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<p>In this paper, we study the <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(L_p\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>L</mi> <mi>p</mi> </msub> </math></EquationSource> </InlineEquation> curvature problem in Gaussian probability space. The problem is to solve a partial differential equation on the unit sphere, involving the curvature operators <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(\sigma _{n-k}/\sigma _{n}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>σ</mi> <mrow> <mi>n</mi> <mo>-</mo> <mi>k</mi> </mrow> </msub> <mo stretchy="false">/</mo> <msub> <mi>σ</mi> <mi>n</mi> </msub> </mrow> </math></EquationSource> </InlineEquation>. The existence and uniqueness of solution to the equation where <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(p\ge k+1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>≥</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> are established in the smooth category.</p>

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The Prescribed \(L_p\) Curvature Problem in Gaussian Probability Space

  • 期刊论文

  • 发表日期：

    2025年12月8日

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  • Yibin Feng,
  • Yuanyuan Li

摘要

In this paper, we study the \(L_p\) L p curvature problem in Gaussian probability space. The problem is to solve a partial differential equation on the unit sphere, involving the curvature operators \(\sigma _{n-k}/\sigma _{n}\) σ n - k / σ n . The existence and uniqueness of solution to the equation where \(p\ge k+1\) p ≥ k + 1 are established in the smooth category.