## Boundary Integral Equations for the Dirichlet and Neumann Problems

In what follows we introduce the area potential and use it to reduce the boundary value problems for the non-homogeneous equilibrium equation to analogous ones for the homogeneous equation.

Let q £ Co°(M2). The area potential of density q is defined by

Since the convolution of any distribution with a function of class Cq°(M2) is an infinitely differentiable function that can be differen tiated under the convolution sign [12], it follows that Uq £ C°°(M2); in addition,

in the distributional sense. We must now establish what restrictions need to be imposed on q to ensure that Uq £

3.1. Theorem . Uq £ if and only if

Proof. We need to examine the asymptotic behavior of (Uq)\x) as \x\ -» oc. We start with the asymptotic behavior of D(x). From the

explicit form of D(x) we easily see that, as |x| -> oo,

This yields

From these equalities it follows that Uq E if and only if

54 DIRICHLET AND NEUMANN PROBLEMS

in other words, if and only if q e ^ _ i )W(R2).