Local pressure for confined systems

We derive a general closed expression for the local pressure exerted onto the corrugated walls of a channel confining a fluid medium. When the fluid medium is at equilibrium the local pressure is a functional of the shape of the walls. It is shown that, due to the intrinsic non-local character of the interactions among the particles forming the fluid, the applicability of approximate schemes such as the concept of a surface of tension or morphometric thermodynamics is limited to wall curvatures small compared to the range of particle-particle interactions.