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

Ahmed Ellithy

TL;DR

This work proves local well-posedness of the Bartnik static extension problem near Schwarzschild spheres by recasting the static vacuum equations in a geodesic gauge, which yields a coupled elliptic-transport system governed by a lapse function u = ln f and radial second fundamental form data. A novel Bochner-type functional framework is developed to accommodate the transport component, and a nonlocal elliptic operator is shown to be invertible on these spaces, enabling an implicit function argument. To overcome obstructions from conformal Killing fields, an artificial vector field X is introduced and subsequently shown to vanish for genuine solutions, yielding a rigorous local solution map from boundary data to static vacuum extensions. The approach provides a flexible, 3D-specific framework that strengthens prior results and may extend to related quasi-local mass problems and generalized Bartnik-type extensions.

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.

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

TL;DR

This work proves local well-posedness of the Bartnik static extension problem near Schwarzschild spheres by recasting the static vacuum equations in a geodesic gauge, which yields a coupled elliptic-transport system governed by a lapse function u = ln f and radial second fundamental form data. A novel Bochner-type functional framework is developed to accommodate the transport component, and a nonlocal elliptic operator is shown to be invertible on these spaces, enabling an implicit function argument. To overcome obstructions from conformal Killing fields, an artificial vector field X is introduced and subsequently shown to vanish for genuine solutions, yielding a rigorous local solution map from boundary data to static vacuum extensions. The approach provides a flexible, 3D-specific framework that strengthens prior results and may extend to related quasi-local mass problems and generalized Bartnik-type extensions.

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 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.

Paper Structure

This paper contains 16 sections, 25 theorems, 327 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Properties of Static Vacuum Extensions
  4. Function Spaces
  5. The Main Theorem
  6. Solvability of Elliptic BVP in ${\mathcal{A}_{H}}^{(2,k)}_{\delta}(M)$ and ${\mathcal{A}_{C}}^{(2,k)}_{\delta}(M)$
  7. Reduction of the problem
  8. Proof of The Main Theorem
  9. Definition of the Artificial Vector Field $X$
  10. Definition and Existence of the Modified Solution
  11. The Vanishing of $X$ for Modified solutions $(g,u,X)$
  12. Proof that $D\Phi_{sc}$ is an Isomorphism
  13. Appendix
  14. A PDE of Finite Type for Conformal Killing Vector Fields
  15. A Hardy-type Inequality
  16. ...and 1 more sections

Key Result

Proposition 2.3

There exists $\tau'=\tau'(n,m_0)>0$ small enough such that the following is true for any $0<\tau<\tau'$. If an asymptotically flat metric $g$ on $M$ of order $\eta>0$ satisfies in Cartesian coordinates where $|\cdot|$ is with respect to the Euclidean metric $\delta$, then:

Theorems & Definitions (68)

  • Remark 1.1
  • Remark 1.2
  • Definition 1
  • Remark 1.3
  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • ...and 58 more