Discontinuous boundary conditions and the Dirichlet problem
Norbert Wiener · Transactions of the American Mathematical Society · 1923
It has been suggested that in the Dirichlet problem there is something essentially antagonistic between the utmost degree of generality attainable as regards the geometrical character of the boundary and the utmost of generality attainable as regards the boundary values assumed over this boundary.The purpose of this paper is to show that the solution of the Dirichlet problem merely for continuous boundary conditions at once entrains the unique determination of a harmonic function correlated with discontinuous boundary conditions of a very general sort.The logical tool employed to this end is the Daniell intégrait In the stress laid on generalized types of integration this paper is closely akin to one of G. C. Evans,t but it appears to the author that though the theory of Evans gives more detailed information concerning the character of the solutions of the Dirichlet problem in the neighborhood of the boundary, it is less direct than the present theory, and less extensible to regions of infinite connectivity or higher dimensionality. I. THE POTENTIAL AT A POINT AS A LINEAR FUNCTIONAL OF THE BOUNDARY CONDITIONSLet B be any open set of points in »-space connected in the sense that any two of its points form extremities of a polygonal line lying entirely within it, and not extending to infinity.Let the Dirichlet problem be solvable over B for any continuous boundary conditions on C. That is, if U(P) is defined for everv point P on C, and if Jim U(P) = U(Q) PQ-H>