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We prove that, in a two-dimensional strip, a steady flow of an idealincompressible fluid with no stationary point and tangential boundaryconditions is a shear flow. The same conclusion holds for a bounded steady flowin a half-plane. The proofs are based on the study of the geometric propertiesof the streamlines of the flow and on one-dimensional symmetry results forsolutions of some semilinear elliptic equations. Some related rigidity resultsof independent interest are also shown in n-dimensional slabs in any dimensionn.

Shear flows of an ideal fluid and elliptic equations in unbounded domains