Underdetermined-elliptic PDE on asymptotically Euclidean manifolds, and generalizations

Abstract

We study underdetermined-elliptic linear partial differential operators $P$ on asymptotically Euclidean manifolds, such as the divergence operator on 1-forms or symmetric 2-tensors. Suitably interpreted, these are instances of (weighted) totally characteristic differential operators on a compact manifold with boundary whose principal symbols are surjective but not injective. We study the equation $P u=f$ when $f$ has a generalized Taylor expansion at $r=\infty$, that is, a full asymptotic expansion into terms with radial dependence $r^{-i z}(\log r)^k$ with $(z,k)\in\mathbb{C}\times\mathbb{N}_0$ up to rapidly decaying remainders. We construct a solution $u$ whose asymptotic behavior at $r=\infty$ is optimal in that the index set of exponents $(z,k)$ arising in its asymptotic expansion is as small as possible. On the flipside, we show that there is an infinite-dimensional nullspace of $P$ consisting of smooth tensors whose expansions at $r=\infty$ contain nonzero terms $r^{-i z}(\log r)^k$ for any desired index set of $(z,k)\in\mathbb{C}\times\mathbb{N}_0$. Applications include sharp solvability results for the divergence equation on 1-forms or symmetric 2-tensors on asymptotically Euclidean spaces, as well as a regularity improvement in a gluing construction for the constraint equations in general relativity recently introduced by the author.