Generalized (n+1) dimensional dirichlet boundary value problem

Abstract

Let f(x) be a piecewise continuous and bounded function on the real line R. It is a classical result that the solution to the Dirichlet boundary value problem:

in the upper half plane is:

We extend the notion of the harmonic function to the space Wecall U(x,y)a harmonic function in an open region of R^{n+1, +}, if it is an infinitely differentiable at each point of the region and satisfies

In this paper we exploit the above definition of harmonic functions to solve the Dirichlet boundary value problem in the space R^n+1,1 with a distributional boundary condition. As it turns out, our solution is quite constructive and its two dimensional case is an extension of the corresponding classical Dirichlet-Boundary-Val ue Problem.