Introduction
Hilbert noted that there existed methods for solving partial differential equations where the function's values were given at the boundary, but the problem asked for methods for solving partial differential equations with more complicated conditions on the boundary (e.g., involving derivatives of the function), or for solving calculus of variation problems in more than 1 dimension (for example, minimal surface problems or minimal curvature problems)Problem statement
The original problem statement in its entirety is as follows:An important problem closely connected with the foregoing eferring_to_Hilbert's_nineteenth_problem.html" ;"title="Hilbert's_nineteenth_problem.html" ;"title="eferring to Hilbert's nineteenth problem">eferring to Hilbert's nineteenth problem">Hilbert's_nineteenth_problem.html" ;"title="eferring to Hilbert's nineteenth problem">eferring to Hilbert's nineteenth problemis 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?Hilbert, David, "Mathematische Probleme" Göttinger Nachrichten, (1900), pp. 253-297, and in Archiv der Mathematik und Physik, (3) 1 (1901), 44-63 and 213-237. Published in English translation by Dr. Maby Winton Newson,Bulletin of the American Mathematical Society The ''Bulletin of the American Mathematical Society'' is a quarterly mathematical journal published by the American Mathematical Society. Scope It publishes surveys on contemporary research topics, written at a level accessible to non-experts. I ...8 (1902), 437-47
. fuller title of the journal Göttinger Nachrichten is Nachrichten von der Königl. Gesellschaft der Wiss. zu Göttingen./ref>
Boundary value problems
In the field ofReferences
*. *. *. {{Hilbert's problems #20 Calculus of variations