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\centerline{\bf Boundary integral method for the}
\centerline{\bf Laplace equation in a slit domain}

\centerline{M. Avigal and J.Steinberg, Technion, Haifa}

\ \\ {\bf Abstract.} Let $D$ be a domain whose boundary is composed of a
simple closed curve $L$ and a slit $S$ inside $L$. The bounded solution $w$
of the Laplace equation, satisfying $\partial w/\partial n=0$ on $S$ and
given conditions on $L$ is represented as an integral transform
  $$ w(P)=\int \limits_L \{K(P,Q)\partial w(Q)/\partial n-w(Q)
     \partial K(P,Q)/\partial n\}\,ds;\quad P\in D, $$
where $K$ is a fundamental solution satisfying $\partial K/\partial n_P=0$,
$P\in S$; $\partial K/\partial n_Q=0$, $Q\in S$. $K$ can be constructed
with the help of a biorthogonal system $\{w_j,\tilde w_j\}$, i.e. a pair of
sequences of harmonic functions in $D$, whose normal derivative vanish on
$S$, and satisfying $(w_j,\tilde w_j)=\alpha_j \delta_{jk}$;
$\alpha_j\ne 0$, with the skew-symmetric bilinear form
  $$ (f,g)=\int \limits_L (f\partial g/\partial n-g\partial f/\partial n)
      \,ds. $$
Choosing for $S$ the interval $|x|<1$, the solution $w$ possesses an harmonic
continuation across $S$ on the two-sheet Riemann surface $R$ of the
function $\zeta=(z^2-1)^{1/2}$, and it can be proved that the expansion
$w=\sum c_j w_j$; $c_j=\alpha_j^{-1}(w,\tilde w_j)$, holds true on that
part of $R$ cut by the largest Cassini-oval contained in $D$, whose foci
are the points $x=\pm 1$; $y=0$. The biorthogonal system is composed of four
subsystems each of which is symmetric or anti-symmetric w.r.t. the
$x,y$-axes and yields a contribution to $K$ of the form
  $$ K_i=-\frac{1}{8\pi}\log \overline{PQ}+H_i(P,Q);\quad (i=1,2,3,4), $$
where $H_i(P,Q)$ is a harmonic function of $Q$ for any fixed $P$ (not at
the ends of $S$, which are branching points) and vice-versa.

All functions are expressed adequately in elliptic coordinates on $R$.
While $K_1$ and $K_2$ are elementary functions, $K_3$ and $K_4$ can be
defined in terms of two of the four hypergeometric functions $F_i$ of two
variables considered by P. Appell in 1880, namely
  $$ F_1(a,b,b',c;X,Y) \mbox{ and } F_3(a,a',b,b',c;X,Y). $$
The integral representation of the solution $w$ now allows to obtain an
integral equation for the unknown values of $w(Q)$ or of
$\partial w(Q)/\partial n$ on $L$, while the path of integration is only $L$,
thus avoiding the singular points at the ends of $S$.
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