Partial regularity for minima of higher-order quasiconvex integrands with natural Orlicz growth

A partial regularity theorem is presented for minimisers of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>k</mml:mi> <mml:mtext>th</mml:mtext></mml:msup> </mml:math> -order functionals subject to a quasiconvexity and general growth condition. We will assume a natural growth condition governed by an <i>N</i>-function satisfying the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>Δ</mml:mi> <mml:mn>2</mml:mn></mml:msub> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>∇</mml:mi> <mml:mn>2</mml:mn></mml:msub> </mml:math> conditions, assuming no quantitative estimates on the second derivative of the integrand; this is new even in the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>k</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn></mml:mrow> </mml:math> case. These results will also be extended to the case of strong local minimisers.