Elliptic equations in weighted Besov spaces on asymptotically flat Riemannian manifolds

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 and the momentum constraints with constant mean curvature and with a background metric that satisfies very low regularity assumptions. These results extend the regularity results of Holst et al. (Commun Math Phys 288:547–613, 2009) 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 et al. in Chin Ann Math Ser B 27(1), 31–52, 2006; Maxwell in Commun Math Phys 253(3):561–583, 2005), as well as they enable us to construct the initial data for the Einstein–Euler system.