  The Zaremba boundary-value problem is a boundary value problem for
Laplace-type second-order partial differential operators acting on smooth
sections of a vector bundle over a smooth compact Riemannian manifold with
smooth boundary but with non-smooth (singular) boundary conditions, which
include Dirichlet conditions on one part of the boundary and Neumann ones on
another part of the boundary. We study the heat kernel asymptotics of Zaremba
boundary value problem. The construction of the global parametrix of the heat
equation is described in detail and the leading parametrix is computed
explicitly. Some of the first non-trivial coefficients of the heat kernel
asymptotic expansion are computed explicitly.
