  We discuss the initial boundary value problem for a heat equation in a domain
surrounded by a layer. The main features of this problem are twofold: on one
hand, the layer is thin compared to the scale of the domain, and on the other
hand, the thermal conductivity of the layer is drastically different from that
of the bulk; moreover, the bulk is isotropic, but the layer is anisotropic and
``optimally aligned" in the sense that any vector in the layer normal to the
interface is an eigenvector of the thermal tensor. We study the effects of the
layer by thinking of it as a thickless surface, on which ``effective boundary
conditions" (EBCs) are imposed. In the three-dimensional case, we obtain EBCs
by investigating the limiting solution of the initial boundary value problem
subject to either Dirichlet or Neumann boundary conditions as the thickness of
the layer shrinks to zero. These EBCs contain not only the standard boundary
conditions but also some nonlocal ones, including the Dirichlet-to-Neumann
mapping and the fractional Laplacian. One of the main features of this work is
to allow the drastic difference in the thermal conductivity in the normal
direction and two tangential directions within the layer.
