On the regularity of solutions to the Hamilton-Jacobi equations for the N-body problem

TL;DR

This paper studies the regularity and singularities of solutions to Hamilton-Jacobi equations derived from the N-body problem, providing geometric estimates on the singular set and insights into solution structure.

Contribution

It establishes that certain renormalized value functions are viscosity solutions and characterizes their singularities, including rectifiability and dimension bounds.

Findings

Singularities correspond to non-unique minimizers in the variational problem.

The singular set is proven to be rectifiable with a specific Hausdorff measure.

An upper bound on the Hausdorff dimension of regular conjugate points is provided.

Abstract

We prove that certain suitably renormalized value functions associated with the $d$-dimensional ($d\geq2$) $N$-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 $\mathcal{H}^{d(N-1)-1}$-rectifiability, and we provide an upper bound on the Hausdorff dimension of the set of regular conjugate points.