  The boundary-value problem for Laplace-type operators acting on smooth
sections of a vector bundle over a compact Riemannian manifold with generalized
local boundary conditions including both normal and tangential derivatives is
studied. The condition of strong ellipticity of this boundary-value problem is
formulated. The resolvent kernel and the heat kernel in the leading
approximation are explicitly constructed. As a result, the previous work in the
literature on heat-kernel asymptotics is shown to be a particular case of a
more general structure. For a bosonic gauge theory on a compact Riemannian
manifold with smooth boundary, the problem is studied of obtaining a
gauge-field operator of Laplace type, jointly with local and gauge-invariant
boundary conditions, which should lead to a strongly elliptic boundary-value
problem. The scheme is extended to fermionic gauge theories by means of local
and gauge-invariant projectors. After deriving a general condition for the
validity of strong ellipticity for gauge theories, it is proved that for
Euclidean Yang--Mills theory and Rarita--Schwinger fields all the above
conditions can be satisfied. For Euclidean quantum gravity, however, this
property no longer holds, i.e. the corresponding boundary-value problem is not
strongly elliptic. Some non-standard local formulae for the leading asymptotics
of the heat-kernel diagonal are also obtained. It is shown that, due to the
absence of strong ellipticity, the heat-kernel diagonal is non-integrable near
the boundary.
