Hi,
I am currently working on the Fenics Tutorial (https://fenicsproject.org/pub/tutorial/html/._ftut1009.html) for solving NSE with IPCS. I do not understand how the variational form for pressure is derived. In the tutorial, it says that the variational problem

results from

Why did the boundary term when doing integration by parts vanish everywhere even though we only had a dirichlet bc at the outflow?

When you derive a pressure poisson equation you are surely missing a boundary condition on p, because the original NS equations only have a first derivative of p, while the pressure Poisson equation has two. You have then to come up with a BC for p, and coming up with the BC that makes those boundary terms disappear is one possible choice.