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<p>For any subharmonic function <i>u</i>, we prove that <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(|\partial _j u|\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mo stretchy="false">|</mo> </mrow> <msub> <mi>∂</mi> <mi>j</mi> </msub> <mrow> <mi>u</mi> <mo stretchy="false">|</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> (<InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(j=1, \ldots , n\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>j</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> </mrow> </math></EquationSource> </InlineEquation>) is upper semi-continuous, provided that the super-level sets of <i>u</i> can be touched from the exterior by uniform <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(C^{1,\text {Dini}}\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mrow> <mn>1</mn> <mo>,</mo> <mtext>Dini</mtext> </mrow> </msup> </math></EquationSource> </InlineEquation> domains at every point. This idea extends to a class of general operators, as well as to the boundary behaviour of the gradient of solutions of the Dirichlet problem in a domain whose boundary satisfies this geometric condition.</p>

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Partial Regularity of the Gradient for Subsolutions

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年6月27日

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  • A. Hakobyan,
  • M. Poghosyan,
  • H. Shahgholian

摘要

For any subharmonic function u, we prove that \(|\partial _j u|\) | ∂ j u | ( \(j=1, \ldots , n\) j = 1 , … , n ) is upper semi-continuous, provided that the super-level sets of u can be touched from the exterior by uniform \(C^{1,\text {Dini}}\) C 1 , Dini domains at every point. This idea extends to a class of general operators, as well as to the boundary behaviour of the gradient of solutions of the Dirichlet problem in a domain whose boundary satisfies this geometric condition.