  We report a holographic study of a two-dimensional dilaton gravity theory
with the Dirichlet boundary condition for the cases of non-vanishing and
vanishing cosmological constants. Our result shows that the boundary theory of
the two-dimensional dilaton gravity theory with the Dirichlet boundary
condition for the case of non-vanishing cosmological constants is the
Schwarzian term coupled to a dilaton field, while for the case of vanishing
cosmological constant, a theory does not have a kinetic term. We also include
the higher derivative term $R^2$, where $R$ is the scalar curvature that is
coupled to a dilaton field. We find that the form of the boundary theory is not
modified perturbatively. Finally, we show that a lattice holographic picture is
realized up to the second-order perturbation of boundary cut-off $\epsilon^2$
under a constant boundary dilaton field and the non-vanishing cosmological
constant by identifying the lattice spacing $a$ of a lattice Schwarzian theory
with the boundary cut-off $\epsilon$ of the two-dimensional dilaton gravity
theory.
