Isolated singularities of solutions of non-linear partial differential equations

TLDR: For the special case of the Laplace equation =+OiSw this solution can be uniquely characterized as the only (nonconstant) solution which exhibits radial symmetry as discussed by the authors.

Abstract: A significant role in the theory of linear elliptic second-order partial differential equations in two independent variables has been played by the concept of the fundamental solution. Such a function is a single-valued solution of the equation, regular except for an isolated point at which it possesses a logarithmic singularity. For the special case of the Laplace equation =+OiSwthis solution can be uniquely characterized as the only (nonconstant) solution which exhibits radial symmetry. The requiirement of radial symmetry leads to an ordinary differential equation whose solution (up to a constant) is y log (x2+y2)"12. This function admits an important hydrodynamical interpretation as the velocity potential of an incompressible, nonviscous, two-dimensional source-flow. An amount 2w, of fluid is visualized as flowing in unit time out of a source at the origin (i.e. across every closed curve surrounding the source) and into a corresponding sink at infinity. The requirement of incompressibility may be relaxed by assuming a relation-called the equation of state-between the density p of the fluid and the velocity I V4 I. For adiabatic flows this relation takes the form p = [1((y1)/2) 1 V+| 2]1/(7-1), where y is the ratio of specific heats of the fluid. In this case the potential 4 satisfies not the Laplace equation but the nonlinear equation