  Density functional theory (DFT) is notorious for the absence of gradient
corrections to the two-dimensional (2D) Thomas-Fermi kinetic-energy functional;
it is widely accepted that the 2D analog of the 3D von Weizs\"acker correction
vanishes, together with all higher-order corrections. Contrary to this
long-held belief, we show that the leading correction to the kinetic energy
does not vanish, is unambiguous, and contributes perturbatively to the total
energy. This insight emerges naturally in a simple extension of standard DFT,
which has the effective potential energy as a functional variable on equal
footing with the single-particle density.
