With a lean premix burner (1), a plurality of preferably V-shaped, closely spaced fuel channels (12) is formed into the circumferential fuel-supplied film application surface (8) of the atomizer lip (9) in the direction of flow, in order to achieve a better heat transfer to the fuel due to the uniform fuel distribution and a transverse flux in the plurality of small fuel flows caused by the Marangoni effect, to completely converse the fuel by early vaporization and improved spraying, to reduce nitrogen oxide emission and to suppress combustion-driven pressure oscillations.