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 , 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,
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).