  We give an alternate proof of one of the results given in [16] showing that
initial data sets with boundary for the Einstein equations $(M, g, k)$
satisfying the dominant energy condition can be conformally deformed to the
strict dominant energy condition, while preserving the character of the
boundary (minimal, future trapped, or past trapped) while changing the area of
the boundary and ADM energy of the initial data set by an arbitrarily small
amount. The proof relies on solving an equation that looks like the equation
for spacetime harmonic functions studied in [7], but with a Neumann boundary
condition and non-zero right hand side, which we refer to as a spacetime
Poisson equation. One advantage of this method of proof is that the conformal
deformation is explicitly constructed as a solution to a PDE, as opposed to
only knowing the solution exists via an application of the implicit function
theorem as in [16]. We restrict ourselves to the physically relevant case of a
$3$-manifold $M$, though the proof can be generalized to higher dimensions.
