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<p>In this paper, we study the asymptotic behavior at infinity of solutions to the minimal surface equation in exterior domains of the half space. We prove that the solution <i>u</i> tends to a linear function with rate at least <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2087_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="56" /> </InlineMediaObject> <EquationSource Format="TEX">\(x_{n}|x|^{-n}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>x</mi> <mi>n</mi> </msub> <msup> <mrow> <mo stretchy="false">|</mo> <mi>x</mi> <mo stretchy="false">|</mo> </mrow> <mrow> <mo>-</mo> <mi>n</mi> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation>. We further establish a higher order asymptotic expansion. Compared with existing results for exterior domains of the whole space, we do not require any growth condition or dimensional restriction.</p>

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Asymptotic Behavior of Exterior Minimal Graphs in Half Spaces

  • 期刊论文

  • 发表日期：

    2025年7月7日

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  • Zhenyu Fan,
  • Yugao Ouyang

摘要

In this paper, we study the asymptotic behavior at infinity of solutions to the minimal surface equation in exterior domains of the half space. We prove that the solution u tends to a linear function with rate at least \(x_{n}|x|^{-n}\) x n | x | - n . We further establish a higher order asymptotic expansion. Compared with existing results for exterior domains of the whole space, we do not require any growth condition or dimensional restriction.