  A bosonic Laplacian is a conformally invariant second order differential
operator acting on smooth functions defined on domains in Euclidean space and
taking values in higher order irreducible representations of the special
orthogonal group. In this paper, we study boundary value problems involving
bosonic Laplacians in the upper-half space and the unit ball. Poisson kernels
in the upper-half space and the unit ball are constructed, which give us
solutions to the Dirichlet problems with $L^p$ boundary data, $1 \leq p \leq
\infty$. We also prove the uniqueness for solutions to the Dirichlet problems
with continuous data for bosonic Laplacians and provide analogs of some
properties of harmonic functions for null solutions of bosonic Laplacians, for
instance, Cauchy's estimates, the mean-value property, Liouville's Theorem,
etc.
