  We present a consistent treatment of the quantum Hall effect within the
electrostatic approximation. We derive the form of the density of states (DOS)
which differs from the usual gaussian shape valid for non-interacting
electrons. Below a crossover temperature, this DOS gradually transforms into
the non-interacting DOS near the Fermi energy. We obtain an estimate of this
crossover temperature, which determines the range of validity of this
framework, and allows a reconciliation of the width and temperature dependence
of the quantum Hall plateaus with experiments. The DOS dramatically enhances
the electron-lattice relaxation rate, thus reducing the possibility of electron
overheating.
