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<p>We prove the <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="42543_2025_97_Article_IEq4.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="30" /> </InlineMediaObject> <EquationSource Format="TEX">\(C^{1,1}\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mrow> <mn>1</mn> <mo>,</mo> <mn>1</mn> </mrow> </msup> </math></EquationSource> </InlineEquation>-regularity for stationary <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="42543_2025_97_Article_IEq5.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="33" /> </InlineMediaObject> <EquationSource Format="TEX">\(C^{1,\alpha }\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mrow> <mn>1</mn> <mo>,</mo> <mi>α</mi> </mrow> </msup> </math></EquationSource> </InlineEquation> (<InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="42543_2025_97_Article_IEq6.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="69" /> </InlineMediaObject> <EquationSource Format="TEX">\(\alpha \in (0,1)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>) solutions to the multiple membrane problem. This regularity estimate was essentially used in our recent work on Yau’s four minimal spheres conjecture [<a href="http://arxiv.org/abs/2305.08755">arXiv:2305.08755</a>].</p>

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Improved \(C^{1,1}\) Regularity for Multiple Membranes Problem

  • 期刊论文

  • 发表日期：

    2025年4月24日

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  • Zhichao Wang,
  • Xin Zhou

摘要

We prove the \(C^{1,1}\) C 1 , 1 -regularity for stationary \(C^{1,\alpha }\) C 1 , α ( \(\alpha \in (0,1)\) α ∈ ( 0 , 1 ) ) solutions to the multiple membrane problem. This regularity estimate was essentially used in our recent work on Yau’s four minimal spheres conjecture [arXiv:2305.08755].