This problem is solved in the main by the keen methods of H. A. Schwarz, C. Neumann, and Poincaré for the differential equation of the potential.
These methods, however, seem to be generally not capable of direct extension to the case where along the boundary there are prescribed either the differential coefficients or any relations between these and the values of the function.
This general principle will then perhaps enable us to approach the question: Has not every regular variation problem a solution, provided certain assumptions regarding the given boundary conditions are satisfied (say that the functions concerned in these boundary conditions are continuous and have in sections one or more derivatives), and provided also if need be that the notion of a solution shall be suitably extended?
[1] In the field of differential equations, a boundary value problem is a differential equation together with a set of additional constraints, called the boundary conditions.
Much theoretical work in the field of partial differential equations is devoted to proving that boundary value problems arising from scientific and engineering applications are in fact well-posed.