  The purpose of this note is to give a complete proof of a $C^{0,\alpha}$
regularity result for the pressure for weak solutions of the two-dimensional
"incompressible Euler equations" when the fluid velocity enjoys the same type
of regularity in a compact simply connected domain with $C^2$ boundary. To
accomplish our result we realize that it is compulsory to introduce a new weak
formulation for the boundary condition of the pressure which is consistent
with, and equivalent to, that of classical solutions.
