Uniform distributions in nonuniform systems: Wall potentials generating constant density profiles in classical density functional theory

We study the inverse problem of classical density functional theory for inhomogeneous fluids: finding the wall potential that produces a constant equilibrium density profile, i.e., a perfectly flat density distribution in the accessible region adjacent to a substrate. Within Rosenfeld's fundamental measure theory, we solve this problem for a one-component fluid in planar, spherical, and cylindrical geometries, considering both a hard-sphere fluid and a fluid with an additional truncated Lennard-Jones attraction treated at the mean-field level. Explicit analytical expressions are obtained for planar walls, while spherical walls also admit an analytical treatment in a more cumbersome form. The cylindrical case is treated numerically. The construction provides an explicit microscopic realization of structure-cancelling wall fields related to flat-profile conditions that occur under special matching conditions in interfacial theories of wetting and drying. The theory also yields a compact collection of formulas for weighted densities and one-body direct correlation functions in the three fundamental geometries, providing useful reference expressions for density-functional implementations. The resulting analytic wall potentials are validated in independent density functional calculations, which confirm that the prescribed flat profiles are recovered within numerical accuracy.