Uniqueness in weighted Lebesgue spaces for an elliptic equation with drift on manifolds

TL;DR

This paper studies the uniqueness of solutions to elliptic equations with drift on noncompact Riemannian manifolds within weighted Lebesgue spaces, establishing sharp conditions in certain model cases.

Contribution

It provides new results on the uniqueness of elliptic equation solutions with drift on manifolds, including sharp conditions for polynomial volume growth models.

Findings

Uniqueness results for elliptic equations with drift on manifolds

Sharp conditions identified for polynomial volume growth models

Solutions are unique in weighted Lebesgue spaces under specified conditions

Abstract

We investigate the uniqueness, in suitable weighted Lebesgue spaces, of solutions to a class of elliptic equations with a drift posed on a complete, noncompact, Riemannian manifold $M$ of infinite volume and dimension $N\ge2$. Furthermore, in the special case of a model manifold with polynomial volume growth, we show that the conditions on the drift term are sharp.