On the Regularity of Solutions to the Hamilton–Jacobi Equations for the N-Body Problem
Abstract
Abstract 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.
Fields of Study & Subjects
Spacecraft Dynamics and Control
Optimization and Variational Analysis
Advanced Optimization Algorithms Research
Publication details
- Journal / source
- Archive for Rational Mechanics and Analysis
- Publisher
- Springer Science and Business Media LLC
- Volume
- 250
- Issue
- 5
- Publication date
- 4 August 2026
- Language
- EN