  In the context of the bulk-boundary correspondence we study the correlation
functions arising on a boundary for different types of boundary conditions. The
most general condition is the mixed one interpolating between the Neumann and
Dirichlet conditions. We obtain the general expressions for the correlators on
a boundary in terms of Green's function in the bulk for the Dirichlet, Neumann
and mixed boundary conditions and establish the relations between the
correlation functions. As an instructive example we explicitly obtain the
boundary correlators corresponding to the mixed condition on a plane boundary
$R^d$ of a domain in flat space $R^{d+1}$. The phases of the boundary theory
with correlators of the Neumann and Dirichlet types are determined. The
boundary correlation functions on sphere $S^d$ are calculated for the Dirichlet
and Neumann conditions in two important cases: when sphere is a boundary of a
domain in flat space $R^{d+1}$ and when it is a boundary at infinity of Anti-De
Sitter space $AdS_{d+1}$. For massless in the bulk theory the Neumann
correlator on the boundary of AdS space is shown to have universal logarithmic
behavior in all AdS spaces. In the massive case it is found to be finite at the
coinciding points. We argue that the Neumann correlator may have a dual
two-dimensional description. The structure of the correlators obtained, their
conformal nature and some recurrent relations are analyzed. We identify the
Dirichlet and Neumann phases living on the boundary of AdS space and discuss
their evolution when the location of the boundary changes from infinity to the
center of the AdS space.
