A symmetry problem in potential theory

The proof of this result is given in Section 1 ; in Section 3 we give various generalizations to elliptic differential equations other than (1). Before turning to the detailed arguments it will be of interest to discuss the physical motivation for the problem itself. Consider a viscous incompressible fluid moving in straight parallel streamlines through a straight pipe of given cross sectional form f2. If we fix rectangular coordinates in space with the z axis directed along the pipe, it is well known that the flow velocity u is then a function of x, y alone satisfying the Poisson differential equation (for n = 2) A u = A i n f2