Partial regularity for $\omega$-minimizers of quasiconvex functionals

TL;DR

This paper proves partial regularity results for $ ext{omega}$-minimizers of quasiconvex functionals with power growth, including $BV$ minimizers and cases with subquadratic growth, under certain smallness and Dini conditions.

Contribution

It establishes new partial regularity results for $ ext{omega}$-minimizers of quasiconvex functionals, including $BV$ cases and subquadratic growth, under minimal assumptions.

Findings

Partial regularity for $BV$ $ ext{omega}$-minimizers under Dini condition.

Partial Hölder continuity in subquadratic case with small $ ext{omega}$.

Results applicable to quasiconvex functionals with power growth.

Abstract

We establish partial regularity for the $\omega$-minimizers of quasiconvex functionals of power growth. A first-order partial regularity result of $BV$ $\omega$-minimizers is obtained in the linear growth case under a Dini-type condition on $\omega$. Only assuming the smallness of $\omega$ near the origin, we show partial H\"{o}lder continuity in the subquadratic case by considering a normalised excess.