20. THE GENERAL PROBLEM OF BOUNDARY VALVES.
An important problem closely connected with the foregoing is the question concerning the existence of solutions of partial differential equations when the values on the boundary of the region are prescribed. 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. Nor can they be extended immediately to the case where the inquiry is not for potential surfaces but, say, for surfaces of least area, or surfaces of constant positive gaussian curvature, which are to pass through a prescribed twisted curve or to stretch over a given ring surface. It is my conviction that it will be possible to prove these existence theorems by means of a general principle whose nature is indicated by Dirichlet's principle. 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?[48]
[48]Cf. my lecture on Dirichlet's principle in theJahresber. d. Deutschen Math.-Vereinigung, vol. 8 (1900), p. 184.
[48]Cf. my lecture on Dirichlet's principle in theJahresber. d. Deutschen Math.-Vereinigung, vol. 8 (1900), p. 184.