The Potential Equation

Abstract

The problem

$$ \Delta _n u = 0\quad where\quad \Delta _n u \equiv \sum\limits_{i = 1}^n {u_{x_i x_i } ,} $$

(3.1)

$$ \begin{gathered} u\left( {{x_1},{x_2},...,{x_{n - 1}},0} \right) = {u_0}\left( {{x_1},{x_2},...,{x_{n - 1}}} \right) \hfill \\ {u_{{x_n}}}\left( {{x_1},{x_2},...,{x_{n - 1}},0} \right) = {u_1}\left( {{x_1},{x_2},...,{x_{n - 1}}} \right) \hfill \\ \end{gathered} $$

(3.2)

is called the initial-value problem for the potential equation. By way of examples, we shall show that for this initial-value problem the third requirement cannot, in general, be satisfied, and that the second requirement can in general be satisfied only if the u _i , i = 0, 1 are analytic functions in R _n-1.