We study the $\eta$-invariant of a Dirac operator on a manifold with boundary
subject to local boundary conditions with the help of heat kernel methods. In
even dimensions, we relate this invariant to $\eta$-invariants of a boundary
Dirac operator, while in odd dimension, it is expressed through the index of
boundary operators. We stress the necessity of the strong ellipticity condition
for the applicability of our methods. We show that the Witten–Yonekura
boundary conditions are not strongly elliptic, though they are very close to
strongly elliptic ones.
