  Liouville conformal field theory (LCFT) is considered on a simply connected
domain with boundary, specializing to the case where the Liouville potential is
integrated only over the boundary of the domain. We work in the probabilistic
framework of boundary LCFT introduced by Huang-Rhodes-Vargas (2015). Building
upon the known proof of the bulk one-point function by the first author, exact
formulas are rigorously derived for the remaining basic correlation functions
of the theory, i.e., the bulk-boundary correlator, the boundary two-point and
the boundary three-point functions. These four correlations should be seen as
the fundamental building blocks of boundary Liouville theory, playing the
analogue role of the DOZZ formula in the case of the Riemann sphere. Our study
of boundary LCFT also provides the general framework to understand the
integrability of one-dimensional Gaussian multiplicative chaos measures as well
as their tail expansions. Finally these results have applications to studying
the conformal blocks of CFT and set the stage for the more general case of
boundary LCFT with both bulk and boundary Liouville potentials.
