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<p>We prove that certain renormalized value functions associated with the <i>d</i>-dimensional (<InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(d\geqq 2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo>≧</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>) <i>N</i>-body problem corresponding to different limiting shapes of expanding solutions, under the assumption that the center of mass is at the origin, are viscosity solutions of the associated Hamilton–Jacobi equation. We analyze their singularities, defined as the initial configurations for which the minimizer of the associated variational problem is not unique. Moreover, we estimate the size of the closure of the singular set by proving its <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\mathcal {H}^{d(N-1)-1}\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="script">H</mi> </mrow> <mrow> <mi>d</mi> <mo stretchy="false">(</mo> <mi>N</mi> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>-</mo> <mn>1</mn> </mrow> </msup> </math></EquationSource> </InlineEquation>-rectifiability, and we provide an upper bound on the Hausdorff dimension of the set of regular conjugate points.</p>

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On the Regularity of Solutions to the Hamilton–Jacobi Equations for the N-Body Problem

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年8月4日

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  • Diego Berti,
  • Davide Polimeni,
  • Susanna Terracini

摘要

We prove that certain renormalized value functions associated with the d-dimensional ( \(d\geqq 2\) d ≧ 2 ) N-body problem corresponding to different limiting shapes of expanding solutions, under the assumption that the center of mass is at the origin, are viscosity solutions of the associated Hamilton–Jacobi equation. We analyze their singularities, defined as the initial configurations for which the minimizer of the associated variational problem is not unique. Moreover, we estimate the size of the closure of the singular set by proving its \(\mathcal {H}^{d(N-1)-1}\) H d ( N - 1 ) - 1 -rectifiability, and we provide an upper bound on the Hausdorff dimension of the set of regular conjugate points.