Equilibrium patterns of a Langmuir monolayer of methyl octadecanoate are perturbed on a micrometer length scale using optical tweezers and laser heating. The consecutive electrophoretic motion of circular domains in liquid expanded or gaseous surroundings is investigated. The experimentally observed domain relaxation is described by a balance of the dissipative hydrodynamic force on the domain and an electrostatic dipole force from a neighboring domain. Drag forces derived from the experiments are in agreement with theoretical predictions (Hughes, B. D.; Pailthorpe, B. A.; White, L. R. J. Fluid Mech. 1981, 110, 349.) for the viscous drag on solid domains moving in monolayer surroundings of negligible surface shear viscosity. The dipole interactions are characterized by the surface potential differences between the coexisting phases. On pure water, the relaxation experiments reveal that the surface potential difference between the liquid condensed and the liquid expanded phase depends on the area fractions and the coexistence cannot be described by an ideal first-order phase transition with constant dipole densities of the phases.