Uniformly elliptic boundary value problems

Abstract

We study boundary conditions for elliptic operators on non-compact manifolds with boundary via uniform K-homology, a version of K-homology sensitive to the large-scale geometry of the manifold. To that end, we develop the theory of relative uniform K-homology. We show that boundary conditions for uniformly elliptic differential operators define classes in the relative and non-relative uniform K-homology of the manifold, depending on the assumed regularity of the boundary condition. Moreover, we define and study a relative index map on relative uniform K-homology that combines uniform coarse information on the interior with secondary information on the boundary. As an application, we compute that on a spin manifold with product structure and uniformly positive scalar curvature on the boundary the image of the relative uniform K-homology class of the Dirac operator under this relative index map is closely connected to a uniform version of the higher $\rho$-invariant of the boundary. In particular, a delocalized APS-index theorem of Piazza and Schick is proved in the uniform setting.