Local Well-posedenss of the Bartnik Static Extension Problem near Schwarzschild spheres

TL;DR

This paper proves the local well-posedness of the Bartnik static extension problem near Schwarzschild spheres, introducing a new geodesic gauge framework and handling small mean curvature cases with energy estimates.

Contribution

It develops a novel approach using a geodesic gauge and Bochner-measurable function spaces to establish existence and uniqueness for the problem.

Findings

Established local well-posedness for perturbations of Schwarzschild spheres.

Introduced a new framework reducing the problem to elliptic and transport equations.

Handled small mean curvature cases with rigorous energy estimates.

Abstract

We establish the local well-posedness of the Bartnik static metric extension problem for arbitrary Bartnik data that perturb that of any sphere in a Schwarzschild $\{t=0\}$ slice. Our result in particular includes spheres with arbitrary small mean curvature. We introduce a new framework to this extension problem by formulating the governing equations in a geodesic gauge, which reduce to a coupled system of elliptic and transport equations. Since standard function spaces for elliptic PDEs are unsuitable for transport equations, we use certain spaces of Bochner-measurable functions traditionally used to study evolution equations. In the process, we establish existence and uniqueness results for elliptic boundary value problems in such spaces in which the elliptic equations are treated as evolutionary equations, and solvability is demonstrated using rigorous energy estimates. The precise nature of the expected difficulty of solving the Bartnik extension problem when the mean curvature is very small is identified and suitably treated in our analysis.