  We consider the three-dimensional incompressible Euler equations on a bounded
domain $\Omega$ with $C^3$ boundary. We prove that if the velocity field $u \in
C^{0,\alpha} (\Omega)$ with $\alpha > 0$ (where we are omitting the time
dependence), it follows that the pressure $p \in C^{0, \alpha} (\Omega)$. In
order to prove this result we use a local parametrisation of the boundary and a
very weak formulation of the boundary condition for the pressure, as was
introduced in [C. Bardos and E.S. Titi, Philos. Trans. Royal Soc. A, 380
(2022), 20210073]. Moreover, we provide an example illustrating the necessity
of this new very weak formulation of the boundary condition for the pressure.
This result is of importance for the proof of the first half of the Onsager
Conjecture, the sufficient conditions for energy conservation of weak solutions
to the three-dimensional incompressible Euler equations in bounded domains.
