear growth and $ ext{C}$-elliptic operators.

Abstract

We establish partial H\"older regularity for (local) generalised minimisers of variational problems involving strongly quasi-convex integrands of linear growth, where the full gradient is replaced by a first order homogeneous differential operator $\mathbb{A}$ with constant coefficients. Working under the assumption of $\mathbb{A}$ being $\mathbb{C}$-elliptic, this is achieved by adapting a method recently introduced by Gmeineder (Partial Regularity for Symmetric Quasiconvex Functionals on BD, J. Math. Pures Appl. 145 (2021), Issue 9, pp. 83--129) and Gmeineder & Kristensen (Partial regularity for BV Minimizers, Arch. Ration. Mech. Anal. 232 (2019), Issue 3, pp. 1429--1473).