The strong Legendre condition and the well-posedness of mixed Robin problems on manifolds with bounded geometry

TL;DR

This paper establishes the well-posedness and regularity of mixed Robin boundary value problems for second-order elliptic operators on non-compact manifolds with bounded geometry, extending classical results to more general settings.

Contribution

It introduces a new framework for analyzing Robin problems on non-compact manifolds, including cases with non-smooth coefficients and vector bundle decompositions.

Findings

Proved well-posedness in Sobolev spaces on non-compact manifolds.

Extended Poincaré inequality to manifolds with bounded geometry.

Established equivalence of the uniform Agmon condition and Gårding inequality.

Abstract

Let $M$ be a smooth manifold with boundary $\partial M$ and bounded geometry, $\partial_D M \subset \partial M$ be an open and closed subset, $P$ be a second order differential operator on $M$, and $b$ be a first order differential operator on $\partial M \smallsetminus \partial_D M$. We prove the regularity and well-posedness of the mixed Robin boundary value problem $$Pu = f \mbox{ in } M,\ u = 0 \mbox{ on } \partial_D M,\ \partial^P_\nu u + bu = 0 \mbox{ on } \partial M \setminus \partial_D M$$ under some natural assumptions. Our operators act on sections of a vector bundle $E \to M$ with bounded geometry. Our well-posedness result is in the Sobolev spaces $H^s(M; E)$, $s \geq 0$. The main novelty of our results is that they are formulated on a non-compact manifold. We include also some extensions of our main result in different directions. First, the finite width assumption is required for the Poincar\'{e} inequality on manifolds with bounded geometry, a result for which we give a new, more general proof. Second, we consider also the case when we have a decomposition of the vector bundle $E$ (instead of a decomposition of the boundary). Third, we also consider operators with non-smooth coefficients, but, in this case, we need to limit the range of $s$. Finally, we also consider the case of uniformly strongly elliptic operators. In this case, we introduce a \emph{uniform Agmon condition} and show that it is equivalent to the G\aa rding inequality. This extends an important result of Agmon (1958).