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<p>A partial regularity theorem is presented for minimisers of <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10231_2025_1547_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="22" /> </InlineMediaObject> <EquationSource Format="TEX">\(k^{\textrm{th}}\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>k</mi> <mtext>th</mtext> </msup> </math></EquationSource> </InlineEquation>-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 <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10231_2025_1547_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">\(\Delta _2\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="normal">Δ</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10231_2025_1547_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="22" /> </InlineMediaObject> <EquationSource Format="TEX">\(\nabla _2\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="normal">∇</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation> conditions, assuming no quantitative estimates on the second derivative of the integrand; this is new even in the <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10231_2025_1547_Article_IEq4.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="41" /> </InlineMediaObject> <EquationSource Format="TEX">\(k=1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> case. These results will also be extended to the case of strong local minimisers.</p>

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

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  • 发表日期：

    2025年1月24日

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  • Christopher Irving

摘要

A partial regularity theorem is presented for minimisers of \(k^{\textrm{th}}\) k th -order functionals subject to a quasiconvexity and general growth condition. We will assume a natural growth condition governed by an N-function satisfying the \(\Delta _2\) Δ 2 and \(\nabla _2\) ∇ 2 conditions, assuming no quantitative estimates on the second derivative of the integrand; this is new even in the \(k=1\) k = 1 case. These results will also be extended to the case of strong local minimisers.