We study the competition between the exchange and the direct Coulomb interaction near the edge of a quantum dot in a strong magnetic field using the density-functional theory in a local approximation for the exchange-energy functional. We develop a scheme in which it is possible to describe fractional filling in the extreme quantum limit. Exchange plays a significant role in reducing the spatial extent of the compressible edge channel regions obtained from an electrostatic description. The transition from narrow incompressible edge channels to broad compressible strips occurs within a narrow and experimentally accessible range of confinement strengths.
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