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<p>We prove a representation formula for superharmonic functions on the half-space <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathbb {R}^N_+:=\mathbb {R}^{N-1}\times ]0,+\infty [\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mo>+</mo> <mi>N</mi> </msubsup> <mo>:</mo> <mo>=</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mrow> <mi>N</mi> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>×</mo> <mrow> <mo stretchy="false">]</mo> <mn>0</mn> <mo>,</mo> <mo>+</mo> <mi>∞</mi> <mo stretchy="false">[</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. As a consequence, we derive some comparison principles and various positivity results.</p>

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Characterization of positive superharmonic functions in a half-space

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年7月3日

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  • Lorenzo D’Ambrosio,
  • Enzo Mitidieri

摘要

We prove a representation formula for superharmonic functions on the half-space \(\mathbb {R}^N_+:=\mathbb {R}^{N-1}\times ]0,+\infty [\) R + N : = R N - 1 × ] 0 , + ∞ [ . As a consequence, we derive some comparison principles and various positivity results.