On Some Class of Solutions to the Two-Dimensional Laplace Equation on a Three-Dimensional Manifold

We find the solution to the two-dimensional Laplace equation on a given set of three independentvariables in the three-dimensional Euclidean space. The problem is solved by transforming the two-dimensionalLaplace equation into some equation with the sought function of three independent variables. Thisturns out possible by introducing a spherical coordinate system. The proposed method allows usto find a solution to the two-dimensional Laplace equation in the form of a function of three independentvariables. By way of example, we consider the problem ofan incompressible fluid flow around a three-dimensional body shaped as an “iron.” For thisproblem, we give the detailed arguments that reduce the three-dimensional Laplace equationdescribing the distribution of the scalar potential of the flow velocities near the surface of the bodywhich depends on three independent coordinates to some two-dimensional Laplace equation whose solutionis strictly justified by analysis. We also observe that similar problems arise notonly in hydrodynamics but also in elasticity and electromagnetism theories. The described technique,namely, the possibility of passing from two to three independent variables by a given transformationenables us to find purely physical solutions for a wide range of problems from the various areas of naturalsciences.