Henri Poincaré: Dirichlet problem

The Dirichlet problem is always possible; this principle is known as Dirichlet’s principle and the first proof is due to Riemann. If a function V is subject to taking given values at various points of a certain surface, bounding a certain volume where the function and its derivatives are continuous, the triple integral:

∫∫∫[(dV/dx)² + (dV/dy)² + (dV/dz)²]dxdydz

cannot be canceled; it therefore admits a certain minimum and it is easy to verify that this minimum corresponds to the case where the function V satisfies Laplace’s equation. This proof, which is close to Riemann’s, is not rigorous because it is subject to all the objections relating to the continuity of functions defined by the calculus of variations.

Also a very large number of geometers have been preoccupied with establishing this principle more solidly; I will cite in the first line the researches of M. Schwarz (in the program of the Polytechnic School of Zurich 1869 and in the Monatsberichte of the Academy of Berlin 1870) although they relate more particularly to the case of two variables and may not always extend unmodified to the case at hand.

Neumann has given on his side a general method which makes it possible to completely solve the problem, if the surface where the function V takes given values is convex. It therefore solves the problem of electrical distribution in the case of a convex conductor.
Robin’s method also applies only to convex conductors. However, there are a certain number of more or less complicated methods known under the name of alternating methods and which make it possible to extend the results to the case of a conductor of any shape or of several insulated conductors.

Source: Henri Poincaré, Sur les Equations aux Dérivées Partielles de la Physique Mathématique, American Journal of Mathematics, Mar., 1890, Vol. 12, No. 3 (Mar., 1890), pp. 211-294. Translation by Nicolae Sfetcu. © 2023 Nicolae Sfetcu