Elliptic Equations in Weighted Besov Spaces on Asymptotically Flat Riemannian Manifolds

TL;DR

This paper applies weighted Besov spaces to elliptic equations on asymptotically flat Riemannian manifolds, establishing existence results for Einstein's constraint equations under low regularity conditions and extending prior regularity results.

Contribution

It extends regularity results for Einstein's constraint equations to asymptotically flat manifolds using weighted Besov spaces, allowing for lower regularity assumptions and construction of initial data.

Findings

Existence theorems for Hamiltonian and momentum constraints with low regularity.

Extension of regularity results from compact to asymptotically flat manifolds.

Construction of initial data for Einstein--Euler system.

Abstract

This paper deals with the applications of weighted Besov spaces to elliptic equations on asymptotically flat Riemannian manifolds, and in particular to the solutions of Einstein's constraints equations. We establish existence theorems for the Hamiltonian an momentum constraints with constant mean curvature and with a background metric which satisfies very low regularity assumptions. These results extend the regularity results of Holst, Nagy and Tsogtgerel about the constraint equations on compact manifolds in the Besov space $B_{p,p}^s$, to asymptotically flat manifolds. We also consider the Brill--Cantor criterion in the weighted Besov spaces. Our results improve the regularity assumptions on asymptotically flat manifolds Choquet--Bruhat, Isenberg and Pollack, and Maxwell, as well as they enable us to construct the initial data for the Einstein--Euler system.